Generating the cycle space of planar graphs

We prove that the cycle space of every planar finitely separable 3-connected graph $G$ is generated by some ${\rm Aut}(G)$-invariant nested set of cycles. We also discuss the situation in the case of smaller connectivity.


Introduction
Dicks and Dunwoody [2] showed that the cut space of every connected graph is generated by some nested set of cuts that is invariant under the automorphisms of the graph, where a set of cuts is nested if no cut separates any two edges of any other cut.
In this note, we present a similar result for the cycle space of a planar graph. Roughly speaking, two cycles in a planar graph are nested if their embeddings in the plane do not cross (see Section 2 for a precise definition). We call a graph finitely separable if (1) there are only finitely many pairwise internally disjoint paths between any two vertices. With these definitions, we are able to state our main theorem: Theorem 1.1. Every planar finitely separable 3-connected graph G has an Aut(G)-invariant nested set of cycles that generates its cycles space.
Note that Theorem 1.1 is rather trivial for finite graphs: the face boundaries form a nested set of cycles that generate the cycle space and is invariant under the automorphisms due to a result of Whitney [12] (cf. Theorem 2.3). 1 Theorem 1.1 can be seen as a dual version of Dicks' and Dunwoody's theorem [2,Theorem 2.20], and indeed, our strategy to prove our main theorem is to deduce it from the fact that Dicks' and Dunwoody's theorem holds for its dual graph. Of course, it is necessary to have a (unique) dual graph. But duals of infinite planar graphs are not straightforward analogues of the finite ones, see Thomassen [10]. Bruhn and Diestel [1] showed that the additional assumption that the graphs are finitely separable implies the existence of unique duals in the 3-connected case. Theorem 1.1 fails for graphs that have lower connectivity -regardless which embedding in the plane is considered: take a graph G consisting of two vertices x, y and four distinct paths of length 2 between them. Obviously, this graph is planar. But all cycles in G, each of which has length 4, lie in the same Aut(G)-orbit and it is easy to find two crossing cycles in G.
Even though Theorem 1.1 fails for graphs of lower connectivity, it is possible to extend the graph canonically to a graph G ′ such that its connectivity remains the 1 The same is proof holds for VAP-free planar graphs. These are graphs that admit an embedding in the plane without any accumulation point of the vertices. same, such that Aut(G) acts canonically on G ′ , and such that G ′ has an Aut(G ′ )invariant nested set of cycles generating the cycle space of G ′ . We will discuss this in Section 4.
It is not clear whether Theorem 1.1 remains true if we omit the assumption of finite separability: Problem. Does every planar 3-connected graph G have an Aut(G)-invariant nested set of cycles that generates its cycles space?
Our main theorem has applications for infinite graphs and groups: in [6], the author uses it to show that locally finite quasi-transitive planar graphs are accessible, and in [5], Georgakopoulos and the author apply it to obtain planar presentations for planar groups. (Planar presentations of planar groups are presentations that directly tell that the group has a planar Cayley graph.)

Preliminaries
Throughout the paper, a graph may have loops and multiple edges (what is usually called a multigraph). 2 Let G be a graph. A one-way infinite path is a ray. Two rays are equivalent if they lie eventually in the same component of G − S for every finite S ⊆ V (G). This is an equivalence relation whose equivalence classes are the ends of G. By Ω(G) we denote the set of ends of G. Now we define a basis of a topology on G ∪ Ω(G). In order to do so, we view every edge as an isometric copy of the unit interval. For every v ∈ V (G) and every n ∈ N, the set of all points on edges incident with v that have distance less than 1/n to v are open. For every end ω and every finite vertex set S, let C(S, ω) be the component of G − S that contains all rays in ω eventually and letĈ(S, ω) be C(S, ω) together with all ends of G that contains rays of C(S, ω). The setĈ(S, ω) together with the interior points of the edges between S and C(S, ω) is open. We denote by |G| the topological space on G ∪ Ω(G) defined by all these open sets.
A vertex v dominates an end ω if for some ray R in ω there are infinitely many v-R paths in G that pairwise meet only in v. We call the end ω dominated. We defineG to be the quotient space obtained from |G| by identifying every dominated end with its dominating vertices. Note that for finitely separable graphs every end is identified with at most one vertex and thatG and |G| coincide if G is locally finite.
A circle is a homeomorphic copy of the unit circle inG and its circuit is the set of edges that it contains. The sum (over F 2 ) of finitely many cycles (C i ) i∈I in G is the set of those edges that occur in an odd number of C i . The set of all sums of cycles form a vector space over F 2 , the cycle space C(G) of G. For every n ∈ N, let C n (G) be the subspace of the cycle space C that is generated by the cycles of length at most n. So we have C(G) = n∈N C n (G). The automorphisms of G act canonically on C(G).
A cut is a subset B of E(G) that is, for some bipartition {X, Y } of V (G), the set of all those edges that have one of its incident vertices in X and the other in Y . We call two cuts B 1 and B 2 nested if for bipartition {X 1 , Y 1 } and {X 2 , Y 2 } that correspond to B 1 and B 2 , respectively, one of the following holds: 2 Note that all cited results that we need for our main result, particularly Theorem 3.1, are valid for this general notion of graphs and are stated in this way.
A cut B is tight if G − B consists of two components. The sum (over F 2 ) of finitely many cuts in G is the set of those edges that occur in an odd number of these cuts. The set of all sums of finite cuts form a vector space over F 2 , the cut space B(G) of G. For n ∈ N let B n (G) be the subspace of B(G) induced by the tight cuts of size at most n. So we have B(G) = n∈N B n (G). Note that no element of B(G) is infinite and that Aut(G) acts canonically on B(G).
Let G and G * be graphs such that G is finitely separable and let * : be a bijection. We call G * the dual of G if for every (finite or infinite) F ⊆ E(G) the following holds: A face of a planar embedding ϕ : G → R 2 is a component of R 2 ϕ(G). The boundary of a face F is the set of vertices and edges of G that are mapped by ϕ to the closure of F . A path in G is facial if it is contained in the closure of a face. The following theorem is due to Whitney [12,Theorem 11] for finite graphs and due to Imrich [7] for infinite ones.
Let G be a 3-connected graph embedded in the sphere. Then every automorphism of G maps each facial path to a facial path.
In particular, every automorphism of G extends to a homeomorphism of the sphere.
So every planar 3-connected graph has, basically, a unique planar embedding into the plane. In the remainder of this paper we always assume this implicitly if we talk about planar embeddings of planar 3-connected graphs.
Duals of finite graphs, can be easily found using the plane, which we will just recall here, as we will use this well-known fact (see e.g. [3, Section 4.6]).
Remark 2.4. Let G be a finite planar graph with planar embedding ϕ : G → R 2 . Let G ′ be the graph whose vertices are the faces of ϕ(G) and where two vertices are adjacent if their boundaries contain a common edge of G. Then G ′ is the dual of G.
For a planar graph G with planar embedding ϕ : G → R 2 , we call a set D of cycles nested if for no C, D ∈ D both faces of R 2 ϕ(C) contain vertices of D. So ϕ(D) lies in one face of R 2 ϕ(C) together with its boundary ϕ(C). We call two

Planar 3-connected graphs
Our main tool in the proof of our main theorem will be the following theorem: Basically, our proof goes as follows. We consider the dual graph G * of G and apply Theorem 3.1 to G * . So we find nested sets of cuts. These cuts then will define us our sequence of nested sets of cycles in G. In preparation of that proof, we shall prove two propositions first.
Proposition 3.2. Let G be a planar finitely separable 3-connected graph and let G * be its dual. If C 1 and C 2 are cycles in G such that the cuts C * i := {e * | e ∈ C i } are nested, then C 1 and C 2 are nested.
Proof. Let ϕ : G → R 2 be a planar embedding of G. Let us suppose that C 1 and C 2 are not nested. Let H ⊆ G * be a finite connected subgraph that contains C * 1 and C * 2 as tight cuts. Then H * contains the two cycles C 1 and C 2 . For i = 1, 2, let A i be the unbounded face of R 2 ϕ(C i ) and let B i be its bounded face. As C 1 and C 2 are not nested, each of A 1 and B 1 contains edges from C 2 . Due to Remark 2.4, the tight cut C * 1 separates edges from C * 2 in H. So each of the two components of G * − C * 1 contains vertices of each of the two components of G * − C * 2 . Thus, C * 1 and C * 2 are not nested in H and hence neither in G * . Proposition 3.3. Let G be a planar finitely separable 3-connected graph and let G * be its dual. If C 1 and C 2 are two cycles in G in the same Aut(G)-orbit, then the cuts C * i := {e * | e ∈ C i } are in the same Aut(G * )-orbit. Proof. Let ϕ : G → R 2 be a planar embedding of G. Consider two cycles C 1 , C 2 in G with C 2 = C 1 α for some α ∈ Aut(G). For i = 1, 2, let G i 1 be the subgraph of G that lies in the closure of the bounded face of R 2 ϕ(C i ). (In particular, we have C i ⊆ G i 1 .) Let G i 2 be defined analogously but with the unbounded face of R 2 ϕ(C i ) instead of the bounded face. So we have G i 1 ∩ G i 2 = C i . For i, j ∈ {1, 2}, letĜ i j be defined by adding one new vertex to G i j that is adjacent to every vertex of C i that has neighbours in G that lie in G i 3−j C i , that is, that has neighbours in G that lie in the face of R 2 ϕ(C i ) that avoids G i j . Let us show the following (3) The graphsĜ i j are 3-connected. Let v be the unique vertex inĜ i j G i j . To prove (3), consider any two vertices x, y ofĜ i j . Let a, b ∈ V (Ĝ i j ) {x, y}. First we look at the case that x and y lie in G i j . If a and b that are distinct from v, then we find a path P between a and b in G that avoids x and y, since G is 3-connected. We look at its canonical image inĜ i j , where we replace every vertex in G G i j by v: this image is an a-b walk, so contains some a-b path. If either a or b is v, say a, then we pick any vertex a ′ in G G i j . Again, we find in G some a ′ -b path that defines us some a-b path inĜ i j . If either x or y is v, the unique vertex inĜ j i G i j , say x, then we find three pairwise disjoint a-b paths in G by Menger's theorem, as G is 3-connected. At least one of them does not contain y. Let P be such a path. If P avoids G Ĝ i j , then this is also an a-b path inĜ i j avoiding x and y. So let us assume that P contains some vertex from G Ĝ i j . Then it has a first and a last vertex on C i , call them u and w. At most one of the two u-w paths on C i contains y, as y does not lie on P and so y is neither u nor w. This path on C i together with aP u and wP b, the subpaths of P from a to u and from w to b, respectively, induce a connected subgraph ofĜ i j that avoids x and y and contains a and b. So we have proved in all situations that no two vertices disconnectsĜ i j , which proves (3). The automorphism α, that maps C 1 to C 2 , induces an isomorphism α ′ fromĜ 1 1 either toĜ 2 1 or toĜ 2 2 due to Theorem 2.3, sayĜ 2 k . Note that the graphs G i j are still finitely separable. Thus, they have a unique dual by Theorem 2.1 and the duals ofĜ 1 1 andĜ 2 k are isomorphic, too. The same holds forĜ 1 2 andĜ 2 3−k . Let H i j be obtained from (Ĝ i j ) * by removing e * for every edge e ∈ E(Ĝ i j ) that is incident with the unique vertex ofĜ i j not in G. To obtain G * from (Ĝ i 1 ) * and (Ĝ i 2 ) * , take H i 1 and H i 2 and identify the edges of C * i in those graphs. It is straight-forward to see that the cuts of the resulting graph are the circuits of G, that is, that the resulting graph is G * . In addition the isomorphism from (Ĝ 1 1 ) * to (Ĝ 2 k ) * induces an isomorphism β 1 : H 1 1 → H 2 k and similarly we obtain an isomorphism β 2 : H 1 2 → H 2 3−k . These isomorphisms define canonically an automorphism β of G * whose restriction to H 1 j is β j . In particular, we have C * 1 β = C * 2 . Now we are able to prove Theorem 3.4, a sharpend version of Theorem 1.1.
Theorem 3.4. For every planar finitely separable 3-connected graph G and all n ∈ N there exists an Aut(G)-invariant nested set D n of cycles of length at most n such that D n generates C n (G).
In addition, we may choose the sets D n such that D n−1 ⊆ D n .
Note that Theorem 1.1 is a direct consequence of Theorem 3.4: just take the set n∈N D n . Obviously, this is nested and generates the whole cycle space of G. Proof of Theorem 3.4. Let G be a planar finitely separable 3-connected graph with planar embedding ϕ : G → R 2 . Due to Theorem 2.1, it has a unique dual G * , which is 3-connected, too. So we can apply Theorem 3.1 and obtain a sequence E 1 ⊆ E 2 ⊆ . . . of subsets of B(G * ) such that each E n is an Aut(G * )-invariant nested set of tight elements of order at most n that generates B n (G * ).
We shall prove that the sequence of the sets with B * := {e * | e ∈ B} satisfies the assertion. Note that D n is a set of cycles of length at most n as E n is a set of finite tight cuts of size at most n. The set D n is nested due to Proposition 3.2 and its Aut(G)-invariance of D n follows directly from Proposition 3.3. As E n generates B n (G * ), the set D n generates C n (G): it suffices to show that every cycle C of length at most n is generated by D n , but as the tight cut C * is the sum of finitely many elements of E n , the dual sets in E(G) of those summands sum to C. This finishes the proof.

Planar graphs of small connectivity
Tutte [11] proved for finite graphs that 2-connected graphs can be decomposed into '3-connected parts' and cycles. Later, his result was extended by Droms et al. [4] to locally finite graphs and by Richter [8] to graphs of arbitrary degree. In order to state this result formally, we need some definitions.
Let G be a graph. A tree-decomposition of G is a pair (T, V) of a tree T and a family V = (V t ) t∈T of vertex sets V t ⊆ V (G), one for each vertex of T , such that (T1) V = t∈T V t ; (T2) for every e ∈ E(G) there exists some t ∈ V (T ) such that V t contains both end vertices of e; (T3) V t1 ∩ V t3 ⊆ V t2 whenever t 2 lies on the t 1 -t 3 path in T .
The sets V t are the parts of the tree-decomposition (T, V) and we also call the graph induced by V t a part of (T, V). The intersections V t1 ∩ V t2 for edges t 1 t 2 of T are its adhesion sets and the maximum size of the adhesion sets is the adhesion of (T, V). For a part V t , its torso is the graph whose vertex set is V t and whose edge set is The automorphisms of G induce a canonical action on vertex sets of G. If every automorphism of G maps every part of the tree-decomposition to another of its parts and this map induces an automorphism of T then we call the tree-decomposition Aut(G)-invariant.
[8] Every 2-connected graph G has an Aut(G)-invariant tree-decomposition of adhesion 2 each of whose torsos is either 3-connected or a cycle or a complete graph on two vertices.
We call a tree-decomposition as in Theorem 4.1 a Tutte decomposition.
Theorem 4.2. Let G be a planar finitely separable 2-connected graph. If G has a Tutte decomposition (T, V) whose adhesion sets of (T, V) are complete graphs, then G has an Aut(G)-invariant nested set of cycles that generates its cycles space.
In addition, we may choose the generating cycles so that each of them lies in a unique part of (T, V).
Proof. Due to the assumption on the adhesion sets, we know that not only the torsos but also the parts theirselves are either cycles or 3-connected. First, we show: (4) Cycles of distinct parts are nested.
Let ϕ be the planar embedding of G into R 2 and let C 1 , C 2 be cycles in distinct parts. If C 1 and C 2 are not nested, then both faces of R 2 ϕ(C 1 ) contain vertices of C 2 and thus C 1 and C 2 have two common vertices. As they lie in distinct parts and the intersection of distinct parts lies in some adhesion set, which has size 2, the cycles C 1 and C 2 have precisely two common vertices x, y and these form an adhesion set of (T, V). Let V i be the part that contains C i for i = 1, 2. If one of the parts, say V 1 , is a cycle, then C 1 contains the edge xy. As it contains no other vertex of C 2 , it cannot contains vertices of both faces of R 2 ϕ(C 2 ). Thus, each of the two parts V i is 3-connected. In particular, we find in V 1 a third path P from every vertex of the bounded face of R 2 ϕ(C 2 ) to every vertex of the unbounded face. But then ϕ(P ) crosses ϕ(C 2 ), so there is a third vertex in V 1 ∩ V 2 . This contradiction shows (4).
Since (T, V) is Aut(G)-invariant, parts that are cycles are mapped to parts that are cycles. So if we take the set D 1 of cycles that appear as a part in (T, V), then this set is Aut(G)-invariant and nested due to (4). For every 3-connected part H of (T, V) we find, by Theorem 1.1, a nested Aut(G)-invariant set D H of cycles generating the cycles space of H. Now consider a set H of 3-connected parts of (T, V) that contains precisely one element of each Aut(G)-orbit on the 3-connected parts. Then the set is obviously Aut(G)-invariant and it is nested by (4). We claim that D := D 1 ∪ D 2 satisfies the assertion. The additional part is obvious by definition of D. As D is nested and Aut(G)-invariant, it remains to prove that it generates the cycle space of G. So let C be any element of the cycles space of G. It suffices to consider the case that C is a cycle, as all cycles generate the cycle space. If C lies in a part of (T, V), then it is generated by D by defintion of D. If C does not lie in any part of (T, V), then there is some adhesion set {x, y} of (T, V) that disconnects C. Let P 1 , P 2 be the two distinct x-y paths on C. Then C is the sum of P 1 + yx and P 2 + yx. By induction on the length of C, the cycles P 1 + yx and P 2 + yx are generated by D and so is C.
Note that we can embed any planar 2-connected graph G in a planar 2-connected graph G ′ such that G ′ is obtained from G by making the adhesion sets of a Tutte decomposition (T, V) of G complete. This is always possible as any adhesion set separates the graph, so lies on the boundary of some face and disconnects this face boundary; thus, it is not possible that two new such edges 'cross' 3 since no adhesion set of any tree-decomposition can separate any other of its adhesion sets.
As each cycle of a graph lies in a unique maximal 2-connected subgraph, an immediate consequence of Theorem 4.2 using the well-known block-cutvertex tree and the fact that cycles in distinct components are disjoint is the following: Theorem 4.3. Let G be a planar finitely separable graph such that for each of its maximal 2-connected subgraphs the adhesion sets of some Tutte decomposition are complete graphs on two vertices. Then G has an Aut(G)-invariant nested set of cycles that generates its cycles space.