The Noncrossing Bond Poset of a Graph

The partition lattice and noncrossing partition lattice are well studied objects in combinatorics. Given a graph $G$ on vertex set $\{1,2,\dots, n\}$, its bond lattice, $L_G$, is the subposet of the partition lattice formed by restricting to the partitions whose blocks induce connected subgraphs of $G$. In this article, we introduce a natural noncrossing analogue of the bond lattice, the noncrossing bond poset, $NC_G$, obtained by restricting to the noncrossing partitions of $L_G$. Both the noncrossing partition lattice and the bond lattice have many nice combinatorial properties. We show that, for several families of graphs, the noncrossing bond poset also exhibits these properties. We present simple necessary and sufficient conditions on the graph to ensure the noncrossing bond poset is a lattice. Additionally, for several families of graphs, we give combinatorial descriptions of the M\"obius function and characteristic polynomial of the noncrossing bond poset. These descriptions are in terms of a noncrossing analogue of non-broken circuit (NBC) sets of the graphs and can be thought of as a noncrossing version of Whitney's NBC theorem for the chromatic polynomial. We also consider the shellability and supersolvability of the noncrossing bond poset, providing sufficient conditions for both. We end with some open problems.

Other methods of generalizing the noncrossing partition lattice have been studied and shown to be fruitful. For example, in [13] Edelman introduced the k-divisible noncrossing partition lattice, the subposet of NC n where each partition has all of its block sizes divisible by k. Later, Armstrong [2] introduced and studied the k-divisible noncrossing partition lattice for each finite Coxeter group. More recently, motivated by the connection between the noncrossing partition lattice and parking functions, Bruce, Dougherty, Hlavacek, Kudo, and Nicolas [12] introduced a subposet of the noncrossing partition lattice obtained by removing chains that corresponded to parking functions with certain restrictions. To solve a conjecture put forward in that article, Mühle [23] defined two new subposets of the noncrossing partition lattice obtained by removing partitions which do not contain certain blocks. Mühle [23] showed that these new posets are graded, shellable, and supersolvable.
In this article, we introduce a new generalization of the noncrossing partition lattice which is based on the structure of finite graphs. This generalization can be thought of as the intersection of the noncrossing partition lattice and a bond lattice. Given a graph with vertex set [n], its bond lattice is a subposet of the partition lattice obtained by restricting it to the set of partitions such that for each block B in the partition, the induced subgraph of G with vertex set B is connected. Note that the bond lattice of the complete graph is the partition lattice since any induced subgraph of the complete graph is connected. The bond lattice carries important combinatorial information about the graph. For example, it encodes exactly the same information as the cycle matroid associated to the graph. In fact, the bond lattice is (isomorphic to) the lattice of flats of this cycle matroid. Moreover, its characteristic polynomial is (essentially) the chromatic polynomial of the graph and its chromatic symmetric function can be computed from the lattice as well. Since the partition lattice is the bond lattice of the complete graph, one can consider the noncrossing partition lattice as a noncrossing version of a bond lattice. It is this idea that is the starting point of our work. The noncrossing bond poset of a graph is the intersection of the noncrossing partition lattice and its bond lattice, i.e. the poset obtained from its bond lattice by removing any partition that is crossing.
While many of the generalizations of the noncrossing partition lattice discussed above exhibit the nice properties of NC n , the situation for the noncrossing bond poset is a bit more nuanced. In general, many of these properties do not hold for the noncrossing bond poset of generic graphs. We note that this might be expected as the structure of graphs can vary widely. Nevertheless, we are able to identify families of graphs for which some of these nice properties still hold. We present simple necessary and sufficient conditions on the graph for the noncrossing bond poset to be a lattice (see Theorem 2.6). The Möbius function and the characteristic polynomial of the bond lattice of a graph can be interpreted combinatorially in terms of non-broken circuit (NBC) sets via Whitney's NBC theorem. We show that, for several families of graphs, similar interpretations hold for the noncrossing bond poset in terms of what we call noncrossing NBC sets. We obtain our noncrossing version of Whitney's NBC theorem in two different ways, one by using the theory of non-bounded below (NBB) sets introduced by Blass and Sagan [8] (see Theorem 3.5) and the other by using the minimum EL-labeling for geometric lattices introduced by Björner [6] (see Theorem 4.4). Both approaches are necessary as there are some graphs which can be handled by only one of these methods. Moreover, the EL-labeling approach allows us to provide shellability results for the noncrossing bond poset of several families of graphs. Additionally, we show that the noncrossing bond poset of perfectly labeled graphs, which arise from chordal graphs, admit S n EL-labelings (see Proposition 5.10). Using a result of McNamara [22], we get that when these are lattices they are supersolvable lattices. These results on perfectly labeled graphs parallel the results for chordal graphs in realm of bond lattices.
We also give two algorithms for non-crossing bond posets in the appendix. Algorithm A.1 determines if the noncrossing bond poset of a graph is a lattice. Algorithm A.6 determines if a graph belongs to a family of graphs for which the noncrossing NBC interpretation of the Möbius function and characteristic polynomial hold. Our algorithms both run in time polynomial in n, the number of vertices of the graph. This is of interest because brute-force algorithms that do not take advantage of the theory we develop can, in the worst case, take time super-exponential in n.
The rest of the paper is organized as follows. In Section 2 we discuss the basic structure of the noncrossing bond poset. After this, we consider the Möbius function and characteristic polynomial in Section 3. Edge labelings are then studied in Section 4. Next, we look at the properties of several families of graphs in Section 5. Section 6 contains several open problems. We finish with the appendix on algorithms we previously mentioned.
Some of these results appeared in an earlier version of this paper published in the Proceedings of Formal Power Series and Algebraic Combinatorics 2019 [14].

The Structure of the Noncrossing Bond Poset
We assume the reader is familiar with the basic concepts of graph theory (see [30,Graph Theory Appendix] for any undefined terms) as well as basic concepts related to posets (see [30, Chapter 3] for background and notation). Let G be a graph. For the remainder of this paper, unless otherwise noted, we will assume that the vertex set of G is [n]. We will use the notation V (G) for the vertex set of G and E(G) for the edge set of G. When we write out edges, we will write them in the form ij where i < j. Moreover, we will always draw our graphs so that the vertices lie on a circle with vertex 1 at the top and the remaining vertices appearing in clockwise order around the circle. Edges will always be drawn so that they are the line segments between their endpoints. We will refer to this as the graphical representation of G. We say that two edges of G cross if their respective line segments intersect in the graphical representation, i.e. a 1 a 2 and b 1 b 2 cross if and only if a 1 < b 1 < a 2 < b 2 or b 1 < a 1 < b 2 < a 2 . See Figure 1 for several examples of graphical representations.
A subgraph H of a graph G is called spanning if V (H) = V (G). Note that when considering a spanning subgraph H of G, it is enough to just know E(H). Because of this, we will often make no distinction between subsets of E(G) and spanning subgraphs of G.
A subgraph H of a graph G is called induced if whenever u and v are vertices in H and uv ∈ E(G), then uv ∈ E(H). Given a subset of the vertices, S, let G[S] denote the induced subgraph of G with vertex set S. Similarly, if E is a set of edges, we will use G[E] to denote the induced subgraph on the vertices which are endpoints of the edges in E. We say a spanning subgraph of G is a bond if every connected component of the subgraph is induced. As an example, consider the graph G in Figure 1. The subgraphs H, K, and H ∩ K are To each bond H, one can associate a set partition, π(H), so that i and j are in the same block of π(H) if and only if i and j are in the same connected component of H. For example, for the bond H in Figure 1, we have that π(H) = 12345/6. Similarly, for each partition π = B 1 /B 2 / · · · /B k , we associate the corresponding spanning subgraph G[π] whose edge set is the disjoint union of the edges in G[ For example, the partition 1248/56/37 is crossing since we can pick 2, 4 ∈ 1248 and 3, 7 ∈ 37. A partition is noncrossing if it is not crossing. We say the bond H is crossing (resp. noncrossing) if π(H) is crossing (resp. noncrossing). It is not hard to verify the following proposition. Note that a noncrossing bond H can contain crossing edges, as long as every pair of edges that cross belong to the same connected component of H. For example, the bond H in Figure 1 is noncrossing since it corresponds to 12345/6, but it has crossing edges, namely 14 and 35. Definition 2.2. Let G be a graph. The bond lattice of G, denoted by L G , is the collection of bonds of G ordered by inclusion. The noncrossing bond poset, denoted by NC G , is the collection of noncrossing bonds of G ordered by inclusion.
See Figure 2 for an example of a graph, its bond lattice, and noncrossing bond poset. Unlike the bond lattice, the noncrossing bond poset is not always a lattice. Note that for the graph G in Figure 2, the bond 13/24 is crossing and so is not in NC G . Thus for this graph G, NC G not only fails to be a lattice, it even fails to have a maximum element.
Note that NC G is not necessarily a meet semi-lattice either; the graph G in Figure 1 gives one such example. The bonds H and K in Figure 1 are noncrossing and thus are in NC G . However, they do not have a meet in NC G since the noncrossing bonds X that are below H and K (i.e. are contained in H and K) are those with E(X) = ∅, {14}, or {35}. These do not contain a unique maximal element, so H and K do not have a meet.  e and f that is minimal among all such subgraphs with respect to containment. If such a subgraph exists, we denote it by J(e, f ). We say G is crossing closed if every pair of crossing edges in G are crossing closed.
Note that the graph G in Figure 1 is not crossing closed since 14 and 35 are crossing but not crossing closed. There are two distinct minimal, induced, connected subgraphs of G that contain 14 and 35, namely the bond H = G[12345/6] and the bond K = G[13456/2] shown in Figure 1. On the other hand, trees and complete graphs are crossing closed. If G is a tree and e and f are two crossing edges, then J(e, f ) is the unique path in G with end-edges e and f . If G is a complete graph, then J(e, f ) = G[e∪f ]. However, this is far from a complete list of families of crossing closed graphs. We will save the discussion of more families which are crossing closed for later sections after we have developed a few more concepts.
Before we continue, let us explain the choice of the letter "J" in the notation J(e, f ). As we will see in Theorem 2.5, in the case that J(e, f ) exists it is (essentially) the join of e and f in NC G . We also wish to explain the reason we use the term "crossing closed". As was mentioned in the introduction, the bond lattice is the lattice of flats of the cycle matroid of the graph. One of the many equivalent ways to define a matroid is through a closure operator. In terms of the lattice of flats, the closure of a subset of the ground set is the join of the elements in the lattice of flats. As we mentioned before, crossing closed edges are exactly the crossing edges which have a join in the graph's noncrossing bond poset and crossing closed graphs are exactly the graphs whose noncrossing bond poset is a lattice. Thus, a graph being crossing closed implies the existence of a closure operator on the crossing edges that behaves in a similar way that the closure operator does in the cycle matroid. Note that this does not give us a matroid structure as our closure operator is not the same as the one for matroids. Lemma 2.4. Let G be a graph and let e and f be two crossing closed edges of G. Then J(e, f ) is of one of the following two forms, depending on whether or not there is an edge in G connecting a vertex in e to a vertex in f . Now suppose there are no edges in G connecting a vertex of e to a vertex of f . Let T be a spanning tree of J(e, f ) that contains e and f . We claim that T cannot have a vertex of degree more than 3 with respect to T . To see why, suppose this was not the case. Then since T is a tree, it must have at least 3 leaves. Both endpoints of e cannot be leaves of T as then T would then be e and T must also contain f . The same can be said of f , so e and f together can contain at most two leaves of T . Thus there must be a third leaf w of T with w not in e or f . J(e, f ) \ w is then induced, connected (since it contains spanning tree T \ w), and contains e and f , a contradiction to the minimality of J(e, f ). Thus T is a path containing e and f . The edges e and f must be the two edges at the ends of T . Otherwise at least one end-vertex of T , say w, would not be on either e or f and J(e, f ) \ w would again contradict the minimality of J(e, f ).
Let the vertices of J(e, f ) be given in the order v, v ′ , x 0 , . . . , x k , w ′ , w that they come along the spanning path T , where e = vv ′ , f = ww ′ , and k ≥ 0. E(J(e, f )) is the set of edges these vertices induce. We claim that, besides the edges in T , there are no additional edges in E(J(e, f )) except possibly for the edges x 0 v and x k w should either of them be present in G.
Suppose, for the sake of obtaining a contradiction, that the claim at the end of the last paragraph is not true. Then k > 0. If vx i with i > 0 is present, then v ′ vx i x i+1 . . . x k w ′ w is a path containing e and f whose vertex set is strictly contained in V (J(e, f )) contradicting the minimality of J(e, f ). Similarly, we cannot have an edge v ′ x i with i > 0. By assumption we have no edges from a vertex of e = vv ′ to a vertex of f = ww ′ . Thus we have no edges from v or v ′ to any vertex other than x 0 . Similarly we have no edges from w or w ′ to any vertex other than x k . The only remaining possibility is then that we have an edge x i x j with j − i ≥ 2. But then vv ′ x 0 . . . x i x j . . . x k w ′ w is a path containing e and f whose vertex set is strictly contained in J(e, f ), again contradicting the minimality of J(e, f ).
We now claim that each x i is a cut-vertex of G and separates e and f . Suppose instead that G \ x i has a component C that contains e and f . Then by the minimality of J(e, f ), C must contain J(e, f ) which contains x i , a contradiction.  Proof. First, suppose that NC G is a lattice. Then joins exist and in particular the join e ∨ f exists for each pair of crossing edges e, f in G. Thus, by the previous theorem, J(e, f ) exists for every pair of crossing edges e, f and G is crossing closed. Now suppose G is crossing closed. Then G must be a noncrossing bond of itself. If G had two connected components C 1 and C 2 that cross, then there must be edges e ∈ C 1 and f ∈ C 2 that cross. There can be no connected subgraph of G that contains both e and f , so J(e, f ) cannot exist, contradicting the assumption that G was crossing closed. Thus G is the unique maximum element of NC G . Since NC G is finite and contains a maximum element, to show NC G is a lattice, it suffices to show that it is a meet-semilattice.
We claim that if H, In the appendix, we present Algorithm A.1 that decides if NC G is a lattice in time polynomial in n, the number of vertices of the graph G. It does this by deciding the equivalent question of whether G is crossing closed. We note in the appendix that a brute-force algorithm for this problem could, in the worst case, take time super-exponential in n.
A lattice is called atomic if every element is the join of a particular subset of atoms, wherê 0 is considered to be the empty join. Moreover, we say the lattice is (upper) semimodular if whenever x ∧ y ⋖ x, y, then x, y ⋖ x ∨ y. Here and throughout the paper we use the notation x ⋖ y to denote that y covers x in the poset. A finite lattice which is both atomic and semimodular is called geometric.
It is well-known that there is a bijection between geometric lattices and (simple) matroids. Since the bond lattice of a graph is the lattice of flats for its cycle matroid, every bond lattice is geometric. The situation for the noncrossing bond poset differs.   Parts (b) and (c) of the previous proposition were proved in Theorem 2.6 and Theorem 2.5 respectively. The proofs of (a), (d), and (e) are straightforward.
By Proposition 2.7 part (e), the noncrossing bond poset of any crossing closed graph has â 1. However, this is not the only way to have a maximal element. For example, the noncrossing bond poset of the graph G in Figure 1 is not a lattice since G is not crossing closed but still has a1, which is G itself. The following proposition provides several characterizations for when a1 exists. The proof is straightforward and thus omitted.  The following lemma will help us prove results on general graphs by reducing them to the case of connected graphs. Lemma 2.9. Suppose that G consists of connected components C 1 , C 2 , . . . , C k such that no edges of C i and C j cross for all i = j. Then NC G ∼ = NC C 1 × NC C 2 × · · · × NC C k . Proof. Using induction, it suffices to show this result when k = 2. In that case it is easy to check that the map ϕ : is an isomorphism.
In the previous example, we were able to find a saturated0-1 chain in which each cover is obtained by merging exactly two blocks of the corresponding partition. It turns out that for graphs whose connected components do not cross (such as connected graphs and crossing closed graphs) such a chain can always be found. In the following proposition and throughout the remainder of the paper, we use cc(H) to denote the number of connected components of a graph H. Proposition 2.10. Let G be a graph on [n] which is a noncrossing bond of itself. Then NC G is graded if and only for every cover relation H ⋖ H ′ , there are exactly two blocks of H that merge to get H ′ . Moreover, in the case that NC G is graded, the rank function is given by ρ(H) = n − cc(H).
Proof. By Proposition 2.8, NC G has a1 = G and the connected components of G do not cross. By Lemma 2.9, if G has more than one connected component, then NC G is the product of noncrossing bond posets, one for each connected component. Since the product of graded posets is graded and the rank function of the product is the sum of the rank functions, it suffices to prove the result for connected graphs and so we may assume G is connected.
Let T be a spanning tree of G. Let e 1 , e 2 , . . . , e n−1 . be a sequence of edges of T such that for each forest in the sequence {e 1 }, {e 1 , e 2 }, . . . , {e 1 , e 2 , . . . , e n−1 } there is a unique nontrivial connected component. For each 1 ≤ i ≤ n − 1, let H i be the induced subgraph on {e 1 , e 2 , . . . , e i }. Since each H i has a unique nontrivial connected component, each H i is noncrossing. Moreover,0 ⋖ H 1 ⋖ · · · ⋖ H n−1 = G is a maximal chain of length n − 1. Since the minimum element of NC G has n blocks, the maximum element has 1 block, and there is a maximal chain of length n − 1, NC G is graded if and only if for every cover relation H ⋖ H ′ , there are exactly two blocks of H that merge to get H ′ . Now the last statement of the theorem follows immediately since when we merge two blocks, the number of connected components decreases by one.

The Möbius Function and Characteristic Polynomial
In this section, we introduce a family of graphs called upper crossing closed graphs. The motivation is that this is a class of graphs for which we are able to provide combinatorial interpretations of the Möbius functions and characteristic polynomials of the corresponding noncrossing bond posets. We briefly recall the definitions of the Möbius function and characteristic polynomial. For a more information, we refer the reader to [30,Chapter 3].
We will be dealing with the one-variable version of the Möbius function, defined for posets P with0. This can be recursively defined by Moreover, if P has0 and also is graded with rank function ρ, then the characteristic polynomial of P is given by Let ch(G, t) be the chromatic polynomial of the graph G, the polynomial p(t) such that for all positive integers t, p(t) is the number of proper colorings of G using at most t colors The chromatic polynomial of a graph G and characteristic polynomial of its bond lattice are related in the following way. [29,Theorem 2.7] ) For all finite graphs G, ch(G, t) = t cc(G) χ(L G , t).
In [33], Whitney gave a combinatorial interpretation for the coefficients of the chromatic polynomial in terms of non-broken circuit sets or NBC sets, which are defined as follows. Let G be a graph. Let ✂ be a total ordering on the edges of G. A broken circuit of G with respect to ✂ is a collection of edges of G obtained by removing the smallest edge of a cycle of G with respect to that ordering. We say a subset S of E(G) is a non-broken circuit set or NBC set if S contains no subsets which are broken circuits. Let nbc k (G) be the number of k-edge NBC sets of G with respect to ✂. Whitney showed the following.
Theorem 3.2 (Whitney [33]). Let G be a finite graph on [n]. Then for any total ordering  Part of the interest of this theorem is its assertion of the non-obvious fact that that the number of NBC sets of size k does not depend on the ordering of the edges. By Theorem 3.1 and the fact that L G has rank function ρ(H) = n − cc(H) we have the following. Theorem 3.3 (Whitney [33]). Let G be a finite graph. Then for any total ordering ✂ of As an example, consider the twisted 4-cycle graph G in Figure 5. It is not hard to calculate that This can be done by calculating the Möbius function of L G and then using the definition of the characteristic polynomial or by calculating ch(G, t) and then using Theorem 3.1. We can also use Theorem 3.3. Suppose ✂ is lexicographic order, i.e. 12 ✁ 13 ✁ 24 ✁ 34. Since G is a cycle, the only broken circuit is {13, 24, 34} and hence every subset of E(G) is an NBC set except for {13, 24, 34} and {12, 13, 24, 34}. Thus the absolute values of the coefficients of χ(L G , t) are indeed nbc 0 (G) = 1, nbc 1 (G) = 4, nbc 2 (G) = 6, nbc 3 (G) = 3 and nbc k (G) = 0 for k > 3. This agrees with the coefficients one finds by using the Möbius function or the chromatic polynomial. Now let us compare this with the characteristic polynomial of NC G . From the Möbius values shown in Figure 5, we see that Since the absolute value of the coefficients of χ(NC G , t) are less than or equal to the corresponding values in χ(L G , t), it is plausible that the coefficients of χ(NC G , t) might count a subset of the NBC sets of G.
If ✂ is a total order of E(G) we say that S is a noncrossing NBC set with respect to ✂, if S is an NBC set with respect to that order and contains no edges which cross in the graphical representation of S. We define N CN BC k (G) to be the set of noncrossing NBC sets of size k and ncnbc k (G) to be the number of such sets.
If ✂ is again the lexicographic order on the edges of the twisted 4-cycle, then all NBC sets are also noncrossing NBC sets except for {13, 24} and {12, 13, 24}. Thus the sequence ncnbc 0 (G) = 1, ncnbc 1 (G) = 4, ncnbc 2 (G) = 5, ncnbc 3 (G) = 2 and ncnbc k (G) = 0 for k > 3 does match the sequence of absolute values of the coefficients of χ(NC G , t). Unfortunately, unlike nbc k (G), ncnbc k (G) does depend on the ordering ✂ (despite the fact that the notation does not reflect this). For example, if we order the edges as 13 ✁ 24 ✁ 12 ✁ 34, there is only one noncrossing NBC set with 3 edges, instead of 2 and so this ordering will not give the correct coefficients. However, the upper crossing closed graphs (defined below), do have edge orderings for which the Möbius values of NC G and the absolute values of the coefficients of χ(NC G , t) are indeed the counts of noncrossing NBC sets with respect to those orderings. Definition 3.4. We say a graph G is upper crossing closed if it is crossing closed and there is a total ordering ✂ on E(G) such that for every pair of crossing edges e and f , J(e, f ) contains an edge h such that h ✁ e, f . If ✂ is one such ordering, we say that G is upper crossing closed with respect to ✂ and also that ✂ is an upper crossing closed ordering of E(G).
Note that the twisted 4-cycle in Figure 5 is upper crossing closed with respect to lexicographic order since J(13, 24) = G contains 12 and 12 ✁ 13, 24. Note also that it is not upper crossing closed with respect to the other ordering 13 ✁ 24 ✁ 12 ✁ 34 we considered previously since 13 is then the minimum edge in J(13, 24). As we saw earlier, when using the lexicographic ordering on the twisted 4-cycle (which is upper crossing closed), the coefficients of the characteristic polynomial count noncrossing NBC sets. This is not a coincidence as we see next. Moreover, if NC G is graded, then Before we can prove Theorem 3.5, we need to discuss the notion of NBB sets developed by Blass and Sagan in [8].
Definition 3.6 (Bass and Sagan [8]). Let L be a lattice and let ✂ be a partial order on the atoms of L. A subset S of the atoms of L is bounded below if there exists an atom a such that We say a subset S of the atoms of L is a non-bounded below (NBB) set for x if S contains no bounded below sets and S = x.
Blass and Sagan's result generalizes Whitney's NBC theorem. In particular, if one considers a graph G and its bond lattice L G , then the edges of G correspond to the atoms of L G and the NBB sets are exactly the NBC sets of the graph with respect to whatever ordering is put on the atoms/edges. Blass and Sagan showed that we can use NBB sets to determine the value of the Möbius function.
Theorem 3.7 (Blass and Sagan [8]). Let L be a lattice and let ✂ be a partial order on the atoms of L. Then for all x ∈ L, where the sum is over NBB sets for x.
In Lemma 3.9 below, we will show that if ✂ is an upper crossing closed ordering of a graph G then the NBB sets and the NCNBC sets are the same. We illustrate this using our running example of the twisted 4-cycle in Figure 5. For this example, we use the lexicographic ordering which was already shown to be upper crossing closed. First, let us note that the empty set and any singleton subset of atoms is NBB since their joins only have themselves below them. Moreover, since G is a 4-cycle, any subset of 2 edges which do not cross is NBB for the same reason. Now let us turn our attention to the two edges that do cross, 13 and 24. Their join is the entire graph and since 12 is lexicographically smaller than 13 and 24, {13, 24} is a bounded below set. So every 2-element subset of the atoms except {13, 24} is NBB. Finally, we consider the subsets of size 3. Since G is a 4-cycle the join of any 3-element subset is the entire graph. Since  Lemma 3.8. Let G be a crossing closed graph and let S ∈ N CN BC k (G). Then the join of the elements in S is the same in L G and NC G Proof. In this proof, we will use L G and N C G to denote the join operators in L G and NC G respectively. First, let us show that L G S is a noncrossing bond. Suppose this was not the case and let C 1 and C 2 be components of L G S which have crossing edges. Let S 1 = S ∩ E(C 1 ) and S 2 = S ∩ E(C 2 ). Then S 1 is spanning tree of C 1 and S 2 is a spanning tree of C 2 . Since C 1 and C 2 cross, there exists edges ac ∈ E(C 1 ) and bd ∈ E(C 2 ) with a < b < c < d which cross. In S 1 there is a path from a to c, but this path must separate b and d. Similarly, in S 2 there is a path between b and d. This path must cross the path between a and c. However, these two paths are in different connected components. This implies that S 1 and S 2 must cross, but then S is crossing which is impossible.
Since S is a collection of edges of G and contains no broken circuits, it forms a spanning forest of G. It is not hard to see that L G S is the bond whose induced connected components are the connected components of S. As we saw, since S is noncrossing, L G S is a noncrossing bond in L G and hence is an element of NC G . It follows that the partition associated with L G S is noncrossing. It is not hard to see that this is exactly the same partition associated with N C G S. Thus, the result holds. Lemma 3.9. Let G be an upper crossing closed graph with total ordering ✂ on E(G). Suppose G is upper crossing closed with respect to ✂. Order the atoms of NC G by ✂.
Proof. Suppose that S ∈ N BB k (G), but that S / ∈ N CN BC k (G). If S is not an NBC set, then it contains a broken circuit C. Let e be the edge removed from the cycle to obtain C. Then it is not hard to see that C is a bounded below set with e as the atom which is below all the elements of C. This would imply that S / ∈ N BB k (G). Thus, S must be an NBC set. Now suppose that S has crossing edges. Let S ′ be a set consisting of two such crossing edges. Since G is upper crossing closed, S ′ contains an edge smaller than all the edges of S ′ . It follows that S ′ is a bounded below set, but then S is not an NBB set as it contains S ′ . Thus S ∈ N CN BC k (G).
Next, suppose that S ∈ N CN BC k (G), but that S / ∈ N BB k (G). Then S contains a bounded below set, T . Note that since T ⊆ S, T is a noncrossing NBC set. Moreover, since T is bounded below, there exists an atom e such that e ✁ t for all t ∈ T and e < T . By Lemma 3.8, T is the same in L G and NC G . Thus, e < T and e / ∈ T , implies that e must be in some cycle C of T . To see why, note that T is the induced subgraph on T . The fact that e < T , implies e is in T . So if e = uv, there is a path from u to v in T . The set T must contain a spanning tree for the component containing u and v. But since e / ∈ T , e is not on this spanning tree, so e must be on a cycle. Moreover, since e is smaller than all the elements of T , C \ e would be a broken circuit of T . But then S is not an NBC set which is impossible.
We are now ready to prove Theorem 3.5.
(Proof of Theorem 3.5). Order the atoms of NC G by ✂. Using the fact that G is upper crossing closed, Lemma 3.9 shows that a subset of atoms of NC G is NBB if and only if it is a noncrossing NBC set of G. Then using Blass and Sagan's result, we have that for each where the sum is over all the noncrossing NBC sets B such that B = H. Since B is a noncrossing NBC set, Lemma 3.8 implies that B is the same in L G and NC G . We claim that for a fixed H, all the NBC sets whose join is H in L G have the same size, namely n − cc(H). To see why, suppose that S is an NBC set and S = H. It must be the case that the edges in S form a subgraph of G so that its connected components are exactly the connected components of H. Moreover, since NBC sets cannot contain cycles, S must be minimal with respect to spanning the connected components. So each connected component of S must be a tree. Thus the number of edges in S is n − cc(S) = n − cc(H). It now follows that To finish, note that if NC G is graded, then since G is crossing closed, it has a1 and so Proposition 2.10 implies that the rank function of In the statement of Theorem 3.5, we had to make the assumption that NC G is graded in order to describe the characteristic polynomial. We must do so because there exists upper crossing closed graphs with the property that their noncrossing bond posets are not graded. To construct such an example, we will take a subdivision of the graph G given in Figure 6. As mentioned earlier, the noncrossing bond poset of the graph of G is not graded. Additionally, since every edge of the graph is crossed, it cannot be upper crossing closed. This is because if e is the smallest edge and it crosses some edge f , J(e, f ) would need to have an edge smaller than e.
Consider the graph H given in Figure 6 which is obtained from subdividing the edges 24 and 15 and labeling the new vertices 2 ′ and 6 ′ . H is also a tree and hence is crossing closed. Any ordering of the edges in which 16 ′ and 22 ′ come first is an upper crossing closed ordering since every J(e, f ) contains one of these edges. To see why NC H is not graded, consider the bond corresponding to the partition 16 ′ /22 ′ 6/35/4. If we try to add any edge to this bond, we create a new crossing. Since H is a tree, if NC H was graded, each covering relation would be obtained by adding a single edge, a contradiction. In this section, we saw that the notion of upper crossing closed allows us to give a combinatorial description of the Möbius function and characteristic polynomial in terms of noncrossing NBC sets. As we will see in a later section, the assumption of upper crossing closed is not always necessary. Indeed there are graphs, which are not even crossing closed which have such an interpretation. However, there are also graphs for which no such interpretation is possible. Consider the 5-pointed star in Figure 7. The characteristic polynomial is given by The coefficient of t is 0, but the coefficient of t 0 is nonzero. If the characteristic polynomial was the generating function for noncrossing NBC sets, then this would imply that there is 1 noncrossing NBC set of size 4, but none of size 3. However, every subset of a noncrossing NBC set is a noncrossing NBC set so this is impossible. This argument generalizes. Say a polynomial P (t) = c n t n + c n−1 t n−1 + · · · + c 1 t + c 0 has an internal zero if there exists a k such that c k = 0, but c k+1 , c k−1 = 0. By the hereditary property of noncrossing NBC sets, the generating function for noncrossing NBC sets cannot have internal zeros. We give an algorithm, Algorithm A.6 in the appendix, that when given a graph G will either produce an specific upper crossing closed ordering of E(G) or will produce what we term an obstruction, a specific subgraph of G that clearly shows there can be no such ordering. This gives a forbidden subgraph characterization of upper crossing closed graphs, Theorem A.4: a graph is upper crossing closed if and only if it contains no such subgraph.
We also prove that Algorithm A.6 will run in time polynomial in n, the number of vertices of G and note that a brute-force algorithm will, in the worst case, take time super-exponential in n.

Edge Labelings and Shellability
In this section we discuss edge labelings of the noncrossing bond poset. We pay particular attention to the minimum labeling, introduced by Björner, which is an EL-labeling for every geometric lattice. While the minimum labeling is not an EL-labeling for every noncrossing bond poset, we give a sufficient condition which guarantees it is an EL-labeling. We do this for two reasons. First, this will show that the poset is shellable. Second, it will allow us to show that the combinatorial interpretation for the Möbius function in terms of noncrossing NBC sets holds for more than just upper crossing closed graphs.
We briefly review edge labelings of posets. We refer the reader to [32] for more information. Let P be a graded poset. An edge labeling of P is a function λ : E(P ) → Λ where E(P ) is the set of edges of the Hasse diagram of P and Λ is a set of labels which is partially ordered. We note here that although the labels are allowed to be partially ordered, in this article they will always be totally ordered. Now suppose that P is a graded poset with edge labeling λ. Let c : Moreover, we say c is decreasing if Before we move on, let us note that while the inequalities for decreasing are allowed to be weak, in this paper, they are always strict. Thus, there is no need to add the adjective "weak" to decreasing.
Let λ be an edge labeling of P . We say λ is an EL-labeling if every interval has a unique increasing maximal chain and this chain precedes every other maximal chain in the interval in lexicographic order. Björner [6] and Björner and Wachs [7] showed that there are several nice topological consequences of a poset having an EL-labeling. For example, they showed that given a poset with an EL-labeling, the order complex of P is shellable and has the homotopy type of a wedge of spheres. Because of this connection with the topology of the order complex, EL-labelings also have implications for the Möbius function. In particular, we have the following simple combinatorial interpretation for the Möbius function for graded EL-labeled posets. We will now consider an edge labeling for the bond lattice that we can also can apply to the noncrossing bond poset.
where the minimum is taken with respect to ✂. Figure 8 contains an example of the minimum labeling where the edges are ordered lexicographically. We note that the usual definition of the minimum labeling is phrased in terms of the join of elements. This makes sense for the bond lattice, however, because the noncrossing bond poset need not be a lattice, we have phrased it in a different (but equivalent) way. Björner showed the following concerning the minimum labeling of the bond lattice. In the following theorem and throughout the section, we will use the term spanning NBC set of a graph X to mean an NBC set S of X such that the induced subgraph S is X. Theorem 4.3 (Björner [6]). Let G be a graph and let ✂ be a total ordering of E(G). Then we have the following (where the NBC sets are taken with respect to ✂).
(a) The minimum labeling with respect to ✂ is an EL-labeling of L G and so L G is shellable.
(b) The labels along any decreasing saturated chain from0 to X form a spanning NBC set of X.
(c) For every X ∈ L G , each spanning NBC set of X appears exactly once as a saturated decreasing chain from0 to X.
Note that Theorem 4.1 and Theorem 4.3 together imply Whitney's theorem (Theorem 3.3) concerning the NBC set interpretation of the Möbius function. Even though the minimum labeling is an EL-labeling for the bond lattice of any graph, it need not be an ELlabeling for the noncrossing bond poset. In the next theorem, we give sufficient conditions for the labeling to be an EL-labeling. We also show that in this setting, we get noncrossing analogues of Björner's result on NBC sets described in the previous theorem. As a result, we get the same combinatorial interpretation for the Möbius function and characteristic polynomial as we did with upper crossing closed graphs, see Theorem 3.5. (a) NC G is graded and ρ(X) = n − cc(X).
(b) The minimum labeling with respect to ✂ is an EL-labeling and so NC G is shellable.
(c) The labels along any decreasing saturated chain from0 to X form a spanning noncrossing NBC set of X.
(d) For every X ∈ NC G , each spanning noncrossing NBC set of X appears exactly once as a saturated decreasing chain from0 to X. and Proof 10, NC G is graded and ρ(X) = n − cc(X). Now let us show (b). Let λ be the minimum labeling with respect to ✂. Since NC G is a graded subposet of L G with the same rank function, the sequence of labels along any saturated chain in NC G also appears as the labels along that chain in L G . Since λ is an EL-labeling of L G , every interval has at most one increasing maximal chain and this chain lexicographically precedes all other chains in that interval. Thus it suffices to show that each interval of NC G has an increasing maximal chain.
Consider has an increasing maximal chain which starts with a label larger than e. Concatenating this chain with H ⋖ H ′′ will produce an increasing maximal chain of [H, H ′ ]. It follows that the minimum labeling is an EL-labeling and so NC G is shellable.
Next we show part (c). Let c be a decreasing saturated chain from0 to X in NC G . By Theorem 4.3 part (b), the labels along c form an NBC set. Thus, it suffices to show these NBC sets are noncrossing. Suppose this was not the case and that there were crossing edges f 1 and f 2 with f 1 ✁ f 2 in some NBC set of X which appears along a saturated chain from 0 to X in NC G . We may assume that X is minimal among elements of NC G that have crossing edges in one of its NBC sets.
We claim that f 1 is the smallest edge of E(X). First note that every spanning NBC set of X contains the smallest edge of E(X). To see why, note that if the smallest edge was a bridge of X, it must be in this spanning set. If it is not a bridge, it is contained in some cycle and so must be in the spanning set since otherwise X would contain a broken circuit. Thus, the smallest edge of X is in every spanning NBC set of X. It follows that the labels along any saturated chain from0 to X must contain the smallest edge. Thus, the labels along every decreasing saturated chain from0 to X must end with the smallest edge. Since X is minimal with respect to having a crossing, the last label must either be f 1 or f 2 . Since f 1 ✁ f 2 , f 1 is the smallest edge in E(X). Now consider the interval [f 2 , X] in NC G . Then f 1 = min E(X) \ {f 2 } and so by assumption the bond induced on f 1 and f 2 is a noncrossing bond. Since f 1 and f 2 are crossing edges, they cannot share an endpoint. It follows that the bond induced on f 1 and f 2 is {f 1 , f 2 }. But then the fact that this bond is noncrossing, contradicts the fact that f 1 and f 2 cross.
Next, we prove (d). Let F = {e 1 , e 2 , . . . , e k } be a spanning noncrossing NBC set of X with e 1 ✄ e 2 ✄ · · · ✄ e k . Let H 0 =0 and for 1 ≤ i ≤ k, let F i be the forest with vertex set V (G) and edge set {e 1 , e 2 , . . . , e i }. Moreover, let H i be the bond induced on F i . We claim that each of these bonds is noncrossing. To see why, note that the partitions associated to F j and H j are the same for all 0 ≤ j ≤ k. Since each F j is noncrossing, Proposition 2.1 implies that the partition for F j is noncrossing and so H j is a noncrossing bond. By Theorem 4.3, each spanning NBC set of X appears exactly once as a saturated decreasing chain in L G . By construction, H 0 ⋖ H 1 ⋖ · · · ⋖ H k is the saturated chain chain from0 to X which produces the NBC set F = {e 1 , e 2 , . . . , e k }. The claim now follows since this chain also exists in NC G .
Finally, for (e), note that by Proposition 2.10 and part (a), ρ(H) = n − cc(H) for all H ∈ NC G . Parts (c) and (d) provide a bijection between the saturated decreasing chains from0 to H and noncrossing NBC sets. Then to finish, apply part (b) and Theorem 4.1.
After giving this sufficient condition for the minimum labeling to be an EL-labeling, the next obvious question is which graphs satisfy this condition. We will leave this discussion for the next section where we will explore several families of graphs which have this property. In particular, we will show the previous theorem applies to perfectly labeled graphs (Definition 5.1), upper crossing closed graphs that are tightly closed (Definition 5.13), and strongly upper crossed graphs (Definition 5.18). The perfectly labeled and strongly upper crossed graphs include graphs that are not crossing closed (and hence not upper crossing closed). On the other hand, not every upper crossing closed graph satisfies the conditions of the previous theorem. Because of this, we genuinely require both Theorem 3.5 and Theorem 4.4 to get the results concerning the noncrossing NBC interpretation for the Möbius function.
We should note that not all graphs G have a shellable NC G . The 5-pointed star in Figure 7 is one such example, as one can check in SageMath [31].

Families of Graphs
In this section, we will consider three families of graphs: perfectly labeled graphs in Subsection 5.1, tightly closed graphs in Subsection 5.2, and strongly upper crossed graphs in Subsection 5.3. We present some of the nice properties that the noncrossing bond posets of these families of graphs have such as gradedness, shellability, and combinatorial formulas  Figure 9: Two labeled graphs. G is perfectly labeled whereas H is not.
for Möbius values. We finish the section by gathering all the results about the structure of the noncrossing bond poset for the families of graphs studied within this article. This information can be found in Table 1.

Perfectly Labeled Graphs
In this subsection we present two main results, Theorem 5.8 and Theorem 5.11, concerning perfectly labeled graphs. We start with the definition of these graphs.
Definition 5.1. Let G be a graph. We say G is perfectly labeled if whenever ik, jk ∈ E(G) with i < j < k, ij ∈ E(G). 1 The graph G in Figure 9 is perfectly labeled. However, H is not perfectly labeled, since, for example, 14 and 24 are edges, but 12 is not an edge. It turns out that there is a classification of graphs which can be perfectly labeled. The reader may have noticed that the graphs in Figure 9 are both chordal, but only G is perfectly labeled. It is well-known that a graph can be perfectly labeled if and only if it is chordal (see, for example, the note immediately following Corollary 4.10 of [29]). However, as we saw not every labeling of a chordal graph gives rise to a perfectly labeled graph. The distinction between perfectly labeled and chordal is immaterial to the structure of the bond lattice since the lattice does not depend on the labeling of the vertex set. However, in the case for the noncrossing bond poset, the structure of the poset can depend on the labeling of the graph. Because of this, we focus on perfectly labeled graphs as opposed to just chordal graphs. This is also why we use the term "perfectly labeled" as opposed to saying "G has a perfect elimination order" which is more common in the literature.
As we will see throughout this subsection, increasing spanning trees and forests play an important role of the combinatorics of the noncrossing bond posets of perfectly labeled graphs. Let T be a tree with vertices which are distinct integers. Let r be the smallest vertex of T . We say T is an increasing tree if the vertices along any path from r to any other vertex form an increasing sequence. We say a spanning subgraph of a graph G is an increasing spanning forest of G if each connected component is an increasing tree.
The forests F 1 and F 2 in Figure 10 are increasing spanning forests of the graph G in Figure 9, whereas F 3 is not since the path from 1 to 2 is not increasing.
We now present a series of lemmas aimed at showing that perfectly labeled graphs satisfy the conditions of Theorem 4.4. This will allow us to show that the noncrossing bond poset of a perfectly labeled graph is graded, shellable, and has a combinatorial interpretation for the Möbius function and characteristic polynomial.
Lemma 5.4. Let G be a connected perfectly labeled graph. Let r be the smallest vertex in G and let v be any other vertex of G. There exists an increasing path from r to v in G. That is, there is a path ru 1 u 2 . . . u k v where r < u 1 < u 2 < · · · < u k < v.
Proof. Suppose this was not the case. Let P be a shortest path from r to v. By assumption, P is not an increasing path. Thus it must contain contain a sequence of vertices ikj where i < j < k. But then since G is perfectly labeled, ij ∈ E(G), so we may replace the edges ik, jk with ij and get a shorter path from r to v. This contradicts the minimality of P .
Lemma 5.5. Let G be a perfectly labeled graph. Suppose that H ≤ H ′ in NC G . Moreover, suppose that B 1 , B 2 , . . . , B k where min B 1 < min B 2 < · · · < min B k are the connected components of H that are merged together to get H ′ . Then merging B 1 and B 2 in H creates a noncrossing bond of G.
If a, c ∈ B 1 ∪ B 2 and b, d ∈ B i , then min B 1 < min B 2 < b < c < d which implies either B 1 and B i cross or B 2 and B i cross (depending on if c ∈ B 1 or c ∈ B 2 ). Neither is possible since H is noncrossing. A similar argument shows that it is not possible that there exists a, c ∈ B i and b, d ∈ B 1 ∪ B 2 with a < b < c < d. Hence merging B 1 and B 2 does not cause a crossing with B 3 , B 4 , . . . , B k . Moreover, merging B 1 and B 2 cannot create a crossing with any of other connected components of G since that would mean that H ′ was crossing.
Thus it suffices to show that merging B 1 and B 2 in H forms a bond of G. It is not hard to see that any induced subgraph of a perfectly labeled graph is perfectly labeled. It follows that H ′ is perfectly labeled. By Lemma 5.4 there is an increasing path in H ′ from min B 1 to min B 2 . Except for min B 2 , this path must only contain vertices from B 1 since otherwise it would not be increasing. So there is an edge, e, between a vertex in B 1 and a vertex in B 2 . Then the bond induced on on E(H) ∪ {e} is exactly the spanning subgraph obtained by merging B 1 and B 2 in H. The result now follows.
In the following lemma we will order the edges colexicographically. That is, we say ab✁a ′ b ′ if and only if b < b ′ or b = b ′ and a < a ′ . Note that colexicographic order is just lexicographic order reading right to left instead of left to right. Proof. Let B 1 , B 2 , . . . , B k be the blocks which merge together to get H ′ where min B 1 < min B 2 < · · · < min B k . Since G is perfectly labeled and H ′ is an induced subgraph, it is perfectly labeled. By Lemma 5.4, in the connected component containing B 1 and B 2 there is an increasing path from min B 1 to min B 2 . The last edge on this path must be of the form a min B 2 where a ∈ B 1 . Let a ′ be the smallest vertex in B 1 for which there is an edge a ′ min B 2 . Since we are ordering edges colexicographically, a ′ min B 2 = min E(H ′ ) \ E(H). The bond induced on E(H) ∪ {a ′ min B 2 } is the bond obtained by merging B 1 and B 2 . By Lemma 5.5 this bond is noncrossing.
As with noncrossing NBC sets, we say an increasing spanning forest is noncrossing if none of the edges cross. For example, the forests F 1 and F 2 in Figure 10 are increasing spanning forests of G in Figure 9, but only F 1 is noncrossing. It turns out that when we use the colexicographic order, there are the same number of noncrossing increasing spanning forests and noncrossing NBC sets. We use the notation ncisf k (G) for the number of noncrossing increasing spanning forest of G with k edges. In [17,Theorem 2.4], it was shown that when G is perfectly labeled and we order the edges lexicographically, the NBC sets are exactly the increasing spanning forests. The proof given there can easily be modified to allow for the case when the edges are ordered colexicographically. Thus we have the following.
Lemma 5.7. Let G be a perfectly labeled graph with the edges ordered colexicographically. Then for all k ≥ 0, ncisf k (G) = ncnbc k (G).
We now present the first main theorem of this section. Applying Lemma 5.6, Lemma 5.7, and Theorem 4.4 we get the following. Theorem 5.8. Let G be a perfectly labeled graph on [n] such that NC G has a1. Then we have the following.
(a) NC G is graded and for H ∈ NC G , the rank of H is given by ρ(H) = n − cc(H).
(b) The minimum labeling with respect to the colexicographic ordering on E(G) is an ELlabeling of NC G and so NC G is shellable.  and where the NBC sets are with respect to the colexicographic ordering on E(G).
We note that in [8], Blass and Sagan used NBB sets to show that the Möbius function of the noncrossing partition lattice counts noncrossing increasing trees and hence is a Catalan number. The previous theorem generalizes this result since the noncrossing partition lattice is the noncrossing bond poset of the complete graph which is perfectly labeled.
Let us now turn our attention to other the main result of this subsection. In [27] Stanley introduced the notion of a supersolvable lattice. A lattice L is called supersolvable if there exists a maximal chain with the property that it along with any other chain in L generates (by taking joins and meets of all elements in the chains) a distributive lattice. It is wellknown that a graph is chordal if and only if its bond lattice is supersolvable (see, for example, Corollary 4.10 and the note that follows it in [29]). As we will see in Theorem 5.11, there is an analogue of this result for noncrossing bond posets. However, we should point out that when we pass to the noncrossing case things become a bit more complicated. First, as we noted earlier, the noncrossing bond poset depends on the labeling of the vertices. Because of this there are chordal graphs which are not perfectly labeled and such that the noncrossing bond poset is not supersolvable. Second, we no longer have an if and only if statement as there exist noncrossing bond posets that are supersolvable lattices, but do not come from chordal graphs. Finally, not every perfectly labeled graph is crossing closed and so the noncrossing bond poset of a perfectly labeled graph may not even be a lattice.
In order to study supersolvability of the noncrossing bond poset we will consider a special type of EL-labeling of graded posets. Let λ be an EL-labeling of a poset of rank n. We say λ is an S n EL-labeling if every maximal chain of P is labeled by a permutation of [n] (with natural ordering on [n]). The edge labeling in Figure 11 is an example of an S n EL-labeling. We note that the condition that the unique maximal chain in each interval is lexicographically first is automatically implied if the maximal chains are labeled by permutations and thus, we only need to check that each interval has a unique increasing maximal chain. In addition to the properties that EL-labeled posets possess, there are special properties that a lattice with an S n EL-labeling possesses. In particular, McNamara [22] showed that if L is a graded lattice then L is supersolvable if and only if it has an S n EL-labeling. We will show that if G is perfectly labeled and connected, then it has an S n EL-labeling. To do this, we use a labeling introduced by Björner and Edelman. See the poset in Figure 11 for an example of this labeling. Björner and Edelman [6] showed that the max-min edge labeling gives an EL-labeling of the noncrossing partition lattice (which is also the noncrossing bond poset of the complete graph). As McNamara points out in [22], this labeling is in fact an S n EL-labeling. It turns out that among connected graphs, perfectly labeled graphs are exactly the graphs where the max-min edge labeling is an S n EL-labeling of the noncrossing bond poset. Note that in the hypothesis of the following proposition, we assume G has n + 1 vertices and is connected. This guarantees that NC G has rank n.
Proposition 5.10. Let G be a connected graph on [n + 1] such that NC G is graded. The max-min edge labeling is an S n EL-labeling of NC G if and only if G is perfectly labeled.
Proof. Given that G has n + 1 vertices, NC G is a subposet of NC K n+1 . Since NC K n+1 is the noncrossing partition lattice on [n + 1], we have that NC G is a subposet of the noncrossing partition lattice. It follows from the assumption that NC G is graded with the same rank function as the noncrossing partition lattice (see Proposition 2.10), we have that the set of maximal chains in NC G is a subset of the maximal chains of the noncrossing partition lattice. Since the cover relations in NC G are the same as that in the noncrossing partition lattice, we also have that the label sequences that appear along the maximal chains in NC G are the same as those that appear in the noncrossing partition lattice. Since it is known that the max-min edge labeling is an S n EL-labeling of the noncrossing partition lattice, to finish the proof we can show that each interval of NC G has an increasing maximal chain if and only if G is perfectly labeled.
Suppose that G is not perfectly labeled. Then there exists edges ik, jk such that i < j < k and ij / ∈ E(G). Let H be the bond of G where i, j, k are in the same connected component and every other connected component is trivial. Consider the interval [0, H]. Since ij / ∈ E(G), this interval has two maximal chains both labeled by k − 1, j − 1. Thus, the interval has no increasing chain.  Figure 12: A perfectly labeled graph which is not crossing closed Next, suppose that G is perfectly labeled. Suppose that [X, Y ] is an interval in NC G and suppose that B 1 , B 2 , . . . , B k are the connected components of X that will merge together to get Y . Moreover, assume that min B 1 < min B 2 < · · · < min B k . It is not hard to see that if there is an increasing maximal chain in [X, Y ], the first step must be to merge B 1 and B 2 . Let Z be the bond obtained by merging B 1 and B 2 in X. We can apply Lemma 5.5 to see that Z ∈ NC G . Now we can use induction to prove that [Z, Y ] has an increasing maximal chain which can be concatenated with the label from X to Z to give an increasing maximal chain in [X, Y ].
In Proposition 5.10, we assumed that G is connected. By Lemma 2.9, if G is not connected and its connected components do not cross, NC G is the product of smaller noncrossing bond posets, one for each connected component. If G is perfectly labeled, each connected component of G must be perfectly labeled. Thus, if G perfectly labeled, the noncrossing bond poset of each of its connected components has an S n EL-labeling. Using McNamara's [22] result about S n EL-labelings and supersolvability implies each is noncrossing bond poset is supersolvable. Moreover, if G is crossing closed its connected components cannot cross. Putting this altogether and using the fact that the product of supersolvable lattices is a supersolvable lattice, we get the following. Theorem 5.11. Let G be a perfectly labeled graph. If G is crossing closed, then NC G is a supersolvable lattice.
We mention here that not every perfectly labeled graph is crossing closed (hence the necessity of the crossing closed hypothesis in Theorem 5.11). The graph in Figure 12 is perfectly labeled, but not crossing closed. This is because there are two minimal induced connected components containing 16 and 57, namely the one containing the vertices 1, 3, 5, 6, 7 and the one containing 1, 2, 4, 5, 6, 7. Nevertheless, when G is crossing closed and perfectly labeled, it is upper crossing closed as we see next.
Proposition 5.12. Let G be a perfectly labeled graph which is crossing closed. Then G is upper crossing closed with respect to the colexicographic and lexicographic order.
Proof. Suppose that ac and bd cross with a < b < c < d. Then in J(ac, bd), there is a path from a to b. Let P : av 1 v 2 . . . v k b be a path from a and b which is minimal with respect to length. If P is not increasing, then there is a an index i with v i−1 , v i+1 < v i . But then since G is perfectly labeled, there is an edge v i−1 v i+1 contradicting the minimality of P . So P must be increasing. Then av 1 is an edge in E(G) and is smaller in colexicographic and  Figure 13: A 2-connected graph that is not crossing closed lexicographic order than ac and bd. It follows that G is upper crossing closed with respect to colexicographic and lexicographic order.
The reader may be wondering if NC G being supersolvable implies that G is chordal since this is the case for the bond lattice of a graph. The graph in Figure 5 shows this is not true. It is a 4-cycle and thus is not chordal. Nevertheless, its noncrossing bond poset is a supersolvable lattice.

Tightly Closed Graphs
As we saw in Section 2, crossing closed graphs need not have graded noncrossing bond posets. However, it turns out that if we restrict what J(e, f ) can look like, we can guarantee the noncrossing bond poset is graded. Moreover, if we further assume the graph is upper crossing closed, we obtain more properties of the poset. We explore these ideas next.
Definition 5.13. Let G be a graph. We say G is tightly closed if it is crossing closed and for all edges e and f that cross, J(e, f ) is a subgraph of K 4 .
The complete bipartite graphs are a family of tightly closed graphs. To see why, note that if two edges cross in a complete bipartite graph, they must connect the two parts of the graph and so must lie on a (twisted) 4-cycle. The 5-pointed star depicted in Figure 7 gives a different an example of a tightly closed graph. Note that since the 5-pointed star is a cycle, it is 2-connected. It turns out that any 2-connected crossing closed graph is tightly closed.
Proposition 5.14. If G is 2-connected and crossing closed, then G is tightly closed.
Proof. By Lemma 2.4, if G was not tightly closed, G would have cut vertices. This is impossible as G is 2-connected.
Before we move on, we wish to mention that 2-connected does not imply crossing closed. For example, the 6-cycle in Figure 13 is 2-connected, but not crossing closed. Theorem 5.15. If G be a tightly closed graph, then NC G is graded.
For each edge f ∈ E(H ′ ) \ E(H), we will count crossings of f with the edges in E(H). We will call such a crossing bad if the edges in the crossing are in different components of the graph with edge set E(H) ∪ {f } and vertex set V (G).
We claim that there is an edge in E(H ′ ) \ E(H) with no bad crossings. To see why suppose that this was not the case and let e = ac be an edge of E(H ′ ) \ E(H) with a minimum number of bad crossings. By assumption, e has at least one bad crossing, say with edge e ′ = bd in H where a < b < c < d. Since e and e ′ are in H ′ and are crossing and H ′ is noncrossing, they must lie in the same connected induced component of H ′ . Thus J(e, e ′ ) is a subgraph of this component. Since G is tightly closed, J(e, e ′ ) is a subgraph of K 4 and so one of the edges ab, bc, cd, ad must be present in H ′ . Without loss of generality we may assume that ab is present. Note that ab is not in H as e and e ′ is assumed to be a bad crossing and e and e ′ thus lie in different components of the graph with edge set E(H) ∪ {e}.
Since there are no edges in E(H ′ ) \ E(H) with no bad crossings, ab must have a bad crossing with some edge, say vw. We claim that vw must cross e. If this was not the case, then a < v < b < w ≤ c < d. This implies that vw crosses e ′ = bd in H and since H is noncrossing, vw and e ′ = bd must be in the same component. But then vw and ab do not form a bad crossing. Thus, vw crosses e and so any edge that crosses ab to form a bad crossing will form a bad crossing with e. This means that e has strictly more bad crossings than ab which is impossible as e was chosen to be minimal. Thus, there is an edge of E(H ′ ) \ E(H) which has no bad crossings.
Let e be an element of E(H ′ ) \ E(H) which has no bad crossings and let G ′ be the graph on V (G) with edge set E(H) ∪ {e}. Since e has no bad crossings, the partition associated to G ′ is noncrossing. Now let H ′′ be the bond induced on E(H) ∪ {e}. Then G ′ and H ′′ have the same connected components and so correspond to the same partition. It follows that H ′′ is a noncrossing partition. Since H < H ′′ ≤ H ′ and H ⋖ H ′ , H ′ = H ′′ . Moreover, by construction there are exactly two components that merge together from H to H ′ and so by Proposition 2.10, NC G is graded.
The notion of tightly closed may seem artificial at first glance. However, there is an ordertheoretic way to define tightly closed that is much like the notion of semimodularity. Recall that a lattice L is (upper) semimodular, if it is graded and for all x, y ∈ L, ρ(x∨y)+ρ(x∧y) ≤ ρ(x) + ρ(y). In the case that a 1 and a 2 are distinct atoms of L, semimodularity implies that ρ(a 1 ∨ a 2 ) = 2. Tightly closed graphs can be defined by slightly relaxing this idea. Theorem 5.16. Let G be a graph which is crossing closed. G is tightly closed if and only if for all distinct atoms a 1 , a 2 ∈ NC G , ρ(a 1 ∨ a 2 ) = 2 or ρ(a 1 ∨ a 2 ) = 3. While the noncrossing bond poset of a tightly closed graph is graded, other nice ordertheoretic properties do not always hold. For example, they need not be shellable or have a noncrossing NBC interpretation. This can be seen by noting that the 5-pointed star is tightly closed, but it is not shellable and also does not have the NCNBC interpretation for its Möbius function. However, if we make the further assumption that the graph is upper crossing closed, we get many nice properties.
Proposition 5.17. Let G be an upper crossing closed graph which is tightly closed. Then the conclusions of Theorem 4.4 hold. In particular, NC G is graded, shellable, and the Möbius function and characteristic polynomial have a combinatorial interpretation in terms of NC-NBC sets.
Proof. First note that since G is crossing closed, Proposition 2.7 part (e) implies that NC G has a1. Now suppose that H < H ′ . Let e = min E(H ′ ) \ E(H). Adopting the terminology from the proof of Theorem 5.15, we will show that e has no bad crossings in H ∪{e}. Before we finish this subsection, let us give an example of a tightly closed upper crossing closed graph. Let K n even,odd be the complete bipartite graph on [n] whose parts are the even and odd numbers. As mentioned at the beginning of this subsection, any complete bipartite graph is tightly closed, so K n even,odd is tightly closed. Recall that our graphs lie on a circle with evenly spaced vertices. Given vertices x < y of K n even,odd , we let dist(x, y) = min(y − x − 1, n − y + x − 1) be the minimum number of vertices between x and y. Define a partial order on E(K n even,odd ) by declaring ij is less than i ′ j ′ if and only if dist(i, j) < dist(i ′ , j ′ ) and let ✂ be any linear extension of this order. It is not hard to show K n even,odd is upper crossing closed with respect to this ordering.

Strongly Upper Crossed Graphs
In this subsection we consider another family of (not necessarily crossing closed) graphs and show their noncrossing bond posets are graded and shellable. We also show the noncrossing NBC set interpretation for the Möbius function and characteristic polynomial still hold in this setting.
Definition 5.18. Let G be a graph with a total ordering, ✂, on the edge set of G. We say that a graph G is strongly upper crossed with respect to ✂ if whenever ac, bd are crossing edges, there is at least one minimal induced connected component of G containing ac and bd and every edge in each minimal induced connected component of G containing ac and bd precedes ac and bd in the ordering ✂.
One may think of strongly upper crossed graphs as a relaxing of the crossing closed condition, but at the cost of requiring a stronger condition on the ordering of edges as compared to that given for upper crossing closed graphs. As an example of a strongly upper crossed graph, consider the graph G in Figure 1. If we order the edges so that 14 and 35 are the largest, G is strongly upper crossed with respect to this order. More generally, any connected graph with a single crossing is strongly upper crossed with ordering where the crossing edges are ordered so that they are the largest. Since the graph in Figure 1 is not chordal nor crossing closed, the family of strongly upper crossed graphs is distinct from the families presented in the previous two subsections.
We should note that not all upper crossing closed graphs are strongly upper crossed. For example, K 5 is upper crossing closed with respect to the lexicographic ordering, but there is no ordering which makes it strongly upper crossed. To see why, note that since 14 and 25 cross and 24 ∈ J(14, 25), 24 must be smaller than 14 and 25. However, 24 and 35 cross and 25 ∈ J(24, 35) so 25 must be smaller than 24 which is impossible.
The following lemma will allow us to apply Theorem 4.4 to strongly upper crossed graphs. Using the previous lemma and the fact that noncrossing bond posets of strongly upper crossed graphs have a1, we get the main theorem of this subsection. Theorem 5.20. Let G be a strongly upper crossed graph. Then the conclusions of Theorem 4.4 hold. In particular, NC G is graded, shellable, and the Möbius function and characteristic polynomial have a combinatorial interpretation in terms of NCNBC sets.

A Summary of Results on Families of Graphs
To finish this section, we gather all the information about families of graphs that we have seen throughout this paper. This data appears in Table 1. The rows of Table 1 refer to the families of graphs and the columns to the properties of the graphs or their noncrossing bond poset. The term "NCNBC interpretation" refers to if the Möbius function and characteristic polynomial (if applicable) have the noncrossing NBC set interpretation of Theorem 3.5 or not. Every instance of "sometimes" is genuine in the sense that there are graphs in that family which do and do not posses the prescribed property.

Open Problems
As we have seen, several of the nice properties of the noncrossing partition lattice and the bond lattice have analogues in the noncrossing bond poset. Given the multitude of nice properties that these lattices enjoy, we encourage the reader to see if their favorite properties have an analogue in the noncrossing bond poset. We collect a few open problems that we have found interesting below. The list is in no way to be considered complete.
Recall that the Whitney numbers of the first kind of a graded poset are the numbers w 0 , w 1 , . . . , w n where w i is the sum of the Möbius values of elements of P of rank i. In other words, they are the coefficients of the characteristic polynomial. Moreover, recall that a sequence a 0 , a 1 , . . . , a n of real numbers is called log-concave if for all 1 ≤ i ≤ n − 1 we have that a i−1 a i+1 ≤ a 2 i . Gian-Carlo Rota conjectured that the Whitney numbers of the first kind for geometric lattices (which include bond lattices) are log-concave. In [19] Huh proved that the Whitney numbers of the first kind for bond lattices are log-concave and further work of Adiprasito, Huh, and Katz [1] proved the more general conjecture concerning the Whitney numbers of the first kind for geometric lattices. Since the noncrossing bond poset is a (relatively) well-behaved subposet of a geometric lattice, it seems natural to ask if the corresponding conjectures hold for the noncrossing bond poset. Question 6.1. For which graphs are the Whitney numbers of the first kind of NC G unimodal or log-concave?
We should note that, unlike the case for the bond lattice, the Whitney numbers of the first kind of the noncrossing bond poset do not need to alternate in sign and can have internal zeros (e.g. the characteristic polynomial of the 5-pointed star in Figure 7 has an internal zero). As a result, it is not the case that the absolute values of the Whitney numbers of the first kind are log-concave or unimodal in general. We mention this since, if the sequence did alternate and have no internal zeros, the log-concavity would imply unimodality.
The noncrossing partition lattice is well-known to be rank-symmetric (see, for example [20]). That is, for NC n+1 , the number of elements of rank k is the same as the number of elements of rank n − k. It seems that it is rare for the noncrossing bond poset to be rank-symmetric. This should not be that surprising as the bond lattice is also rarely ranksymmetric. However, for n ≥ 5, computations suggest that if we let C n denote the cycle on n vertices with edges 12, 23, . . . , n − 1n, 1n, then the complement C n has a noncrossing bond poset which is rank-symmetric. This leads us to a broader question. Question 6.2. When is NC G rank-symmetric?
As we saw in the discussion preceding the previous question, the graph C n seems to have a rank-symmetric noncrossing bond poset. Despite this nice property, it seems that the poset is not shellable. Naturally, this leads us to the following. We note here that C n is tightly closed (but not upper crossing closed). Thus, we know that tightly-closed (and hence crossing closed) does not imply shellability. There is some hope that upper crossing closed graphs produce shellable noncrossing bond posets. However, since they are not always graded, this will require considering non-pure shellings.
Given a graph (or more generally a matroid) one can consider the collection of nonbroken circuits. This set forms a simplicial complex called the broken circuit complex or NBC complex. It has several nice properties, its f -vector encode the coefficients of the chromatic polynomial of the graph and the complex is known to be shellable. A related complex called the independence complex is formed considering all the subsets of the edges sets which form acyclic subgraphs. Since subsets of noncrossing sets are noncrossing, we can also consider the simplicial complex of noncrossing NBC sets and noncrossing independent sets. Question 6.4. What is the structure of the noncrossing NBC complex and noncrossing independence complex of a graph?

A Appendix: Algorithms
In this appendix, we give two important algorithms on NC G , Algorithms A.1 and A.6. We also give a forbidden subgraph characterization of upper crossing closed graphs in Theorem A.4.
Algorithm A.1 decides if NC G is a lattice in time on the order of n 7 where n the number of vertices of the graph G. This is proved in Theorem A.2. Note that a brute-force algorithm to test if NC G is a lattice can take time super-exponential in n. For example, an algorithm that checks if every pair of elements in NC G has a meet and a join, could take time at least on the order of the number of elements of NC G . For an n vertex graph, that may be as large as the Bell number B n of the number of set partitions of [n] and B n > (n/e log(n)) n [4]. Note that we call Algorithm A.1 the "crossing-closed" algorithm as it is actually checking if G is crossing closed. Of course, NC G being a lattice is equivalent to G being crossing closed, see Theorem 2.6.
Algorithm A.6 determines if a graph G is an upper crossing closed graph. Recall that if G is upper crossing closed then the Möbius function and characteristic polynomial of NC G have nice interpretations in terms of noncrossing NBC sets, see Theorem 3.5. When given a graph G, Algorithm A.6 will either produce a specific upper crossing closed ordering of E(G) or will produce what we term an obstruction (see Definition A.3), a specific subgraph of G that clearly shows there can be no such ordering. This also gives a forbidden subgraph characterization of upper crossing closed graphs, Theorem A.4.
In Theorem A.7, we prove that the Algorithm A.6 will run in time on the order of n 8 where again n is the number of vertices of G. Note that a brute force algorithm could again take time super-exponential in n, if it is forced to test some positive fraction of the n 2 ! possible orderings on the edges of G.
We first present our algorithm that decides if G is crossing closed, i.e. if NC G is a lattice.
Algorithm A.1. Crossing Closed Algorithm Input: A graph G on [n]. Output: A yes/no decision as to whether G is crossing closed, or, equivalently, whether NC G is a lattice. Method: For each pair of crossing edges e and f find a shortest path P (e, f ) = x 0 x 1 . . . x k with e = x 0 x 1 and f = x k−1 x k and k ≥ 3. If for some crossing pair e and f , P (e, f ) fails to exist or has k ≥ 4 and has some vertex x i with 2 ≤ i ≤ k − 2 such that x i does not separate e and f then return "No, G is not crossing closed." Otherwise return "Yes, G is crossing closed." Theorem A.2. Algorithm A.1 is a correct algorithm that runs in time O(n 7 ) where n is the number of vertices of G.
Proof. First, we will compute the complexity of the algorithm. The Floyd-Warshall algorithm gives a shortest path between all pairs of vertices in O(n 3 ) time [15]. With that pre-processing done, there are at most Next, we show that the algorithm always gives the correct output. Suppose G is crossing closed. We will show that the algorithm will return a "yes". For every pair of crossing edges e and f , J(e, f ) exists. If there is an edge incident to e and f (where incident means having a common vertex), then P (e, f ) will be a path (e, g, f ) for some edge g connecting e and f and the algorithm will not give a "no" answer based on this pair. If there is no edge incident to both e and f , then by Lemma 2.4, J(e, f ) will be a dumbbell graph, see Figure 3, a graph induced by e, f and a path Q with one end adjacent to e and the other end adjacent to f with all vertices in Q separating e and f . Thus the vertices {x 2 , . . . , x k−2 } on P (e, f ) must be the vertices of Q. The algorithm will not give a "no" answer based on this pair either. Thus the algorithm will return a "yes".
Suppose now that the algorithm returns a "yes". We will show that G is crossing closed. Let e and f be a pair of crossing edges. Since the algorithm returned a "yes", a shortest path P (e, f ) must exist. If the path has 3 vertices, i.e. there is an edge connecting e and f , then J(e, f ) = G[e ∪ f ] exists. Suppose now that the path contains at least 4 vertices. Let M be the subgraph of G induced on the vertices of P (e, f ). We claim that M is contained in every connected, induced subgraph of G that contains e and f and so J(e, f ) = M. Let x be any vertex in P (e, f ) that is not in e or f and let H be an connected induced subgraph of G containing e and f . If x is not in H then it cannot separate e and f . Since the algorithm returned "yes", x must separate e and f and so x must be in H. Thus every vertex of M is in H and since they are both induced, M is in H. Now we turn our attention to the problem of deciding whether a graph is upper crossing closed. First, let us note that not all crossing closed graphs are upper crossing closed. As an example, consider the 5-pointed star in Figure 7. It is not hard to verify that if e and f cross in the 5-pointed star, J(e, f ) is a subgraph of K 4 and so is crossing closed. However, every edge of the graph is crossed and so it is impossible to have an ordering that is upper crossing closed as the smallest edge must be noncrossing. As it turns out, this kind of issue is the only obstacle to a crossing closed graph being upper crossing closed. Definition A.3. Let G be a crossing closed graph. We say a subgraph H of G is an obstruction to G being upper crossing closed if for every edge e in H there is an edge f in H which crosses e such that J(e, f ) ⊆ H.
Theorem A.7 proves that Algorithm A.6 below will, when given a graph G, either produce an upper crossing closed ordering on E(G) or an obstruction. It also proves that an obstruction demonstrates no such ordering is possible. Thus we get the following structural characterization of upper crossing closed graphs. Note also that if every edge e in G crosses some other edge of G, then G itself is an obstruction of G. Thus we also have the following.
Corollary A.5. If G is a graph with every edge crossing some other edge, then G is not upper crossing closed.
Note that Corollary A.5 shows that the graph G of Figure 6 and the 5-pointed star of Figure 7 are not upper crossing closed.
Algorithm A.6. Upper Crossing Closed Algorithm Input: A graph G on [n]. Output: A yes/no decision on whether G is crossing closed. Then if G is crossing closed, a yes/no decision on whether G is upper crossing closed. If G is upper crossing closed, an upper crossing closed ordering is produced, and if G is crossing closed but not upper crossing closed, an obstruction is produced. Method: 1. Run the crossing closed algorithm on G, Algorithm A.1. If the answer is no, return "No. G is not crossing closed and hence not upper crossing closed." and terminate. If the answer is yes, return "Yes. G is crossing closed." and continue.
2. Let L = ∅ and let σ = ∅. (Throughout the algorithm, L will be a subset of E(G) and σ will be an ordering on L.) 3. Let L ′ be the set of edges e in E(G) \ L such that for every edge f ∈ E(G) \ L that crosses e, E(J(e, f )) ∩ L = ∅.
4. If L ′ = ∅ update L to be L ∪ L ′ and update σ to be the ordering on L ∪ L ′ that orders L according to σ and then puts all the edges of L ′ after the edges of L. The ordering within L ′ can be arbitrary. Go back to step 3.
5. If L ′ = ∅, decide on the output of the algorithm. If L = E(G), return "Yes, G is upper crossing closed, and σ is an upper crossing closed ordering on E(G)." If L = E(G), return "No. G is not upper crossing closed, and the spanning subgraph of G with edge set E(G) \ L is an obstruction." We will now show how the algorithm runs on two graphs, one upper crossing closed and the other not. First, let G be the twisted 4-cycle in Figure 5. As we have already seen, G is crossing closed and upper crossing closed with respect to the lexicographic order on its edges. The algorithm will thus correctly conclude that G is crossing closed in step 1 and will set L = ∅ and σ = ∅ in step 2. Next, it will go to step 3. Since L is empty, L ′ is the set of edges which cross no other edges. So L ′ = {12, 34}. Then the algorithm passes to step 4 where L is set to be {12, 34} and σ is set to be some total ordering of {12, 34}. Now we return to step 3. Now, E(G) \ L = {13, 24}. Since 13 and 24 form the only crossing in G and J(13, 24) = G intersects L, the algorithm sets L ′ = {13, 24}. Next, we go to step 4, where L is set to be {12, 34, 13, 24} and σ is some total ordering where the first two elements are 12 and 34 and the last two elements are 13 and 24. Then we return to step 3, where L ′ is set to be empty. Finally, we go to step 5 and since L = E(G), the algorithm returns that G is upper crossing closed with respect to the ordering σ, which indeed it is. The reader may have noticed that the ordering the algorithm produces is not the lexicographic ordering. This is because the algorithm always puts edges with no crossing before any edge with a crossing. Thus the algorithm is not always capable of producing all possible upper crossing closed orderings.
Finally, let us show (e). Suppose that G is not upper crossing closed. Then the algorithm must find an obstruction. If it did not, by part (d), σ would be an upper crossing closed ordering. By part (b), what the algorithm produces is really an obstruction and by part (c) this obstruction demonstrates that G is not upper crossing closed. Thus the algorithm will return an obstruction and correctly returns that G is not upper crossing closed. Now suppose that G is upper crossing closed. Then by the contrapositive of part (c) and by part (b) the algorithm produces no purported obstruction. So then by part (d), it returns an upper crossing closed ordering. It will then correctly return that the graph is upper crossing closed.