In absence of long chordless cycles, large tree-width becomes a local phenomenon

TL;DR

This paper proves that graphs with sufficiently large tree-width necessarily contain either large bipartite subgraphs or long chordless cycles, revealing a local-global structural property.

Contribution

It establishes a new relationship between large tree-width and the presence of specific subgraphs, specifically long chordless cycles or complete bipartite graphs.

Findings

Graphs with large tree-width contain long chordless cycles or large bipartite subgraphs.

The result links global tree-width to local subgraph structures.

Provides a structural characterization relevant for graph theory and algorithms.

Abstract

We prove that, for all $\ell$ and $s$, every graph of sufficiently large tree-width contains either a complete bipartite graph $K_{s,s}$ or a chordless cycle of length greater than $\ell$.

Equations

$$\forall k:$$

$$\forall \ell ,r\exists k:$$

$$\mathrm{tw}(\mathrm{H})\le |A|+\mathrm{tw}(\mathrm{H}-\mathrm{A})\le \mathrm{k}+(\mathrm{\ell }-2)(\mathrm{d}-2)<t.$$

$$|{\mathcal{Q}}^{xy}|>k-p_0-(m-2)\ge \left(\genfrac{}{}{0ex}{}{m}{2}\right)\frac{\ell }{2}.$$

$$\left(\genfrac{}{}{0ex}{}{|Y|-3}{\ell -3}\right)\le \frac{|Y{|}^{\ell -3}}{(\ell -3)!}$$

$$\frac{|Y{|}^{\ell }}{{\ell }^{\ell }}\le \left(\genfrac{}{}{0ex}{}{|Y|}{\ell }\right)\le f\frac{|Y{|}^{\ell -3}}{(\ell -3)!}.$$

$$d(H)\ge \frac{2f}{|H|}\ge \frac{4(\ell -3)!}{r{\ell }^{\ell }}|Y|\ge d.$$

Full text

1 Introduction

In an effort to make the statement in the title precise, let us call a graph parameter $P$ global if there is a constant $c$ such that for all $k$ and $r$ there exists a graph $G$ for which every subgraph $H$ of order at most $r$ satisfies $P(H)<c$, while $P(G)>k$. The intention here is that $P$ being small, even bounded by a constant, on subgraphs of bounded order does not provide a bound on $P(G)$.

Tree-width is a global parameter (we may take $c=2$), as is the chromatic number (with $c=3$). Indeed, it is a classic result of Erdős [6] that for all $k$ and $r$ there exists a graph of chromatic number $>k$ for which every subgraph on at most $r$ vertices is a forest.

It is well-known (see [4]) that the situation changes when we restrict ourselves to chordal graphs, graphs without chordless cycles of length $\geq 4$:

[TABLE]

Hence the only obstruction for a chordal graph to have small tree-width is the presence of a large clique. Since the chromatic number of a graph is at most its tree-width plus one ([4]), the same is true for the chromatic number. In particular, tree-width and chromatic number are local parameters for the class of chordal graphs.

In 1985, Gyárfás [8] made a famous conjecture which implies that chromatic number is a local parameter111Indeed, in terms of our earlier definition, (2) implies that given any integer $c$, there exists a $k$ such that every $\ell$-chordal graph of chromatic number $>k$ has a subgraph of order $\leq c$ and chromatic number $\geq c$. for the larger class of $\ell$-chordal graphs, those which have no chordless cycle of length $>\ell$:

[TABLE]

This conjecture remained unresolved for 30 years and was proved only recently by Chudnovsky, Scott and Seymour [3]. In view of (1), it is tempting to think that an analogue of (2) might hold with tree-width in place of chromatic number. Complete bipartite graphs, however, are examples of triangle-free 4-chordal graphs of large tree-width. Therefore a verbatim analogue of (2) is not possible and any graph whose presence we can hope to force by assuming $\ell$-chordality and large tree-width will be bipartite.

On the positive side, Bodlaender and Thilikos [2] showed that every star can be forced as a subgraph in $\ell$-chordal graphs by assuming large tree-width (see Section 3). However, since stars have tree-width 1, this does not establish locality of tree-width in the sense of our earlier definition. Our main result is that in fact any bipartite graph can be forced as a subgraph:

Theorem 1**.**

Let $\ell\geq 4$ be an integer and $F$ a graph. Then $F$ is bipartite if and only if there exists an integer $k$ such that every $\ell$-chordal graph of tree-width $\geq k$ contains $F$ as a subgraph.

This shows that tree-width is local for $\ell$-chordal graphs: Given any integer $c$, there exists an integer $k$ such that every $\ell$-chordal graph of tree-width $\geq k$ has a subgraph isomorphic to $K_{c,c}$, which has order $2c$ and tree-width $c$.

Theorem 1 also has an immediate application to an Erdős-Pósa type problem. Kim and Kwon [9] showed that chordless cycles of length $>3$ have the Erdős-Pósa property:

Theorem 2** ([9]).**

For every integer $k$ there exists an integer $m$ such that every graph $G$ either contains $k$ vertex-disjoint chordless cycles of length $>3$ or a set $X$ of at most $m$ vertices such that $G-X$ is chordal.

They also constructed, for every integer $\ell\geq 4$, a family of graphs showing that the analogue of Theorem 2 for chordless cycles of length $>\ell$ fails. We complement their negative result by proving that the Erdős-Pósa property does hold when restricting the host graphs to graphs not containing $K_{s,s}$ as a subgraph.

Corollary 3**.**

For all $\ell,s$ and $k$ there exists an integer $m$ such that every $K_{s,s}$-free graph $G$ either contains $k$ vertex-disjoint chordless cycles of length $>\ell$ or a set $X$ of at most $m$ vertices such that $G-X$ is $\ell$-chordal.

The paper is organised as follows. Section 2 contains some basic definitions. Theorem 1, our main result, is proved in Section 3. In Section 4 we formally introduce the Erdős-Pósa property, restate Corollary 3 in that language and give a proof thereof. Section 5 closes with some open problems.

2 Notation and definitions

All graphs considered here are finite and undirected and contain neither loops nor parallel edges. Our notation and terminology mostly follow that of [4].

For two graphs $G$ and $H$, we say that $G$ is $H$-free if $G$ does not contain a subgraph isomorphic to $H$. Given a tree $T$ and $s,t\in T$, we write $sTt$ for the unique $s$-$t$-path in $T$. Given a graph $G$ and a set $X$ of vertices of $G$, a path $P\subseteq G$ is an $X$-path if it contains at least one edge and meets $X$ precisely in its endvertices. A separation of $G$ is a tuple $(A,B)$ with $V=A\cup B$ such that there are no edges between $A\setminus B$ and $B\setminus A$. The order of $(A,B)$ is the number of vertices in $A\cap B$. We call the separation $(A,B)$ tight if for all $x,y\in A\cap B$, both $G[A]$ and $G[B]$ contain an $x$-$y$-path with no internal vertices in $A\cap B$.

Given an integer $k$, a set $X$ of at least $k$ vertices of $G$ is a $k$-block if it is inclusion-maximal with the property that for every separation $(A,B)$ of order $<\!k$, either $X\subseteq A$ or $X\subseteq B$. By Menger’s Theorem, $G$ then contains $k$ internally disjoint paths between any two non-adjacent vertices in $X$.

A tree-decomposition of $G$ is a pair $(T,\mathcal{V})$, where $T$ is a tree and $\mathcal{V}=(V_{t})_{t\in T}$ a family of sets of vertices of $G$ such that for every $v\in V(G)$, the set of $t\in T$ with $v\in V_{t}$ induces a non-empty subtree of $T$ and for every edge $vw\in E(G)$ there is a $t\in T$ with $v,w\in V_{t}$. If $(T,\mathcal{V})$ is a tree-decomposition of $G$, then every $st\in E(T)$ induces a separation $(G_{s}^{t},G_{t}^{s})$ of $G$, where $G_{x}^{y}$ is the union of $V_{u}$ for all $u\in T$ for which $y\notin uTx$. Note that $G_{s}^{t}\cap G_{t}^{s}=V_{s}\cap V_{t}$. We call $(T,\mathcal{V})$ tight if every separation induced by an edge of $T$ is tight.

Given $t\in T$, the torso at $t$ is the graph obtained from $G[V_{t}]$ by adding, for every neighbor $s$ of $t$, an edge between any two non-adjacent vertices in $V_{s}\cap V_{t}$.

Given graphs $G$ and $H$, a subdivision of $H$ in $G$ consists of an injective map $\eta:V(H)\to V(G)$ and a map $P$ which assigns to every edge $xy\in E(H)$ an $\eta(x)$-$\eta(y)$-path $P^{xy}\subseteq G$ so that the paths $(P^{xy}\colon xy\in E(H))$ are internally disjoint and no $P^{xy}$ has an internal vertex in $X:=\eta(V(H))$. The vertices in $X$ are called branchvertices. For an integer $r$, the subdivision is a $(\leq r)$-subdivision if every path $P^{xy}$ has length at most $r$. When $H$ is a complete graph, the map $\eta$ is irrelevant and we only keep track of the set $X$ of branchvertices and the family $(P^{xy}\colon x,y\in X)$.

3 Proof of Theorem 1

As observed in the introduction, the complete bipartite graphs $K_{s,s}$ show that no bound on the tree-width of $F$-free $\ell$-chordal graphs exists if $F$ is not bipartite. We now prove that $F$ being bipartite is sufficient. Since every bipartite graph is a subgraph of some $K_{s,s}$, it suffices to prove Theorem 1 for the case $F=K_{s,s}$.

Our proof is a cascade with three steps. First, we show that sufficiently large tree-width forces the presence of a $k$-block.

Lemma 4**.**

Let $\ell,k$ and $t\geq 2(\ell-2)(k-1)^{2}$ be positive integers. Then every $\ell$-chordal graph of tree-width $\geq t$ contains a $k$-block.

We then prove that the existence of a $k$-block yields a bounded-length subdivision of a complete graph.

Lemma 5**.**

Let $\ell,m$ and $k\geq 5m^{2}\ell/4$ be positive integers. Then every $\ell$-chordal graph that contains a $k$-block contains a $(\leq 2\ell-3)$-subdivision of $K_{m}$.

In the last step, we show that such a bounded-length subdivision gives rise to a copy of $K_{s,s}$.

Lemma 6**.**

For all integers $\ell$ and $s$ there exists a $q>0$ such that the following holds. Let $m,r$ be positive integers with $m\geq qr$. Then every $\ell$-chordal graph that contains a $(\leq r)$-subdivision of $K_{m}$ contains $K_{s,s}$ as a subgraph.

It is immediate that Theorem 1 follows once we have established these three lemmas.

3.1 Proof of Lemma 4

A trivial obstacle to our search for a copy of $K_{s,s}$ is the absence of vertices of high degree. Bodlaender and Thilikos [2] showed, however, that $\ell$-chordal graphs of bounded degree have bounded tree-width. Their exponential bound was later improved by Kosowski, Li, Nisse and Suchan [10] and by Seymour [17].

Theorem 7** ([17]).**

Let $\ell$ and $\Delta$ be positive integers and $G$ a graph. If $G$ is $\ell$-chordal and has no vertices of degree greater than $\Delta$, then the tree-width of $G$ is at most $(\ell-2)(\Delta-1)+1$.

By demanding large tree-width, we can therefore guarantee a large number of vertices of high degree. We now show that these are not all just scattered about the graph. It was shown by the author in [19] that either there is a $k$-block or there is a tree-decomposition which separates the set of vertices of high degree into small pieces. This also follows, without explicit bounds, from a far more general result of Dvořák [5].

Theorem 8** ([19]).**

Let $k\geq 3$ be a positive integer and $G$ a graph. If $G$ has no $k$-block, then there is a tight tree-decomposition $(T,\mathcal{V})$ of $G$ such that every torso has fewer than $k$ vertices of degree at least $2(k-1)(k-2)$.

In fact, tightness of the tree-decomposition is not explicit in [19, Theorem 1], but is established in the proof as Lemma 6.

Now let $\ell,k$ and $t\geq 2(\ell-2)(k-1)^{2}$ be positive integers. Let $G$ be an $\ell$-chordal graph with no $k$-block. For $k=2$, this means that $G$ is acyclic and therefore has tree-width 1. Suppose from now on that $k\geq 3$. We show that the tree-width of $G$ is less than $t$.

By Theorem 8, there is a tight tree-decomposition $(T,\mathcal{V})$ of $G$ such that every torso has fewer than $k$ vertices of degree at least $d:=2(k-1)(k-2)$. Let $t\in T$ arbitrary, let $N$ be the set of neighbors of $t$ in $T$ and let $H$ be the torso at $t$. We claim that $H$ is $\ell$-chordal.

Let $C\subseteq H$ be a chordless cycle. For every edge $xy\in E(C)\setminus E(G)$, there is some $s\in N$ with $x,y\in V_{s}\cap V_{t}$. Since $(T,\mathcal{V})$ is tight, there exists an $x$-$y$-path $P^{xy}$ in $G_{s}^{t}$ which meets $V_{t}$ only in its endpoints. Observe that for every $s\in N$, $C$ contains at most two vertices of $V_{s}$ and these are adjacent in $C$. Hence we can replace every edge $xy\in E(C)\setminus E(G)$ by $P^{xy}$ and obtain a chordless cycle $C^{\prime}$ of $G$ with $|C^{\prime}|\geq|C|$. Since $G$ is $\ell$-chordal, it follows that $|C|\leq\ell$. This proves our claim.

Now, let $A\subseteq V(H)$ be the set of all vertices of degree $\geq d$ in $H$. Then $H-A$ is $\ell$-chordal and has no vertices of degree $>d-1$. By Theorem 7, the tree-width of $H-A$ is at most $(\ell-2)(d-2)+1$. Therefore

[TABLE]

We have shown that every torso has tree-width $<\!t$. We can then take a tree-decomposition of width $<\!t$ of each torso and combine all these to a tree-decomposition of width $<\!t$ of $G$. ∎

3.2 Proof of Lemma 5

In general, the presence of a $k$-block does not guarantee the existence of any subdivision of $K_{m}$ for $m\geq 5$. For example, take a rectangular $k^{2}\times k$-grid, add $2(k+1)$ new vertices to the outer face and make each of these adjacent to $k$ consecutive vertices on the perimeter of the grid (see Figure 1). These new vertices are then a $k$-block in the resulting planar graph.

Our aim in this section is to show that for $\ell$-chordal graphs, sufficiently large blocks do indeed yield bounded-length subdivisions of complete graphs.

Let $\ell,m$ and $k\geq 5m^{2}\ell/4$ be positive integers. Let $G$ be an $\ell$-chordal graph and $X\subseteq V(G)$ a $k$-block of $G$. Let $L:=2\ell-3$. Assume for a contradiction that $G$ contained no $(\leq L)$-subdivision of $K_{m}$. Let $x,y\in X$ non-adjacent. Then $G$ contains a set $\mathcal{P}^{xy}$ of $k$ internally disjoint $x$-$y$-paths. Taking subpaths, if necessary, we may assume that each path in $\mathcal{P}^{xy}$ is induced. Let $p_{0}:=m+m^{2}(\ell-2)$.

Claim: Fewer than $p_{0}$ paths in $\mathcal{P}^{xy}$ have length $>\ell/2$.

Proof of Claim.

Let $\mathcal{P}_{0}$ be the set of all paths in $\mathcal{P}^{xy}$ of length $>\ell/2$ and $p:=|\mathcal{P}_{0}|$. Assume for a contradiction that $p\geq p_{0}$. Let $P,Q\in\mathcal{P}_{0}$. Then $P\cup Q$ is a cycle of length $>\ell$. Since $G$ is $\ell$-chordal, $P\cup Q$ has a chord. This chord must join an internal vertex of $P$ to an internal vertex of $Q$. Choose such vertices $v^{Q}_{P}\in P$ and $v^{P}_{Q}\in Q$ so that the cycle $D:=xPv^{Q}_{P}v^{P}_{Q}Qx$ has minimum length. Note that $D$ is an induced cycle and therefore has length at most $\ell$. In particular, the segment of $P$ joining $x$ to $v^{Q}_{P}$ has length at most $\ell-2$ and similarly for $Q$ and $v^{P}_{Q}$.

For $P\in\mathcal{P}_{0}$, let $P^{\prime}$ be a minimal subpath of $P$ containing every vertex $v_{P}^{Q}$, $Q\in\mathcal{P}_{0}\setminus\{P\}$. Then $\mathcal{P}:=\{P^{\prime}\colon P\in\mathcal{P}_{0}\}$ is a family of $p$ disjoint paths, each of length at most $\ell-3$, and $G$ contains an edge between any two of them. Fix an arbitrary $\mathcal{Q}\subseteq\mathcal{P}$ with $|\mathcal{Q}|=m$. Since $p\geq p_{0}$, every $Q\in\mathcal{Q}$ contains a vertex $u_{Q}$ which has neighbors on at least $m^{2}$ different paths in $\mathcal{P}\setminus\mathcal{Q}$.

Let $U:=\{u_{Q}\colon Q\in\mathcal{Q}\}$. We iteratively construct a $(\leq L)$-subdivision of $K_{m}$ with branchvertices in $U$. Let $t:=\binom{m}{2}$ and enumerate the pairs of vertices of $U$ arbitrarily as $e_{1},\ldots,e_{t}$. In the $j$-th step, we assume that we have constructed a family $\mathcal{R}^{j}=(R_{i})_{i<j}$ of internally disjoint $U$-paths of length at most $L$, so that $R_{i}$ joins the vertices of $e_{i}$ and meets at most two paths in $\mathcal{P}\setminus\mathcal{Q}$. We now find a suitable path $R_{j}$.

Let $Q^{1},Q^{2}\in\mathcal{Q}$ with $e_{j}=u_{Q^{1}}u_{Q^{2}}$. At most $2(j-1)<m^{2}$ paths in $\mathcal{P}\setminus\mathcal{Q}$ meet any of the paths in $\mathcal{R}^{j}$. Since $u_{Q^{1}}$ is adjacent to vertices on at least $m^{2}$ different paths in $\mathcal{P}\setminus\mathcal{Q}$, there is a $P^{1}\in\mathcal{P}\setminus\mathcal{Q}$ which is disjoint from every $R_{i}$, $i<j$, and contains a neighbor of $u_{Q^{1}}$. We similarly find a path $P^{2}\in\mathcal{P}\setminus\mathcal{Q}$ for $u_{Q^{2}}$. Since either $P^{1}=P^{2}$ or $G$ has an edge between $P^{1}$ and $P^{2}$, $P^{1}\cup P^{2}\cup\{u_{Q^{1}},u_{Q^{2}}\}$ induces a connected subgraph of $G$ and therefore contains a $u_{Q^{1}}$-$u_{Q^{2}}$-path $R_{j}$ of length at most $L$, which meets only two paths in $\mathcal{P}\setminus\mathcal{Q}$.

Proceeding like this, we find the desired subdivision of $K_{m}$ after $t$ steps. This contradiction finishes the proof of the claim. ∎

Let $Y\subseteq X$ with $|Y|=m$. For any two non-adjacent $x,y\in Y$, let $\mathcal{Q}^{xy}\subseteq\mathcal{P}^{xy}$ be the set of all $P\in\mathcal{P}^{xy}$ of length at most $\ell/2$ which have no internal vertices in $Y$. By the claim above, we have

[TABLE]

Pick one path $P\in\mathcal{Q}^{xy}$ for each pair of non-adjacent vertices $x,y\in Y$ in turn, disjoint from all previously chosen paths. Since $|Q^{xy}|\geq\binom{m}{2}\frac{\ell}{2}$ and each path only has at most $\ell/2-1$ internal vertices which future paths need to avoid, we can always find a suitable such path $P$. Together with all edges between adjacent vertices of $Y$, this yields a $(\leq\ell/2)$-subdivision of $K_{m}$ in $G$ with branchvertices in $Y$. ∎

We would like to point out that a modification of the above argument can be used to produce a $(\leq\ell/2)$-subdivision of $K_{m}$ if $k$ is significantly larger.

Indeed, suppose we find a family $\mathcal{P}$ of $p$ disjoint paths, each of length at most $\ell-3$, such that $G$ contains an edge between any two of them. Then the subgraph $H$ induced by $\bigcup_{P\in\mathcal{P}}V(P)$ has at most $(\ell-2)p$ vertices and at least $\binom{p}{2}$ edges. One can then use a classic result of Kövari, Sós and Turán [11] to show that $H$ contains a copy of $K_{m,m^{2}}$ if $p$ is sufficiently large. Since $K_{m,m^{2}}$ contains a $(\leq 2)$-subdivision of $K_{m}$, this establishes an upper bound on the number of paths of length $>\ell/2$ in any $\mathcal{P}^{xy}$. The rest of the proof remains the same.

3.3 Proof of Lemma 6

The combination of Lemma 4 and Lemma 5 already establishes that tree-width is a local parameter for $\ell$-chordal graphs. The purpose of Lemma 6 is merely to narrow the set of bounded-order obstructions down as far as possible. We will use the following theorem of Kühn and Osthus [13].

Theorem 9** ([13]).**

For every integer $s$ and every graph $H$ there exists a $d$ so that every graph with average degree at least $d$ either contains $K_{s,s}$ as a subgraph or contains an induced subdivision of $H$.

In fact, we only need the special case $H=C_{\ell+1}$. This special case has a simpler proof which can be found in Kühn’s PhD-thesis [12]. Fix an integer $d$ so that every $\ell$-chordal graph of average degree at least $d$ contains $K_{s,s}$ as a subgraph. We prove the assertion of Lemma 6 with $q:=d^{2}\frac{\ell^{\ell}}{4(\ell-3)!}$.

Let $m,r$ be positive integers with $m\geq qr$ and let $G$ be an $\ell$-chordal graph containing a $(\leq r$)-subdivision of $K_{m}$. Let $X$ be the set of branchvertices and $(P^{xy}\colon x,y\in X)$ the family of paths of the subdivision. Taking subpaths, if necessary, we may assume that every path is induced.

Assume for a contradiction that $G$ contained no copy of $K_{s,s}$. By Theorem 9, every subgraph of $G$ contains a vertex of degree $<\!d$. In particular, there is an independent set $Y\subseteq X$ with $|Y|\geq m/d$. Let $H$ be the subgraph of $G$ induced by $\bigcup_{x,y\in Y}V(P^{xy})$. Note that $|H|\leq r\binom{|Y|}{2}$.

Call an edge of $H$ red if it joins a vertex $x\in Y$ to an internal vertex of a path $P^{yz}$ with $x\notin\{y,z\}$. Call an edge of $H$ blue if it joins an internal vertex of a path $P^{wx}$ to an internal vertex of a path $P^{yz}$ with $\{w,x\}\neq\{y,z\}$. We will show that $H$ must contain many edges which are either red or blue, so that the average degree of $H$ is at least $d$.

Fix an arbitrary cycle $R$ with $V(R)=Y$. For any $Z\subseteq Y$ with $|Z|=\ell$, obtain the cycle $R_{Z}$ with $V(R_{Z})=Z$ by contracting every $Z$-path of $R$ to a single edge. We then get a cycle $C_{Z}\subseteq H$ by replacing every edge $xy\in R_{Z}$ with the path $P^{xy}$. Since each path $P^{xy}$ has length at least 2 and $H$ is $\ell$-chordal, the cycle $C_{Z}$ must have a chord. Since $Y$ is independent and every path $P^{xy}$ is induced, the chord must be a red or blue edge of $H$.

Consider a red edge $xv\in E(H)$ with $x\in Y$, $v\in P^{yz}$ and $x\notin\{y,z\}$. If this edge is a chord for a cycle $C_{Z}$, then $\{x,y,z\}\subseteq Z$. Hence it can only occur as a chord for at most

[TABLE]

choices of $Z$. Similarly, every blue edge $uv\in E(H)$ with $u\in P^{wx}$, $v\in P^{yz}$ and $\{w,x\}\neq\{y,z\}$ can only be a chord of $C_{Z}$ if $\{w,x,y,z\}\subseteq Z$. This also happens for at most

[TABLE]

choices of $Z$. Let $f$ be the number of edges of $H$ which are either red or blue. Since every $Z\subseteq Y$ with $|Z|=\ell$ gives rise to a chord, it follows that

[TABLE]

This shows that the average degree of $H$ is

[TABLE]

By Theorem 9, $H$ contains a copy of $K_{s,s}$. ∎

4 Erdős-Pósa for long chordless cycles

A classic theorem of Erdős and Pósa [7] asserts that for every integer $k$ there is an integer $r$ such that every graph either contains $k$ disjoint cycles or a set of at most $r$ vertices meeting every cycle. This result has been the starting point for an extensive line of research, see the survey by Raymond and Thilikos [15].

Let $\mathcal{F},\mathcal{G}$ be classes of graphs and $\leq$ a containment relation between graphs. We say that $\mathcal{F}$ has the Erdős-Pósa property for $\mathcal{G}$ with respect to $\leq$ if there exists a function $f$ such that for every $G\in\mathcal{G}$ and every integer $k$, either there are disjoint $Z_{1},\ldots,Z_{k}\subseteq V(G)$ such that for every $1\leq i\leq k$ there is an $F_{i}\in\mathcal{F}$ with $F_{i}\leq G[Z_{i}]$, or there is a $X\subseteq V(G)$ with $|X|\leq f(k)$ such that $F\not\leq G-X$ for every $F\in\mathcal{F}$. When $\mathcal{G}$ is the class of all graphs, we simply say that $\mathcal{F}$ has the Erdős-Pósa property with respect to $\leq$. We write $F\subseteq G$ if $F$ is isomorphic to a subgraph of $G$ and $F\subseteq_{\mathrm{i}}G$ if $F$ is isomorphic to an induced subgraph of $G$.

The theorem of Erdős and Pósa then asserts that the class of cycles has the Erdős-Pósa property with respect to $\subseteq$. This implies that cycles also have the Erdős-Pósa property with respect to $\subseteq_{\mathrm{i}}$. It is known that for every $\ell$, the class of cycles of length $>\ell$ has the Erdős-Pósa property with respect to $\subseteq$, see [18, 1, 14]. Recently, Kim and Kwon [9] proved that cycles of length $>3$ possess the Erdős-Pósa property with respect to $\subseteq_{\mathrm{i}}$:

Theorem 10** ([9]).**

There exists a constant $c$ such that for every integer $k$, every graph $G$ either contains $k$ vertex-disjoint chordless cycles of length $>3$ or a set $X$ of at most $ck^{2}\log k$ vertices such that $G-X$ is chordal.

In contrast, Kim and Kwon [9] showed that, for any given $\ell\geq 4$, cycles of length $>\ell$ do not have the Erdős-Pósa property with respect to $\subseteq_{\mathrm{i}}$. For any given $n$, they constructed a graph $G_{n}$ with no two disjoint chordless cycles of length $>\ell$, for which no set of fewer than $n$ vertices meets every chordless cycle of length $>\ell$ in $G_{n}$. This graph $G_{n}$ contains a copy of $K_{n,n}$. We show that this is essentially necessary:

Corollary 11**.**

For all integers $\ell$ and $s$, the class of cycles of length $>\ell$ has the Erdős-Pósa property for the class of $K_{s,s}$-free graphs with respect to $\subseteq_{\mathrm{i}}$.

This follows from Theorem 1 by a standard argument. Since the proof is quite short, we provide it for the sake of completeness. First, recall the following consequence of the Grid Minor Theorem of Robertson and Seymour [16].

Theorem 12** ([16]).**

For all positive integers $p$ and $q$ there exists an $r$ such that for every graph $G$ with tree-width $\geq r$, there are disjoint $Z_{1},\ldots,Z_{p}\subseteq V(G)$ such that $G[Z_{i}]$ has tree-width $\geq q$ for every $1\leq i\leq p$.

Proof of Corollary 11.

Let $k$ be an integer. By Theorem 1 there exists an integer $t$ such that every $\ell$-chordal graph with tree-width $\geq t$ contains $K_{s,s}$. By Theorem 12, there exists an $r$ such that every graph with tree-width $>r$ has $k$ vertex-disjoint subgraphs of tree-width $\geq t$.

Let $G$ be a $K_{s,s}$-free graph. We show that either $G$ contains $k$ disjoint chordless cycles of length $>\ell$ or there is a set of at most $r(k-1)$ vertices whose deletion leaves an $\ell$-chordal graph.

Suppose first that the tree-width of $G$ was greater than $r$. Let $Z_{1},\ldots,Z_{k}$ be disjoint sets of vertices such that $G[Z_{i}]$ has tree-width $\geq t$ for every $i$. Then, by Theorem 1, every $G[Z_{i}]$ must contain a chordless cycle of length $>\ell$, since $K_{s,s}\not\subseteq G[Z_{i}]$. Therefore $G$ contains $k$ disjoint chordless cycles of length $>\ell$.

Suppose now that $G$ had a tree-decomposition $(T,\mathcal{V})$ of width $<\!r$. For every chordless cycle $C\subseteq G$ of length $>\ell$, let $T_{C}\subseteq T$ be the subtree of all $t\in T$ with $V_{t}\cap V(C)\neq\emptyset$. If there are $k$ disjoint such subtrees $T_{C^{1}},\ldots,T_{C^{k}}$, then $C^{1},\ldots,C^{k}$ are also disjoint and we are done. Otherwise, there exists $S\subseteq V(T)$ with $|S|<k$ which meets every subtree $T_{C}$. Then $Z:=\bigcup_{s\in S}V_{s}$ meets every chordless cycle of length $>\ell$ in $G$ and $|Z|\leq r(k-1)$.

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5 Open problems

A large amount of research is dedicated to the study of $\chi$-boundedness of graph classes, introduced by Gyárfás [8]. Here, a class $\mathcal{G}$ of graphs is called $\chi$-bounded if there exists a function $f$ so that for every integer $k$ and $G\in\mathcal{G}$, either $G$ contains a clique on $k+1$ vertices or $G$ is $f(k)$-colourable. This is a strengthening of the statement that chromatic number is a local parameter for $\mathcal{G}$, with cliques being the only bounded-order subgraphs to look for.

As we have seen, cliques are not the only reasonable local obstruction to having small tree-width. Nontheless, we may still ask

For which classes of graphs is tree-width a local parameter? 2. 2.

What kind of bounded-order subgraphs can we force on these classes? 3. 3.

For which classes can we force large cliques by assuming large tree-width?

We have seen in Section 4 that long chordless cycles have the Erdős-Pósa property for the class of $K_{s,s}$-free graphs. For which other classes is this true? Kim and Kwon [9] raised this question for the class of graphs without chordless cycles of length four.