On fractional fragility rates of graph classes

We consider, for every positive integer $a$, probability distributions on subsets of vertices of a graph with the property that every vertex belongs to the random set sampled from this distribution with probability at most $1/a$. Among other results, we prove that for every positive integer~$a$ and every planar graph $G$, there exists such a probability distribution with the additional property that deleting the random set creates a graph with component-size at most $(\Delta(G)-1)^{a+O(\sqrt{a})}$, or a graph with treedepth at most $O(a^3\log_2(a))$. We also provide nearly-matching lower bounds.


Introduction
Planar graphs "almost" have bounded treewidth, in the following sense: For every assignment of weights to vertices and for every positive integer a, it is possible to delete vertices of at most 1/a fraction of the total weight so that the resulting subgraph has treewidth at most 3a − 3. Equivalently, there exists a probability distribution on subsets of vertices whose complement induces a subgraph of treewidth at most 3a − 3, such that each vertex belongs to a set sampled from this distribution with probability at most 1/a. This property is the key ingredient of a number of approximation algorithms for planar graphs [1]. To study this phenomenon more generally, Dvořák [5] introduced the notion of fractional fragility of a graph class with respect to a graph parameter. Let us give the definitions we need to speak about this notion.
For ε > 0, we say that a probability distribution on the subsets of vertices of a graph G is ε-thin if for each vertex v, the probability that v belongs to a set sampled from this distribution is at most ε. For example, we commonly use the following (1/a)-thin probability distribution. Suppose that sets X 1 , . . . , X a ⊆ V (G) are pairwise disjoint, and that t of these sets are empty. We give to each non-empty set X i the probability 1/a and to the empty set the probability t/a. All other sets are given probability 0. We call this distribution the uniform distribution on {X 1 , . . . , X a }.
Let f be a graph parameter, that is, a function assigning to every graph a non-negative real number such that isomorphic graphs are assigned the same value. We will generally consider parameters that are monotone (satisfying that f (H) f (G) whenever H is a subgraph of G), or at least hereditary (satisfying f (H) f (G) whenever H is an induced subgraph of G). For a real number b and a graph G, let G f b be the set of all subsets X ⊆ V (G) such that f (G[X]) b, and let G f ↓b be the set of all subsets For example, if tw is the function that to every graph assigns its treewidth, then G tw↓3a−3 is the set of vertex sets whose complement induces a subgraph of treewidth at most 3a − 3.
Let r : N → R + 0 be a non-decreasing function. A graph G is fractionally f -fragile at rate r if for every positive integer a, there exists a (1/a)-thin probability distribution on G f ↓r(a) . Of course, every graph is fractionally f -fragile at rate given by the constant function r(a) := f (G); so the notion is more interesting for graph classes. We say that a class of graphs is fractionally f -fragile at rate r if each graph from the class is, and we say that the class is fractionally f -fragile if it is fractionally f -fragile at some rate. Coming back to the introductory example, the class of planar graphs is known to be fractionally tw-fragile at rate r(a) := 3a − 3; see Corollary 11 below for details.
Graphs from fractionally f -fragile classes can be viewed as being close to graphs for which the parameter f is bounded, and this proximity can be useful when reasoning about their structural and quantitative properties. There are also natural links to the theory classes of bounded expansion [5]. Furthermore, as we already mentioned in the introduction, the notion has algorithmic applications, especially in the design of approximation algorithms. Let us give an example of such an application, which was already presented before [5] along with other applications. Consider a property π(G, X) of a graph G and a subset X of its vertices. We say the property is downward-hereditary if π(G, X) being true implies that π(H, V (H) ∩ X) is true for every induced subgraph H of G, and upward-hereditary if π(G, X) being true implies that π(G , X) is true for every graph G such that G is an induced subgraph of G . As an example, the property "X is an independent set in G" is both downward-and upward-hereditary. Let α π (G) be the largest size of a set X ⊆ V (G) such that π(G, X) is true.
Observation 1. Let f be a graph parameter and let r, t : N → R + 0 be non-decreasing functions. Suppose that a class G of graphs is fractionally f -fragile at rate r and, moreover, that there exists an algorithm that for a graph G ∈ G and a positive integer a returns a set sampled from a (1/a)-thin probability distribution on G f ↓r(a) in polynomial time. Let π be a downward-and upward-hereditary property, and suppose that α π (H) can be determined in time t(f (H)) · poly(|H|) for any graph H. Then there exists a randomized algorithm that for a graph G ∈ G and a positive integer a returns in time t(r(a)) · poly(|G|) an integer b α π (G) such that E[b] (1 − 1/a)α π (G).
Proof. Sample a set X from a (1/a)-thin probability distribution on G f ↓r(a) , so that f (G − X) r(a), and let b = α π (G − X). Since π is upward-hereditary, we have b α π (G).
Moreover, consider a set Y ⊆ V (G) such that π(G, Y ) is true and |Y | = α π (G). Since X is sampled from a (1/a)-thin probability distribution, we have E[|Y \ X|] (1 − 1/a)|Y |. Since π is downward-hereditary, π(G − X, Y \ X) is true, and thus For example, the independence number is hard to approximate within a polynomial factor [2] in general, but can be determined in time 3 tw(H) · poly(|H|) for any graph H. Consequently, for any class of graphs that is fractionally tw-fragile at rate r (efficiently in the sense of Observation 1), the independence number can be approximated in an n-vertex graph from the class up to the factor of (1 − 1/a) in time 3 r(a) · poly(n). Let us remark that in essentially all known cases of fractionally fragile classes, it is possible to find a probability distribution as described in Observation 1 with support of polynomial size, and thus the algorithm can be derandomized by trying all sets from the support rather than sampling one of them.
Fractional fragility is also related to generalizations of the (fractional) chromatic number arising from the following scheme introduced by Wood [16, Section 10]. An (f, b)-coloring of a graph G is an assignment ϕ of colors to the vertices such that f (G[ϕ −1 (c)]) b for every color c, that is, such that each color class belongs to G f b . We can now define χ f,b (G) as the least number of colors in an (f, b)-coloring of G. For a class of graphs G, we naturally define χ f (G) as the smallest integer s such that for some positive integer b, all graphs G ∈ G satisfy χ f,b (G) s. For example, let (G) be the maximum of the orders of the components of the graph G. Then χ ,1 (G) is just the ordinary chromatic number of G, while in general, the parameter χ ,b (G) has been studied as the clustered chromatic number [16].
Similarly to the way the fractional chromatic number is derived from the ordinary chromatic number [15], we can also derive the fractional variant of this generalization. A fractional (f, b)-coloring of a graph G is a function κ : the number of colors |κ| used by this coloring is Y ∈G f b κ(Y ). We define χ frac f,b (G) to be the infimum of |κ| over all fractional (f, b)-colorings κ of G. For a class G of graphs, we define χ frac f (G) as the infimum of the real numbers s such that for some positive integer b, all graphs G ∈ G satisfy χ frac f,b (G) s. Note that unlike the ordinary fractional chromatic number case, this can indeed be a proper infimum: as b increases, the fractional (f, b)-coloring may need fewer colors, converging to but never reaching χ frac f (G). This motivates the following definition that captures the rate of the convergence. For a real number c and a function r : N → R + 0 , we say that a class of graphs G is fractionally f -colorable by c colors at rate r if for every integer a 1, every graph G ∈ G satisfies χ frac f,r(a) (G) c + 1/a. As we will see below (Lemma 2), fractional f -fragility is equivalent to fractional f -colorability by 1 color, at a matching rate.

Main results
The previous treatment of fractional fragility [5] was mostly qualitative. In this paper, we focus on the quantitative aspect: the rate of fractional fragility for various parameters and graph classes. Note that the rate is important in the applications, as it determines, e.g., the multiplicative constant in the complexity of the approximation algorithm from Observation 1.
In Section 3, we consider the parameter , the maximum component size. By Lemma 5, only classes of graphs with bounded maximum degree can be fractionally -fragile.
• In Theorem 16, we prove that the fractional -fragility rate r of any class of graphs containing all trees of maximum degree ∆ 3 is exponential, more precisely r(a) (∆ − 1) a−3 .
In Section 4, we turn our attention to another graph parameter, treedepth. This parameter naturally generalizes the component size, but fractional td-fragility does not require bounded maximum degree.
• In Theorem 31, we show that graphs of treewidth at most t are fractionally td-fragile at a polynomial rate r(a) = O(a t ). We also give a matching lower bound. In particular, in Theorem 26, we prove that planar graphs of treewidth at most two have fractional td-fragility rate r(a) = Ω(a 2 ).
• For outerplanar graphs (an important subclass of graphs of treewidth at most two), we show that the fractional td-fragility rate is r(a) = Θ(a log a), in Theorems 33 and 29.
• In Corollary 35, we show that planar graphs are fractionally td-fragile at rate r(a) = O(a 3 log a), in contrast to the lower bound Ω(a 2 log a) for this class.

Preliminaries
In this section, we show some basic properties of the fractional f -fragility, and present several auxiliary results we need in the rest of the paper.

Basic properties of fractional fragility
The relationship between fractional f -colorability and fractional f -fragility is given by the following lemma.

Lemma 2.
Let r : N → R + 0 be a non-decreasing function, and let r : N → R + 0 be defined by setting r (a) := r(a + 1) for every a ∈ N. Let f be a graph parameter whose value is at most r(1) on the empty graph. A class G of graphs is fractionally f -fragile at rate r if and only if it is fractionally f -colorable by 1 color at rate r .
Proof. Suppose first that G is fractionally f -fragile at rate r. Hence, for any positive integer a and any graph G ∈ G, there exists a 1 a+1 -thin probability distribution on G f ↓r(a+1) . Recall that a subset of V (G) belongs to G f ↓r(a) if and only if its complement belongs For each vertex v ∈ V (G) and a set X sampled from the probability distribution, we have and thus κ is a fractional (f, r (a))-coloring of G using |κ| = a+1 a = 1 + 1/a colors. Since this holds for every positive integer a and for all graphs in G, the class G is f -colorable by 1 color at rate r .
Conversely, suppose that G is f -colorable by 1 color at rate r . Consider a positive integer a and a graph G ∈ G. Note that setting Pr(V (G)) := 1 and Pr(Z) := 0 for all Z V (G) gives a 1-thin probability distribution on G f ↓r(1) , since r(1) f (G − V (G)). Hence, we can assume that a 2. Then there exists a fractional (f, r (a − 1))-coloring κ with |κ| = 1 + 1 a−1 , from which one can obtain a (1/a)-thin probability distribution on G f ↓r(a) by setting Pr(Z) := a−1 a κ(V (G) \ Z) for each Z ∈ G f ↓r(a) . Indeed, for any vertex v ∈ V (G) and a set X sampled from this distribution, we have This shows that G is fractionally f -fragile at rate r.
Let us note the following necessary condition for fractional f -fragility. We say that a graph G is f -breakable at rate r if for every positive integer a, there exists a set X ∈ G f ↓r(a) of size at most |V (G)|/a. The next observation readily follows from the definitions by using the linearity of expectation.

Observation 3. If a graph G is fractionally f -fragile at rate r, then it is also f -breakable at rate r.
A seminal result on -breakability dates back to the work of Lipton and Tarjan [10]; they proved it in the special case of planar graphs, however, they proof directly generalizes to any class with sufficiently small balanced separators. A separation in a graph G is a pair (A, B)   We should also note the following property, already observed in an earlier work [5].

Lemma 5. Suppose that f is a monotone graph parameter that is unbounded on stars.
Then every fractionally f -fragile class of graphs has bounded maximum degree.
Proof. Suppose that a class G of graphs is fractionally f -fragile at rate r. Since f is unbounded on stars, there exist an integer k such that f (K 1,k ) > r (3). We show that all graphs in G have maximum degree at most 3k − 3. Suppose, on the contrary, that a graph G ∈ G contains a vertex v of degree at least 3k − 2. Choose a set X ∈ G f ↓r(3) at random from a (1/3)-thin probability distribution. Consider the random variable R : A linear programming dual formulation of fragility leads to the following observation. Proof. Choose a set X ∈ G f ↓r(a) at random from a (1/a)-thin probability distribution. By the linearity of expectation, E[w(X)] w(V (G))/a, and thus there exists a set Z ∈ G f ↓r(a) such that w(Z) w(V (G))/a; i.e., there exists Z ⊆ V (G) such that w(Z) w(V (G))/a and f (G − Z) r(a).

Chordal graphs
Due to the following well-known observation, when considering graphs of bounded treewidth, it is often convenient to work in the setting of chordal graphs, that is, graphs not containing any induced cycles other than triangles.
Observation 7. Every graph has a chordal supergraph with the same set of vertices and the same treewidth. Moreover, if G is chordal, then tw(G) = ω(G) − 1.
Indeed, this equivalent characterization of treewidth can be taken as its definition, and thus we do not provide its usual (somewhat technical) definition, which can be found, e.g., in [14].
Each chordal graph G has an elimination ordering: an ordering of the vertices of G such that the neighbors of each vertex that precede it in the ordering induce a clique. By Observation 7, in an elimination ordering of G, each vertex is preceded by at most tw(G) of its neighbors. Moreover, for every induced path P in G, the last vertex of V (P ) according to the elimination ordering must be an end-vertex of P . In particular, this implies the following property.

Observation 8. Let G be a connected chordal graph and let v be the first vertex in an elimination ordering
The next observation is also based on this fact. Proof. Suppose for a contradiction that K does not induce a clique in G, and thus K contains distinct non-adjacent vertices x and y. In particular, this implies that |K| 2, and thus i > 0, since only v is at distance 0 from v. Since both x and y are at distance Since H is connected and every vertex in K has a neighbor in H, there exists a path between x and y in G with all internal vertices in H; let Q be a shortest such path. It follows that Q is an induced path. Consequently, the last vertex of Q in the elimination ordering is one of its end-points, say y. Let u be the neighbor of y in Q; since xy ∈ E(G), we have u ∈ V (H). Since the distance from v to y is i and the distance to u is greater than i, there exists a shortest path P from v to u passing through y. But both v and u precede y in the elimination ordering, and thus the last vertex of P in the elimination ordering is neither of the ends of P . This is a contradiction, since P is an induced path.

Planar graphs and treewidth
As we have mentioned in the introduction, planar graphs are fractionally tw-fragile. This is a well-known consequence of the fact that the treewidth of planar graphs is at most linear in their radius, which follows from ideas of Robertson and Seymour [13] and Baker [1]. The version we use, together with a short proof, can be found in a work by Eppstein [6,Lemma 4].

Theorem 10. Every planar graph of radius at most d has treewidth at most 3d.
The fractional tw-fragility now follows by a standard layering argument [1,7], which we restate in our notation.
Corollary 11. The class of planar graphs is fractionally tw-fragile at rate r(a) = 3a − 3.
Proof. Let G be a planar graph, without loss of generality connected, and let a be a positive integer, at least 2 since the statement is trivial for a = 1. Let v be an arbitrary vertex of G and for every non-negative integer i, let L i be the set of vertices of G at distance There is some integer j such that C contains only vertices at distance between i+ja+1 and i+ja+a−1 from v. Let G be the graph obtained from G by deleting all vertices at distance at least i + ja + a from v and by contracting all vertices at distance at most max(i + ja, 0) from v to a single vertex x. Clearly, G is a minor of G, and thus G is planar. Moreover, every vertex of G is at distance at most a − 1 from x and C ⊆ G . Consequently, tw(C) tw(G ) 3a − 3 by Theorem 10. Since this is the case for every component of Pilipczuk and Siebertz [12] demonstrated another relationship between planar graphs and graphs of bounded treewidth. Given a partition P of vertices of a graph G, let G/P be the graph obtained from G by contracting each part of P to a single vertex and suppressing the arising loops and parallel edges. A path P in a graph G is geodesic if for every x, y ∈ V (P ), the distance between x and y in G is the same as their distance in P . Pilipczuk and Siebertz [12] proved that every planar graph G admits a partition P of its vertices such that G/P has treewidth at most 8 and each part of P induces a geodesic path in G.
We need a variation which follows from a result proved by Dujmović et al. [4]. In a rooted tree T , a vertical path is an initial segment of a path from a vertex of T to the root; the lower endpoint of a vertical path is its vertex farthest from the root. Suppose T is a spanning tree of a graph G. Then a tripod is the union of the vertex sets of up to three pairwise disjoint vertical paths in T whose lower endpoints induce a clique in G.
Theorem 12 (Dujmović et al. [4,Theorem 16]). For every plane triangulation G and every rooted spanning tree T of G, there exists a partition P of G into tripods such that G/P has treewidth at most 3.
We can in fact say a bit more; note that the minor G/P of G is again a plane triangulation (after suppressing faces of length 1 and 2, but keeping non-facial loops and parallel edges), and thus by the following lemma, it is actually chordal.

Lemma 13. Let H be a multigraph triangulating the plane. If H has treewidth at most 3, then the underlying simple graph of H is chordal.
Proof. Suppose for a contradiction that C is an induced cycle in H of length at least four. By contracting all but four edges of C if necessary, we can assume C to have length exactly four. For each 2-cycle K in H, delete the vertices contained in the face of K disjoint from C and suppress the resulting 2-face, so that we can assume H does not have parallel edges. Since H is a triangulation, it does not have loops, either, and thus H is a simple graph. Consequently, H − C has exactly two components (one drawn in each face of C), and contracting each of these components to a single vertex, we deduce from the fact that H triangulates the plane that H contains the octahedron graph as a minor. However, this graph has treewidth four, which is a contradiction.
If T is a rooted breadth-first search spanning tree in some graph G, then all vertical paths in T are geodesic in G. Hence, applying Theorem 12 to such a spanning tree and using Lemma 13, we obtain the following result. We say that a partition P of the vertices of G is trigeodesic if every part of P induces in G a connected subgraph whose vertex set is covered by at most three geodesic paths of G.

Theorem 14.
For every plane triangulation G, there exists a trigeodesic partition P of V (G) such that G/P is chordal and has treewidth at most 3.

Maximum component size
Recall that (G) is the maximum of the orders of the components of the graph G. The parameter χ ,b has been intensively studied under the name clustered chromatic number [16], and is among the most natural relaxations of the chromatic number. Clustered coloring specializes to the usual notion of vertex coloring, in the sense that χ ,1 (G) = χ(G).
In the special case of planar graphs, clustered chromatic number is in general no better than ordinary chromatic number: for every integer b, there exists a planar graph G b such that χ ,b (G b ) = 4. These graphs G b necessarily have unbounded maximum degree: Esperet and Joret [8] proved that for every ∆, there exists b such that every planar graph G of maximum degree at most ∆ satisfies χ ,b (G) 3. Moreover, the Hex lemma implies that this bound cannot be improved. The situation is different in the fractional setting due to Lemma 2, since planar graphs of bounded maximum degree are fractionally -fragile (the assumption of bounded maximum degree is necessary by Lemma 5). In fact, Dvořák [5] proved fractional -fragility in much greater generality, for all classes of bounded maximum degree with strongly sublinear separators.
Theorem 15 (Dvořák [5]). Let β be a real number in (0, 1]. For every function s(n) = O(n 1−β ) and every integer ∆, there exists a function r such that every graph with balanced s-separators and maximum degree at most ∆ is fractionally -fragile at rate r.
Let us remark that the argument used to prove Theorem 15 gives a very bad bound on the rate r, especially compared to the polynomial -breakability bound from Theorem 4. As shown by Lipton and Tarjan [9], planar graphs have balanced s-separators for s(n) = 3 √ n, and thus they are -breakable at rate O(a 2 ). Considering Observation 3, it is natural to ask whether (subject to a bound on the maximum degree) planar graphs are also fractionally -fragile at quadratic rate O(a 2 ). As our first result, we show that this is not the case, even for much more restricted graph classes.
Theorem 16. Let ∆ 3 be an integer and let G be a class of graphs that contains all trees of maximum degree at most ∆. If G is fractionally -fragile at rate r, then r(a) (∆ − 1) a−3 for every integer a 4.
Proof. Fix an integer a 4. Let T be the complete rooted (∆ − 1)-ary tree of depth d (the root has depth 0 and the leaves have depth d, every non-leaf vertex has exactly ∆ − 1 children), where d 3a − 1. We aim to use Lemma 6. For Let a := 1 1+a/(d+1) · a, and note that a For each non-negative integer i, let n i be the number of vertices in C v whose depth is by i larger than the depth of v, so that w(

Subject to this inequality and to the constraints n
3 (as can be seen by a standard weight-shifting argument). It follows that giving the same lower bound on (T − X). By Lemma 6, we conclude that r(a) (∆ − 1) a −3 . Because this inequality holds for all d 3a − 1 and lim d→∞ a = a, the statement of the lemma follows.
Conversely, many interesting graph classes, including planar graphs, nearly match the lower bound provided by Theorem 16. We start by an argument for graphs with bounded treewidth. We use the following well-known fact [14, (2.6)]. G is a graph of treewidth less than k and Z  a subset of vertices of G, then G has a separation (D, B) of order at most k such that |Z \ D| 2 3 |Z| and |Z \ B| 2 3 |Z|. Iterating this splitting procedure, we obtain the following generalization.

Observation 17. Let k be an integer. If
Lemma 18. Let k, s and p be positive integers such that s 12k. If G is a graph of treewidth less than k and W a subset of vertices of G of order at most ps, then there exists a set C ⊆ V (G) and non-empty sets A 1 , . . . , A t ⊆ V (G) for some t < 6p such that . If no such element X exists, the procedure stops and we set t := i + 1, C := C i and {A 1 , . . . , A t } := A i . Assuming the construction stops, it is clear the conditions (ii), (iii) and (iv) hold. Since |C i+1 \ C i | k for i ∈ {0, . . . , t − 2}, it suffices to argue that the construction stops with t < 6p. Without loss of generality, we can assume that the construction does not stop in the first step, i.e., that |W | > s. If On the other hand, the electronic journal of combinatorics 27(4) (2020), #P4.9 By induction, we conclude that for i ∈ {0, . . . , t − 1}, we have Combining the inequalities, we obtain (i + 1)/3 < (p + i/6), and hence i < 6p − 2. Consequently, the construction stops with t < 6p.
Corollary 19. Let k, s and p be positive integers such that s 12k. If G is a graph of treewidth less than k and W a subset of vertices of G of order at most ps, then there exists a set C ⊆ V (G) \ W and a partition E 1 , . . . , E t Proof. Apply Lemma 18 and replace C by C \ W if necessary, so that C ∩ W = ∅. A tree partition (T, β) of a graph G consists of a tree T and a function β that to each vertex of T assigns a subset of vertices of G, such that  r(a). Consequently, (G − X i ) r(a), and thus X i ∈ G ↓r(a) . Note that X i ∩ X j = ∅ if i = j. Considering the uniform distribution on {X 0 , . . . , X a−1 }, which is (1/a)-thin, we conclude that G is fractionally -fragile at rate r.
For example, if ∆ 3 and T is a tree of maximum degree at most ∆, then we can root T and define β(v) := {v} for every v ∈ V (T ), thereby obtaining a rooted tree partition of T of depth-a order O((∆−1) a−2 ). It thus follows from Lemma 20 that trees of maximum degree at most ∆ are fractionally -fragile at rate O((∆ − 1) a−2 ), essentially matching the bound from Theorem 16. We now construct good tree partitions for graphs of bounded treewidth and maximum degree.
Lemma 21. Let a, b, k and ∆ be positive integers with ∆ 3 and a b. If G is a connected graph of treewidth less than k and maximum degree at most ∆, then G admits a rooted tree partition (T, β) of depth-a order at most 12k(∆ − 1) a+b−1 1 + 6 a/b . Proof. Let s := 12k. We construct the tree partition starting from the root and adding children as described below. It might be helpful to look at Figures 1 and 2 while reading the construction. To every vertex v of T will be associated, in addition to β(v), three sets, namely σ(v), γ(v) and κ(v). When considering a vertex v with parent z in T , two of these will already have been defined in one of the previous steps: the set σ(v) ⊆ V (G) \ β(z), which at the end of the construction will be equal to β(S) for the subtree S of T consisting of v and all its descendants, and the set γ(v) ⊆ β(z), which is of size at most (∆ − 1) b−1 s and such that in G, all neighbors of vertices from σ(v) are contained in σ(v) ∪ γ(v), and each vertex in γ(v) has at most ∆ − 1 neighbors in σ(v). The set κ(v), which must be contained in β(v), is defined when v is considered; its role becomes clear later.
Clearly, we can assume that G has at least three vertices. For the root r of T , we start the construction by letting β(r) consist of two adjacent vertices of G and κ(r) := ∅, adding a child u of r to T , and setting σ(u) := V (G) \ β(r) and γ(u) := β(r). Suppose now that the construction reaches a vertex v of T with parent z. Let W be the set of vertices in σ(v) that have a neighbor (in G) in γ(v).
If |W | (∆ − 1) b−1 s, we let β(v) := W and κ(v) := ∅; when σ(v) = W , then v is a leaf of T , otherwise, we add a child x to v and set σ(x) := σ(v) \ W and γ(x) := W (see Figure 1). Notice that if y ∈ σ(x), then all neighbors of y in G are contained in σ(v) ∪ γ(v), since σ(x) ⊆ σ(v). Moreover, because y / ∈ W we know that y has no neighbor in γ(v), and hence all neighbors of y are contained in σ(v) = σ(x) ∪ γ(x). Let us also point out that a vertex w ∈ γ(x) has less than ∆ neighbors in σ(x), because w has a neighbor in γ(v), which is disjoint from σ(v).
Let us consider the case that |W | > (∆ − 1) b−1 s; in this case, we say that v is a branching vertex.  Figure 2). Let us point out that |γ(u i )| s (∆ − 1) b−1 s by property (ii) from Corollary 19, and that vertices in σ(u i ) only have neighbors in σ(u i ) ∪ γ(u i ) by property (iv) from Corollary 19. Let us also remark that each vertex in γ(u i ) has at most ∆ − 1 neighbors in σ(u i ), since G has maximum degree at most ∆, each vertex in W has a neighbor in γ(v), and due to the property (iii) from Corollary 19 for vertices in C ∩ γ(u i ).
Note that since |γ(v)| s when v is the child of a branching vertex, and |β(v)| (∆ − 1)|γ(v)| when v is not a branching vertex, if x and y are two distinct branching vertices and x is an ancestor of y, then the depth of x is by at least b larger than the depth of y. Note also that every branching vertex x has less than 6(∆ − 1) b children and satisfies |κ(x)| < 6(∆ − 1) b k.
The described construction clearly results in a rooted tree partition of G. Let us now consider any subtree S of T of depth at most a − 2, with root w. The level of a branching vertex x of S is the number of branching vertices on the path from x to w, excluding x itself; hence, each branching vertex has level at most (a − 2)/b a/b . The number of branching vertices of S of level i is at most 6(∆ − 1) b i . If w is the root of T , then let B := β(w), otherwise let B be the set of vertices in σ(w) with a neighbor in γ(w); in either case, we have |B| (∆ − 1) b s. Note that each vertex in β(S) is either at distance at most a − 2 from B, or at distance at most a − 2 − b · i from a vertex in κ(x) for some branching vertex x ∈ V (S) of level i. Therefore, we have The result can be extended to planar graphs using their fractional tw-fragility; that is, by combining Corollaries 11 and 22.
Theorem 23. For every integer ∆ 3, the class of planar graphs with maximum degree at most ∆ is fractionally -fragile at rate r(a) Proof. Let G be a planar graph of maximum degree at most ∆. Consider an integer a > 256, let a := 2 √ a , let a := a + 1 and note that 1/a + 1/a < 1/a. Choose X ∈ G tw↓3a −3 at random from the (1/a )-thin probability distribution given by Corollary 11. Then G − X is a planar graph of treewidth less than 3a − 2 and maximum degree at most ∆.
. Consequently, choosing the random set Z in this way gives a probability distribution on

Treedepth
By Lemma 5, we cannot hope to extend the results on fractional -fragility to any class with unbounded maximum degree. The natural parameter to consider in graphs with unbounded maximum degree is the treedepth [11]: firstly, stars have treedepth at most 2, and secondly, a connected graph of maximum degree at most ∆ and treedepth at most d has at most ∆ d vertices, thus giving us about as good a relationship to as one may hope for in the case where the maximum degree is bounded from above. The treedepth td(G) of a graph G is the minimum integer d for which there exists a rooted tree T of depth at most d − 1 with vertex set V (G) such that every edge of G joins a vertex to one of its ancestors or descendants in T . Given Corollary 22 and the relationship between and td outlined above, one could perhaps hope that graphs of bounded treewidth are fractionally td-fragile at a linear rate. However, this is not the case. For the simplicity of presentation, we only give the counterargument for the case of graphs of treewidth two, but it can be naturally generalized to show that the class of all graphs of treewidth at most t cannot be fractionally td-fragile at rate better than Ω(a t ).
For a graph H and a non-negative integer d, let T d (H) be the graph inductively defined as follows: T 0 (H) is the graph consisting of a single vertex v, which we call the handle of T 0 (H).  Proof. We prove the statement by induction on the non-negative integer p. The case p = 0 being trivial, suppose that p 1. For x ∈ V (S), let S x be the subtree of S induced by x and all its descendants. We can assume that t d (V (S x )) < (2p + 1)t d (x) for every x ∈ V (S) \ {s}, as otherwise we can consider S x instead of S and combine the obtained minor with the path from x to s in S. In particular, for a child . By the induction hypothesis, each of S x 1 and S x 2 contains B p−1 as a minor rooted in x 1 and x 2 , respectively, which combine with s to form B p as a minor rooted in s.
Next, let us lift this result to T d (B d ). For a non-negative integer p, let us define q(p) := 10 √ p + 1. Let us remark that the function q is chosen so that q(p)−2 q(p− (q(p)−4)/8 ) holds for p 1.

Lemma 25. Let d and p be non-negative integers such that d q(p) − 1, let G := T d (B d ) and w := (t d ) d , let s be a vertex of G and let S be a connected induced subgraph of G contained in the jug of s and containing s. If w(V (S)) q(p)w(s), then S contains B p as a minor rooted in s.
Proof. We prove the statement by induction on the non-negative integer p. The case p = 0 being trivial, suppose that p 1. For x ∈ V (S), let S x be the intersection of S with the jug of x. We can assume that w(V (S x )) < q(p)w(x), as otherwise we can consider S x instead of S, find the required minor in S x , and combine it with a path from s to x.
Let T be the subgraph of G induced by the neighbors of s in the jug of s (note that T is a copy of B d ), and let N be the set of neighbors of s in G that belong to S. Notice that N ⊆ V (T ) since S is contained in the jug of s by assumptions. We have and hence Let r be the root of T and set X := (V (T ) \ N ) ∪ {r}. By (1) and the assumption d q(p) − 1, For x ∈ X, let T x be the subtree of the forest T [N ∪ {x}] induced by x and its descendants, and set a( (2) and (3), we have Since the left side of (4) is a weighted average of the values a(x) for x ∈ X, there exists x ∈ X such that a(x) > q(p)/4. Let T x be the smallest subtree of T x containing x and all vertices in of T x belongs to B. Set p := (q(p) − 4)/8 . Lemma 24 ensures that T x contains T p as a minor µ, which can be extended so that for every leaf u, the bag µ(u) contains a leaf y of T x ; this leaf in particular does not belong to B. Hence, w(V (S y )) (q(p) − 2)w(y), which is at least q(p − p )w(y) by the definition of q, since p 1. Consequently, the induction hypothesis implies that S y contains T p−p as a minor rooted in y. Adding these minors of T p−p for each leaf of T p , and replacing x by s in the root bag, we obtain T p in S as a minor rooted in s, as required.
We now use Lemma 6 to give the desired lower bound.
Theorem 26. Let r : N → R + 0 be a non-decreasing function. If all planar graphs of treewidth at most two are fractionally td-fragile at rate r, then r(a) = Ω(a 2 ). a and d such that d a 20. Let G := T d (B d ) and w := (t d ) d . Note that G is planar and has treewidth at most two. Suppose that X is a subset of V (G) such that w(X) w(V (G))/a. Let r be the handle of G and let X = X ∪ {r}; we have w(X ) w(X)

Proof. Consider two integers
and thus there exists x ∈ X such that w(V (S x )) a 2 w(x). Set p := a 2 /400 − 1 , so p is a non-negative integer. Because 10 √ p + 1 a 2 , we deduce from Lemma 25 that S x contains B p as a minor.
Note that B p has treedepth p + 1, that deleting a vertex decreases the treedepth by at most one, and that treedepth is minor-monotone [11].
Since this holds for every set X with w(X) w(V (G))/a, Lemma 6 implies that r(a) p = Ω(a 2 ).
Outerplanar graphs are planar and have treewidth two; however, the graphs T d (B d ) are not outerplanar if d 2. As we will see below, outerplanar graphs are actually fractionally td-fragile at a subquadratic rate. Nevertheless, even for outerplanar graphs the rate is not linear, as we now show. Let us start by showing that T d (P n ) has substantial treedepth, where P n is the n-vertex path. d 0, a 1 and n 2 a be integers. The graph T d (P n ) has treedepth at least ad + 1.

Lemma 27. Let
Proof. We prove the statement by induction on the non-negative integer d. The case d = 0 is trivial, and we thus assume that d 1. Set G := T d (P n ), let v be the handle of G, and let Q be the n-vertex path induced by the neighbors of v. For a subpath Q of Q, we define J Q to be the union of the vertex sets of every jug the handle of which is contained in Q . Suppose that R is a rooted tree witnessing the treedepth of T d (P n ). By finite induction we build a path u 0 . . . u a in R starting at the root u 0 of R and a decreasing sequence Q 0 ⊃ Q 1 ⊃ · · · ⊃ Q a of subpaths of Q such that the following invariants hold for each i ∈ {0, . . . , a}.
(iii) the subtree of R rooted at u i contains all vertices of J Q i .
We proceed by finite induction on i ∈ {0, . . . , a}. The path Q 0 is chosen arbitrarily among the subpaths of Q with 2 a vertices and u 0 is the root of R. For i ∈ {1, . . . , a}, the path Q i is selected as one of the halves of Q i−1 so that J Q i does not contain u i−1 . It follows that Q i satisfies (i) and (ii). By (iii), we know that the subtree of R rooted at u i−1 contains all vertices of J Q i−1 , and thus also all vertices of J Q i . Since G[J Q i ] is connected and u i−1 / ∈ J Q i , we can choose u i as the unique child of u i−1 in R such that the subtree of R rooted at u i contains all vertices of J Q i , so that (iii) is satisfied. This concludes the construction.
Now let x ∈ V (Q a ), let J x be the jug of x and let R x be the subtree of R rooted at u a . We know by (iii) that V (J x ) ⊆ V (R x ). Because J x is isomorphic to T d−1 (P n ), the induction hypothesis implies that R x has depth at least a(d − 1). Since u a has depth a in R, it follows that R has depth at least ad. Consequently, the treedepth of T d (P n ) is at least ad + 1.
We now give an argument analogous to that of Lemma 25. For the d-vertex path P d , let p d : V (P d ) → R + 0 be the mapping that assigns 1 to each vertex of P d . Proof. We prove the statement by induction on the non-negative integer p. The case p = 0 being trivial, we suppose that p 1. For x ∈ V (S), let S x be the intersection of S with the jug of x. We can assume that w(V (S x )) < (4b + 2p + 2)w(x), as otherwise we can consider S x instead of S, find the required minor in S x , and combine it with a path from s to x.
Let P be the subgraph of G induced by the neighbors of s in the jug of s. Note that P is a copy v 1 . . . v d of P d , and let N be the set of neighbors of s in G that are contained in S. Notice that N ⊆ V (P ) by hypothesis. We have < w(s) + (4b + 2p + 2)w(N ), the electronic journal of combinatorics 27(4) (2020), #P4.9 and thus Let B consist of the vertices x in N such that w(V (S x )) < (4b + 2p)w(x). Since we have Given induced by x and the vertices to the right of x. Observe that (V (P x )) x∈X is a partition of V (P ). Consequently, setting a(x) := (w(V (P x )) − w(V (P x ) ∩ B))/w(x), we deduce from (6) and (7) that Since the left side of (8) is a weighted average of the values a(x) for x ∈ X, there exists x ∈ X such that a(x) > (2b + p + 1)/2. Since all vertices of P have the same weight, we deduce that P x − x contains at least (2b + p + 1)/2 − 1 b vertices not belonging to B. For each such vertex y, we have w(V (S y )) (4b + 2p)w(y), and by the induction hypothesis, S y contains T p−1 (P b ) as a minor rooted in y. Since P x − x is a subpath of P [N ] and s is adjacent to every vertex in N , these minors along with s combine to form T p (P b ) as a minor rooted in s and contained in S, as required. Now are ready to give a lower bound for outerplanar graphs.
Theorem 29. Let r : N → R + 0 be a non-decreasing function. If all outerplanar graphs are fractionally td-fragile at rate r, then r(a) = Ω(a log a).
Proof. Consider two integer a and d such that d a 28. Set G := T d (P d ) and w := (p d ) d . Note that G is outerplanar. Suppose that X is a subset of V (G) such that w(X) w(V (G))/a. Let r be the handle of G and let X := X ∪ {r}; we have w(X ) w(X) + 1 = w(X) + w(V (G)) d+1 2 a w(V (G)). For x ∈ X , let J x be the jug of x and let S x be the component of J x − (X \ {x}) containing x. We have and thus there exists x ∈ X such that w(V (S x )) a 2 w(X ). Set c := (a − 4)/12 . We deduce from Lemma 28 that S x contains T c (P c ) as a minor.
As this holds for every set X with w(X) w(V (G))/a, Lemma 6 implies that r(a) c log 2 c = Ω(a log a).
Next, we will give a general upper bound for graphs with bounded treewidth. To this end, we need the following property of treedepth. H, H 1 , . . . , H t be induced subgraphs of a graph G such that We are now ready to give the following upper bound on the rate of td-fragility for graphs with bounded treewidth.

Lemma 30. Let
Theorem 31. For every non-negative integer t, the class of graphs with treewidth at most t is fractionally td-fragile at rate r(a) = 2 t(t+1)/2+1 a t .
Proof. We proceed by induction on the non-negative integer t. Graphs of treewidth 0 have no edges, and thus they have treedepth 1. Hence, suppose that t 1. Let a be a positive integer and let G be a graph of treewidth at most t, which we can assume to be connected and chordal by Observation 7 without loss of generality. Let us fix an elimination ordering of G, and let v be the first vertex in this ordering. For a non-negative integer d, let L d be the set of vertices of G at distance exactly d from v. For i ∈ {0, . . . , 2a − 1}, let X i := s 0 L i+s·2a , and choose X randomly from the uniform distribution on {X 0 , . . . , X 2a−1 }.
Consider a non-negative index j, and note that G[L j ] has treewidth at most t − 1: for j = 0 it is obvious, while for j 1 it follows from the fact that each vertex of L j has a neighbor in L j−1 preceding it in the elimination ordering by Observation 8, and hence in the restriction of the elimination ordering to G[L j ], each vertex is preceded by at most t − 1 of its neighbors. The induction hypothesis thus implies that for each j, we can choose a set If v ∈ V (G), then there exists a unique index j such that v ∈ L j , and hence Consequently, it suffices to show that Z ∈ G td↓2 t(t+1)/2+1 a t . As G[L j \ Y j ] has treedepth at most 2 (t−1)t/2+1 (2a) t−1 = 2 t(t+1)/2 a t−1 for each j, the conclusion follows by repeatedly applying Lemmas 9 and 30. Indeed, let i ∈ {0, . . . , 2a − 1} be the index such that X = X i . It suffices to bound the treedepth of the subgraph of G induced by (L i+1 \ Y i+1 ) ∪ · · · ∪ (L i+2a−1 \Y i+2a−1 ), and of that induced by ∪ i−1 j=0 L j \Y j in the border case -which is omitted as similar to what follows only with different index boundaries yielding fewer applications of the lemmas. To this end, for any j ∈ {1, . . . , , define H C to be the subgraph of G induced by the union of V (C) and the subset of vertices of H with a neighbor (in G) that belongs to V (C). Lemma 9 ensures that H ∩ H C is a clique, and Lemma 30 that the treedepth of H ∪ C H C is at most td(H) + 2 t(t+1)/2 a t−1 . Therefore, the conclusion follows by finite induction on j ∈ {1, . . . , 2a − 1}.
Let us remark that Theorem 31 implies that every fractionally tw-fragile class of graphs is also fractionally td-fragile; for example, this includes all proper minor-closed classes [3]. More precisely, if a graph G is fractionally tw-fragile at rate t, then it is also fractionally td-fragile at rate r(b) = 2 t(2b)(t(2b)+1)/2+1 (2b) t(2b) , as seen by first sampling a set X from a 1 2b -thin probability distribution on G tw↓t(2b) and then applying Theorem 31 with a = 2b to G − X.
As we mentioned before, the bound provided by Theorem 31 can be improved for the special case of outerplanar graphs. Firstly, we note that the following holds.
Observation 32. Suppose that G is an outerplanar graph, that K ⊆ V (G) induces a connected subgraph of G, and let H be a connected subgraph of G − K such that each vertex of H has a neighbor in K. Then H is a path.
Proof. If H is not a path, it either is a cycle or contains a vertex of degree at least three. Contracting K to a single vertex, and considering it along with H, we obtain either K 4 or K 2,3 as a minor of G, contradicting the assumption that G is outerplanar.
We can now modify the argument used to demonstrate Theorem 31. We use the fact that a path with n vertices has treedepth log 2 (n + 1) , see [11].
Proof. Let a be a positive integer. Let G be an outerplanar graph, which without loss of generality can be assumed to be connected. By triangulating the inner faces, we can assume that G is chordal. Let L 0 , L 1 , . . . and X 0 , . . . , X 2a−1 , and X be defined in the same way as in the proof of Theorem 31. From Lemma 9 and Observation 32, we infer that G[L j ] is a disjoint union of paths, for each non-negative integer j. Repeating the same layering argument in G[L j ], for each j, we can choose a set Y j ⊆ L j at random so that log 2 (2a) for every j, and letting Z := X ∪ Y 0 ∪ Y 1 ∪ · · · , we apply Lemmas 9 and 30 similarly as in the proof of Theorem 31 to infer that Z ∈ G td↓2a(1+ log 2 a ) , while Pr[v ∈ Z] 1/a for each v ∈ V (G).
Combining Theorem 31 with Corollary 11 yields that planar graphs are fractionally td-fragile at rate a O(a) . A much better bound can be obtained using Theorem 14. To this end, let us introduce another variation on Theorem 31.
Proof. Let a be a positive integer. Let G be a planar chordal graph, which without loss of generality can be assumed to be connected. Let L 0 , L 1 . . . and X 0 , . . . , X 2a−1 , and X be defined in the same way as in the proof of Theorem 31. For every j 1, Lemma 9 implies the neighborhood of every component of G[L j ] in L j−1 induces a connected subgraph of G, and since G is planar, G[L j ] is outerplanar. By Theorem 33, we can for each j choose a set Y j ⊆ L j at random so that Pr[v ∈ Y j ] 1 2a for each v ∈ V (L j ) and G[L j \ Y j ] has treedepth at most 4a(2 + log 2 a ). Consequently, letting Z := X ∪ Y 0 ∪ Y 1 ∪ · · · , Lemmas 9 and 30 imply that Z ∈ G td↓8a 2 (2+ log 2 a ) similarly as before, while Pr[v ∈ Z] 1/a for each v ∈ V (G).
Let us point out that the rate from Theorem 34 cannot be substantially improved: the graphs T d (T d (P d )) are planar, chordal, and combining the ideas of Theorems 26 and 29, one can show that they cannot be fractionally td-fragile at rate better than Ω(a 2 log a). We can now compose the results to obtain a bound for planar graphs.
Proof. Let a be a positive integer. Let G be a planar graph, which without loss of generality can be assumed to be connected. Let v be a vertex of G, and for d 0, let L d be the set of vertices of G at distance exactly d from v in G. For i ∈ {0, . . . , 2a − 1}, let X i := L i ∪ L i+2a ∪ · · · , and choose X from the uniform distribution on {X 0 , . . . , X 2a−1 }.
Consider any component H of G − X. This component contains only vertices at distance from v between i + 2aj + 1 and i + 2aj + 2a − 1 for some integer j. Let G be the graph obtained from G by deleting all vertices at distance at least i + 2aj + 2a from v and by contracting all vertices at distance at most max(i + 2aj, 0) from v to a single vertex x. Clearly, G is a minor of G, and thus G is planar. Moreover, every vertex of G is at distance at most 2a − 1 from x and H ⊆ G . Let G be a triangulation of G , and let P be a trigeodesic partition of V (G ) such that G /P is chordal, which exists by Theorem 14. By Theorem 34, we can choose a set Y H ⊆ V (G /P) at random such that td(G /P − Y H ) 32a 2 (3 + log 2 a) and Pr[z ∈ Y H ] 1 2a for every z ∈ V (G /P). We can naturally view Y H as a subset of P; with this in mind, let Y H := V (H) ∩ P ∈Y H P . Clearly, for every u ∈ V (H), we have Pr[u ∈ Y H ] 1 2a . Furthermore, note that since G has radius less than 2a, every geodesic path in G has less than 4a vertices, and since P is trigeodesic, if follows that |P | < 12a for every P ∈ P. Consequently, we can turn the tree T witnessing the treedepth of G /P − Y H into one for H − Y H by replacing each vertex P ∈ V (G /P − Y H ) in T by a path consisting of the vertices contained in P ∩ V (H). Therefore td(H − Y H ) < 12a td(G /P − Y H ) 384a 3 (3 + log 2 a ).
Letting Z be the union of X and the sets Y H for each component H of G − X, we conclude that td(G − Z) 384a 3 (3 + log 2 a ) and Pr[u ∈ Z] 1/a for each u ∈ V (G).
As we mentioned before, the planar graphs T d (T d (P d )) cannot be fractionally td-fragile at rate better than Ω(a 2 log a). We leave open the problem of determining the correct rate for planar graphs (between the bounds of Ω(a 2 log a) and O(a 3 log a) we obtained).

Algorithmic remarks
In this paper, we mostly focused on the structural aspects of fractional fragility. Nevertheless, all our upper bound results can be straightforwardly turned into polynomial-time algorithms to sample from the corresponding probability distribution. Moreover, an inspection of the proofs shows that the distributions have supports of bounded size. More precisely: • For a graph G of bounded treewidth and maximum degree, the support of the (1/a)-thin distribution on G ↓r(a) obtained in Corollary 22 has size at most a.
• For a planar graph G of bounded maximum degree, the support of the (1/a)-thin distribution on G ↓r(a) obtained in Theorem 23 has size at most (a + 1) 2 √ a .
• For a graph G of treewidth at most t, the support of the (1/a)-thin distribution on G td↓r(a) obtained in Theorem 31 has size at most f t (a), where f 0 (a) = 1 and f t (a) = 2af t−1 (2a). Note that to obtain this bound, we use the fact that the choices of the sets Y j do not have to be independent; consequently, we can use the same random choices for each j, obtaining only f t−1 (2a) choices for the random set Y 0 ∪ Y 1 ∪ · · · , rather than f t−1 (2a) the number of layers a naive analysis would suggest.
the electronic journal of combinatorics 27(4) (2020), #P4.9 • For an outerplanar graph G, the support of the (1/a)-thin distribution on G td↓r(a) obtained in Theorem 33 has size at most 4a 2 .
• For a planar graph G, the support of the (1/a)-thin distribution on G td↓r(a) obtained in Corollary 35 has size at most 512a 4 .
As noted after Observation 1, this implies that in the algorithmic applications, we obtain deterministic rather than randomized algorithms.