On the block number of graphs

TL;DR

This paper investigates the structure of graphs based on their block number, providing a decomposition theorem for graphs without large blocks, and establishing bounds on block numbers in minor-closed classes and relation to tree-width.

Contribution

It introduces a structure theorem for graphs without a (k+1)-block, linking block number to tree-decomposition and degree constraints, and bounds block numbers in minor-closed classes.

Findings

Graphs without (k+1)-blocks have a specific tree-decomposition structure.

Block number in minor-closed classes is bounded by a cube root of the graph size.

High tree-width guarantees the existence of a minor with a k-block.

Abstract

A $k$-block in a graph $G$ is a maximal set of at least $k$ vertices no two of which can be separated in $G$ by deleting fewer than $k$ vertices. The block number $\beta(G)$ of $G$ is the maximum integer $k$ for which $G$ contains a $k$-block. We prove a structure theorem for graphs without a $(k+1)$-block, showing that every such graph has a tree-decomposition in which every torso has at most $k$ vertices of degree $2k^2$ or greater. This yields a qualitative duality, since every graph that admits such a decomposition has block number at most $2k^2$. We also study $k$-blocks in graphs from classes of graphs $\mathcal{G}$ that exclude some fixed graph as a topological minor, and prove that every $G \in \mathcal{G}$ satisfies $\beta(G) \leq c\sqrt[3]{|G|}$ for some constant $c = c( \mathcal{G})$. Moreover, we show that every graph of tree-width at least $2k^2$ has a minor containing a $k$-block. This bound is best possible up to a multiplicative constant.

Equations

$$\lfloor \frac{\mathrm{ad}{\mathrm{m}}_{\infty }(\mathrm{G})+1}{2}\rfloor \le \beta (G)\le \mathrm{ad}{\mathrm{m}}_{\infty }(\mathrm{G}).$$

$$\underset{H\prec G}{\max }\beta (H)\le 1+\sqrt{2e(G)}.$$

$$|A|+\sum _{s\in S}|B_s|\ge |A|+|B|\ge (m-1)(k-2).$$

$$|\mathcal{Q}|=1+|A|+|S|\ge 1+\frac{|A|+(k-2)|S|}{k-2}\ge m.$$

$$\text{If }\bigcup {\mathcal{Q}}_x^{\prime }\cap \bigcup {\mathcal{Q}}_y^{\prime }\ne \varnothing ,\text{ then }xy\in E(H).$$

$$\text{If }x\in X\text{ stops some }{Q}^{\prime }\in {\mathcal{Q}}_y^{\prime },\text{ then }xy\in E(H).$$

$$e(J)\le d|J|\le d(|X|+(2r-1)|\mathcal{P}|)\le 2r{d}^2|X|.$$

$$\sum _{y\in Y}{q}_y\le e(J)\le 2r{d}^2|X|.$$

$$|G|$$

$$\ge r|X|(\alpha k-2r{d}^2).$$

$$\beta (G)\le \mathrm{ad}{\mathrm{m}}_{\infty }(\mathrm{G}).$$

$$1+\underset{H\subseteq G}{\max }\delta (H)=\mathrm{ad}{\mathrm{m}}_1(\mathrm{G})\le \mathrm{ad}{\mathrm{m}}_2(\mathrm{G})\le \dots \le \mathrm{ad}{\mathrm{m}}_{|G|}(\mathrm{G})=\mathrm{ad}{\mathrm{m}}_{\infty }(\mathrm{G}).$$

$$\mathrm{ad}{\mathrm{m}}_{\mathrm{r}}(\mathrm{G})>\mathrm{ad}{\mathrm{m}}_{\infty }(\mathrm{G})-\frac{|G|}{\mathrm{r}+1},$$

$$F(n,r):=\max \{\mathrm{ad}{\mathrm{m}}_{\mathrm{r}}(\mathrm{G}):\mathrm{G}\in \mathcal{G},|G|=\mathrm{n}\}.$$

Full text

We prove a structure theorem for graphs without a $(k+1)$-block, showing that every such graph has a tree-decomposition in which every torso has at most $k$ vertices of degree $2k^{2}$ or greater. This yields a qualitative duality, since every graph that admits such a decomposition has block number at most $2k^{2}$.

We also study $k$-blocks in graphs from classes of graphs $\mathcal{G}$ that exclude some fixed graph as a topological minor, and prove that every $G\in\mathcal{G}$ satisfies $\beta(G)\leq c\sqrt[3]{|G|}$ for some constant $c=c(\mathcal{G})$.

Moreover, we show that every graph of tree-width at least $2k^{2}$ has a minor containing a $k$-block. This bound is best possible up to a multiplicative constant.

1 Introduction

Given $k\in\mathbb{N}$, a set $X$ of at least $k$ vertices of a graph $G$ is $(<\!k)$-inseparable if no two vertices in $X$ can be separated in $G$ by deleting fewer than $k$ vertices. A maximal such set is a $k$-block and can be thought of as a highly connected part of the graph, although it may draw its connectivity from the ambient graph $G$ rather than just the subgraph induced by $X$ itself. The maximum integer $k$ for which $G$ contains a $k$-block is the block number of $G$, denoted by $\beta(G)$.

The notion of $k$-blocks is a successful concept in the theory of graph-decompositions. Carmesin, Diestel, Hundertmark and Stein [7] showed that $k$-blocks provide a natural model of a “highly connected substructure” into which a graph can be decomposed in a tree-like manner. This was further refined by Carmesin, Diestel, Hamann and Hundertmark [5, 6] and by Carmesin and Gollin [8]. Following a question raised in [7], the study of graphs which do not contain $k$-blocks was initiated by Carmesin, Diestel, Hamann and Hundertmark in [4], with a focus on degree-conditions. Here, our emphasis lies on the structure of these graphs and we relate the block number to other width-parameters for graphs.

Dualities between the occurrence of some highly connected substructure and a tree-like structure of the whole graph, such as between blockages and path-decompositions [3], brambles and tree-decompositions [22] or tangles and branch-decompositions [20], are of particular interest in structural graph theory.

A unified framework for duality theorems for width-parameters in graphs and matroids was developed by Diestel and Oum [12, 13]. Based on this framework, Diestel, Eberenz and Erde [11] proved a duality theorem for $k$-blocks and described a class $\mathcal{T}_{k}$ of tree-decompositions such that a graph has no $k$-block if and only if it has a tree-decomposition in $\mathcal{T}_{k}$. The only downside is that $\mathcal{T}_{k}$ is given rather abstractly and thus seems difficult to work with.

Here, we give a simpler class of tree-decompositions that still acts as an obstruction to the existence of a $k$-block, at the expense of a precise duality: We obtain a qualitative duality theorem with a numerical trade-off.

Theorem 1**.**

Let $G$ be a graph and $k\geq 2$ an integer.

1. (i)

If $G$ has no $(k+1)$-block, then $G$ has a tree-decomposition in which every torso has at most $k$ vertices of degree at least $2k(k-1)$. Moreover, there is such a tree-decomposition of adhesion less than $k$. 2. (ii)

If $G$ has a tree-decomposition in which every torso has at most $k$ vertices of degree at least $k$, then $G$ has no $(k+1)$-block.

This yields a qualitative duality: Every graph either has a $(k+1)$-block or a tree-decomposition that demonstrates that it has no $2k^{2}$-block.

We also study the block number of graphs in classes of graphs that do not contain some fixed graph as a topological minor. Dvořák [15] implicitly characterized those classes $\mathcal{G}$ for which there exists an upper bound on the block number of graphs in $\mathcal{G}$. We shall make this characterization explicit in Section 4.

The absence of an absolute bound does not have to be the end of the story, however. For instance, while the tree-width of planar graphs cannot be bounded by a constant, the seminal Planar Separator Theorem of Lipton and Tarjan [18] implies that $n$-vertex planar graphs have tree-width at most $c\sqrt{n}$ for some constant $c>0$. We prove a bound on the block number in the same spirit. Note that the Planar Separator Theorem can be extended to arbitrary minor-closed classes of graphs, as shown by Alon, Seymour and Thomas [1], but not to classes excluding a topological minor.

Theorem 2**.**

Let $\mathcal{G}$ be a class of graphs excluding some fixed graph as a topological minor. There exists a constant $c=c(\mathcal{G})$ such that every $G\in\mathcal{G}$ satisfies $\beta(G)\leq c\sqrt[3]{|G|}$.

In fact, our proof of Theorem 2 works with a slightly weaker notion of a highly connected substructure which we call a $k$-fan-set: a set $X\subseteq V(G)$ for which every $x\in X$ has a set of $k$ otherwise disjoint paths to $X$. The maximum $k$ for which $G$ contains a $k$-fan-set is called the $\infty$-admissibility of $G$ (sometimes also $\infty$-degeneracy [19]), denoted by $\rm adm_{\infty}(G)$. It turns out that $k$-blocks and $k$-fan-sets are essentially interchangeable concepts.

Theorem 3**.**

For every graph $G$

[TABLE]

We then study the relation between block number and tree-width. It is easy to see that $\beta(G)\leq\rm tw(G)+1$, so the existence of a $k$-block forces large tree-width. However, a graph can have arbitrarily large tree-width and yet have no 5-block: $k\times k$-grids are such graphs. Since tree-width does not increase when taking minors, the tree-width of $G$ (plus one) is even an upper bound for the block number of every minor of $G$. We can prove a converse to this statement, namely that a graph with large tree-width must have a minor with large block number.

Theorem 4**.**

Let $k\geq 1$ be an integer and $G$ a graph. If $\rm tw(G)\geq 2k^{2}-2$, then some minor of $G$ contains a $k$-block. This bound is optimal up to a constant factor.

This paper is organized as follows. Section 2 contains a brief account of definitions, basic facts and terminology used in the rest of the paper. Theorem 1 will be proven in Section 3. In Section 4 we prove a strong form of Theorem 2 about $k$-blocks in classes of graphs excluding a topological minor. Theorem 3 will be proven in Section 5. A more precise version of Theorem 4, relating tree-width to the occurrence of $k$-blocks in a minor, will be proven in Section 6. Section 7 contains some remarks on how our results fit into and extend the existing body of research as well as some open problems.

2 Preliminaries

All graphs considered here are finite and undirected, contain neither loops nor parallel edges and will be written as $G=(V,E)$. Our notation and terminology mostly follow that of [10]. Any graph-theoretic terms not defined here are explained there.

For $X\subseteq V(G)$, an $X$-path is a path of length at least one which meets $X$ precisely in its endvertices. For a vertex $v\in V$ and integer $k$, a $k$-fan from $v$ is a collection $\mathcal{Q}$ of $k$ paths which all have $v$ as a common starting vertex and are otherwise disjoint. It is a $k$-fan to some $U\subseteq V$ if the end-vertex of every path in $\mathcal{Q}$ lies in $U$. We explicitly allow a fan to contain the trivial path consisting only of the vertex $v$ itself. The end-vertex of this path is $v$. A set $X\subseteq V$ is a $k$-fan-set if from every $x\in X$ there exists a $k$-fan to $X$. The $\infty$-admissibility $\rm adm_{\infty}(G)$ is the maximum $k$ for which $G$ contains a $k$-fan-set.

Lemma 5**.**

Let $G$ be a graph and $X\subseteq V(G)$ a $(<\!k)$-inseparable set of vertices for some $k\in\mathbb{N}$. Then $X$ is a $k$-fan-set.

Proof.

Suppose there was an $x\in X$ with no $(k-1)$-fan from $x$ to $X\setminus\{x\}$. By Menger’s Theorem there is a set $S\subseteq V\setminus\{x\}$ with $|S|<k-1$ separating $x$ from $X\setminus(S\cup\{x\})$. Since $|X|\geq k$, there must be some $y\in X\setminus(S\cup\{x\})$. Since $X$ is $(<\!k)$-inseparable, $S$ cannot separate $x$ and $y$, a contradiction. ∎

The converse is not true: If $G$ is a disjoint union of cliques of order $k$, then $V(G)$ is a $k$-fan-set, but not even a 1-block.

If $X\subseteq V$ is $(<\!k)$-inseparable, then every $x\in X$ has degree at least $k-1$ in $G$. Therefore $G$ must have at least $k(k-1)/2$ edges. Since any minor of $G$ has at most $e(G)$ edges, it follows that

[TABLE]

If $T$ is a tree and $s,t\in V(T)$, we denote by $sTt$ the unique path in $T$ from $s$ to $t$. Recall that a tree-decomposition of $G$ is a pair $(T,\mathcal{V})$ of a tree $T$ and a family $\mathcal{V}=(V_{t})_{t\in T}$ of vertex sets $V_{t}\subseteq V(G)$, one for every node of $T$, such that:

1. (T1)

$V(G)=\bigcup_{t\in T}V_{t}$, 2. (T2)

for every edge $uv\in E(G)$ there exists a $t\in T$ with $\{u,v\}\subseteq V_{t}$, 3. (T3)

$V_{t_{1}}\cap V_{t_{3}}\subseteq V_{t_{2}}$ whenever $t_{2}\in t_{1}Tt_{3}$.

The sets $V_{t}$, $t\in T$, in a tree-decomposition are its parts, while the sets $V_{s}\cap V_{t}$, $st\in E(T)$, are its adhesion-sets. For $t\in T$ the torso of $t$ is the graph obtained from $G[V_{t}]$ by adding edges between any two vertices of $V_{t}$ that lie in a common adhesion-set.

The adhesion of $(T,\mathcal{V})$ is the maximum size of an adhesion-set. The width of $(T,\mathcal{V})$ is $\max_{t\in T}(|V_{t}|-1)$ and the tree-width $\rm tw(G)$ of $G$ is the minimum width of any of its tree-decompositions.

3 The structure of graphs without $k$-blocks

Perhaps the most trivial reason a graph $G$ can fail to contain a $(k+1)$-block is if $G$ has at most $k$ vertices of degree at least $k$. These graphs can be used as building blocks for graphs of block number at most $k$

Proof of Theorem 1 (ii).

Let $(T,\mathcal{V})$ be a tree-decomposition in which every torso has at most $k$ vertices of degree at least $k$. Assume that $G$ contained a $(k+1)$-block $X$. Every adhesion-set $V_{s}\cap V_{t}$, $st\in E(T)$, is a clique in the torso of $t$, so $|V_{s}\cap V_{t}|\leq k$ by assumption on the degrees.

Since $X$ is a $(k+1)$-block, it follows from a standard technique that there is a $t\in T$ with $X\subseteq V_{t}$, see [10, Lemma 12.3.4]. We will show that every vertex of $X$ has degree at least $k$ in the torso of $t$, which is a contradiction.

Let $x\in X$ arbitrary and let $A\subseteq V_{t}$ be the set of all neighbors of $x$ in the torso of $t$. If $|A|\geq k$, we are done. Otherwise, let $y\in X\setminus(A\cup\{x\})$. In particular, $x$ and $y$ are non-adjacent in $G$, so by Menger’s Theorem there is a set $\mathcal{P}$ of $k+1$ internally disjoint $x$-$y$-paths in $G$. Since every $P\in\mathcal{P}$ has both end-vertices in $V_{t}$, it has a vertex $z_{P}\in V(P)\cap V_{t}\setminus\{x\}$ which lies closest to $x$ along $P$. Then $x$ and this vertex $z_{P}$ must either be adjacent or lie in a common adhesion-set $V_{s}\cap V_{t}$. Hence $z_{P}\in A$ and, in particular, $z_{P}\neq y$. As the paths in $\mathcal{P}$ are internally disjoint, all these vertices $z_{P}$ are distinct. Thus the degree of $x$ in the torso of $t$ is at least $k+1$. ∎

The converse, decomposing a graph with no $(k+1)$-block into graphs of almost bounded degree, is more intricate.

The fatness of a tree-decomposition $(T,\mathcal{V})$ of an $n$-vertex graph $G$ is the $(n+1)$-tuple $(a_{0},\ldots,a_{n})$ where $a_{i}$ denotes the number of parts of $(T,\mathcal{V})$ of size $n-i$. If $(T,\mathcal{V})$ has lexicographically minimum fatness among all tree-decompositions of adhesion less than $k$, we call $(T,\mathcal{V})$ $k$-atomic. We are going to show that the high-degree vertices of a torso of a $k$-atomic tree-decomposition cannot be separated by deleting fewer than $k$ vertices.

Every edge $t_{1}t_{2}\in E(T)$ yields a separation of $G$ as follows. Let $T_{1},T_{2}$ be the two components of $T-t_{1}t_{2}$, where $t_{1}\in T_{1}$, and let $G_{i}:=\bigcup_{t\in T_{i}}V_{t}$ for $i\in\{1,2\}$. Then $(G_{1},G_{2})$ is a separation of $G$ with separator $X:=G_{1}\cap G_{2}=V_{t_{1}}\cap V_{t_{2}}$.

Lemma 6**.**

Let $(T,\mathcal{V})$ a $k$-atomic tree-decomposition of $G$. For any $t_{1}t_{2}\in E(T)$, there is a component $C$ of $G_{1}-X$ such that every $x\in X$ has a neighbor in $C$.

Proof.

Suppose this was not the case. Let $C_{1},\ldots,C_{m}$ be the components of $G_{1}-X$. Obtain the tree $T^{\prime}$ from the disjoint union of $m$ copies $T_{1}^{1},\ldots,T_{1}^{m}$ of $T_{1}$, where each $t\in T_{1}$ corresponds to $m$ vertices $t^{i}\in T_{1}^{i}$ for $i\in[m]$, and one copy of $T_{2}$ by joining $t_{2}$ to every $t_{1}^{i}$, $i\in[m]$. For $t\in T_{2}$ let $V_{t}^{\prime}=V_{t}$, for $t\in T_{1}$ let $V_{t^{i}}^{\prime}=V_{t}\cap(C_{i}\cup N(C_{i}))$. Observe that the adhesion of $(T^{\prime},\mathcal{V}^{\prime})$ is less than $k$. Let the fatness of $(T,\mathcal{V})$ be $a=(a_{i})_{i}$ and let the fatness of $(T^{\prime},\mathcal{V}^{\prime})$ be $a^{\prime}=(a^{\prime}_{i})_{i}$.

Clearly $|V_{t^{i}}^{\prime}|\leq|V_{t}|$ for every $t\in T_{1}$. If $|V_{t^{i}}^{\prime}|=|V_{t}|$ for some $i\in[m]$, then $V_{t}\subseteq C_{i}\cup N(C_{i})$ and for all $j\neq i$ we have $|V_{t^{j}}^{\prime}|\leq|N(C_{i})|<|X|$. Choose $t\in T_{1}$ with $r:=|G|-|V_{t}|$ minimum under the condition that there is no $i$ with $|V_{t^{i}}|=|V_{t}|$. Since $N(C_{i})\subsetneq X$ for every $i\in[m]$, the node $t_{1}$ satisfies this condition. Thus $r\leq|G|-|V_{t_{1}}|\leq|G|-|X|$. Then $a_{s}=a^{\prime}_{s}$ for all $s<r$ and $a_{r}>a^{\prime}_{r}$, so that $a^{\prime}$ is lexicographically smaller than $a$, a contradiction. ∎

This lemma helps us use the assumption of large degree in a torso.

Lemma 7**.**

Let $(T,\mathcal{V})$ a $k$-atomic tree-decomposition of $G$ for $k\geq 3$. Let $m\geq 1$ be an integer, $t\in T$ and $x\in V_{t}$. If $x$ has degree at least $(m-1)(k-2)$ in the torso of $t$, then there exists an $m$-fan from $x$ to $V_{t}$.

Proof.

Let $A$ be the set of vertices of $V_{t}$ which are adjacent to $x$ in $G$ and let $B$ be the set of vertices that are adjacent to $x$ in the torso of $t$, but not in $G$. Let $\mathcal{Q}_{A}$ be the fan consisting of the trivial path $\{x\}$ and single edges to each $a\in A$. We now construct a fan $\mathcal{Q}_{B}$ from $x$ to $B$ consisting of $V_{t}$-paths.

For every $b\in B$ there is an edge $st\in E(T)$ with $\{b,x\}\subseteq V_{s}\cap V_{t}$. Let $R$ be the set of all neighbors $s$ of $t$ in $T$ with $x\in V_{s}$. Let $S\subseteq R$ minimal such that $B\subseteq\bigcup_{s\in S}V_{s}$. For $s\in S$, let $B_{s}:=B\cap V_{s}$. By minimality of $S$, every $B_{s}$ contains some vertex $b_{s}\notin\bigcup_{s^{\prime}\neq s}B_{s^{\prime}}$.

Let $T_{s}$ be the component of $T-st$ containing $s$ and $G_{s}:=\bigcup_{r\in T_{s}}V_{r}$. By Lemma 6, there is a component of $G_{s}-V_{t}$ that contains neighbors of both $x$ and $b_{s}$. We therefore find a path $P_{s}$ from $x$ to $b_{s}$ in $G_{s}$ that meets $V_{t}$ only in its endpoints. The set $\mathcal{Q}_{B}:=\{P_{s}\colon s\in S\}$ is an $|S|$-fan from $x$ to $B$ in which every path is internally disjoint from $V_{t}$. Thus $\mathcal{Q}:=\mathcal{Q}_{A}\cup\mathcal{Q}_{B}$ is a fan from $x$ to $V_{t}$.

It remains to show $|\mathcal{Q}|\geq m$. As $B\subseteq\bigcup_{s}B_{s}$, we have

[TABLE]

Note that $x\in V_{s}\cap V_{t}$ for every $s\in S$ and $B_{s}\subseteq V_{s}\cap V_{t}\setminus\{x\}$. Since $(T,\mathcal{V})$ has adhesion less than $k$, it follows that $|B_{s}|\leq k-2$. Therefore

[TABLE]

∎

A tree-decomposition is $k$-lean if it has adhesion less than $k$ and for any $s,t\in T$, not necessarily distinct, and any $A\subseteq V_{s}$, $B\subseteq V_{t}$ with $|A|=|B|\leq k$ either there is a set of $|A|$ disjoint $A$-$B$-paths in $G$ or there is an edge $uw\in E(sTt)$ with $|V_{u}\cap V_{w}|<|A|$. As observed in [4], the short proof of Thomas’ theorem [23] given in [2] in fact shows the following.

Theorem 8** ([2]).**

Every $k$-atomic tree-decomposition is $k$-lean.

Lemma 9**.**

Let $(T,\mathcal{V})$ be a $k$-lean tree-decomposition of $G$, $t\in T$ and $u,v\in V(G)$. If from both $u$ and $v$ there are $(2k-1)$-fans to $V_{t}$, then $u$ and $v$ cannot be separated by deleting fewer than $k$ vertices.

Proof.

Suppose there was some $S\subseteq V(G)\setminus\{u,v\}$, $|S|<k$, separating $u$ and $v$. We find a set of $k$ paths of the fan from $u$ to $V_{t}$ which are disjoint from $S$ and let $R_{u}\subseteq V_{t}$ be their endvertices. Note that all vertices in $R_{u}$ lie in the component of $G-S$ containing $u$. Define $R_{v}\subseteq V_{t}$ similarly for $v$.

Since $(T,\mathcal{V})$ is $k$-lean, we find $k$ vertex-disjoint paths from $R_{u}$ to $R_{v}$. All of these paths must pass through $S$, a contradiction. ∎

We now combine all this to complete the proof of Theorem 1.

Proof of Theorem 1 (i).

Let $(T,\mathcal{V})$ be a $(k+1)$-atomic tree-decomposition of $G$. For $t\in T$ let $X_{t}\subseteq V_{t}$ be the set of vertices of degree at least $2k(k-1)$ in the torso of $t$. By Lemma 7, every $x\in X_{t}$ has a $(2k+1)$-fan to $V_{t}$. By Lemma 9, no two vertices of $X_{t}$ can be separated by deleting fewer than $k+1$ vertices. Since $G$ has no $(k+1)$-block, it follows that $|X_{t}|\leq k$. ∎

4 Excluded topological minors and $k$-blocks

When considering $k$-blocks, the topological minor relation is more natural than the ordinary minor relation. For example, it is easy to see that a $(k+1)$-block in a graph $H$ yields a $(<\!k)$-inseparable set in any graph containing $H$ as a topological minor. No such statement is true when considering minors: It is easy to construct a triangle-free graph $G$ of maximum degree 3 that contains the complete graph of order $k$ as a minor. This graph $G$ has no 4-block.

In this section we study the block number of graphs from classes of graphs $\mathcal{G}$ that exclude some fixed graph as a topological minor. Examples of such classes include graphs of bounded genus, bounded tree-width or bounded degree. In general, there exists no upper bound on the block number of graphs in $\mathcal{G}$. In fact, we can explicitly describe a planar graph with block number $k$: take a rectangular $rk\times k$-grid, add $2(r+1)$ vertices to the outer face and join each of these to $k$ vertices on the perimeter of the grid (see Figure 1). If $2(r+1)\geq k$, these new vertices are $(<\!k)$-inseparable.

We are thus faced with two tasks: First, to characterize those classes for which there exists an upper bound on the block number. Second, to obtain a relative upper bound on the block number of graphs in $\mathcal{G}$ when no absolute upper bound exists.

4.1 The bounded case

As indicated in the introduction, Dvořák [15] implicitly characterized the classes for which there exists an upper bound on the block number. Since $k$-blocks are not mentioned in [15], we make this characterization explicit here without adding any ideas not present in [15].

A small modification of the graph depicted in Figure 1 yields a planar graph $H_{k}$ with roughly $k^{3}/2$ vertices and block number $k$ which can be drawn in the plane so that every vertex of degree greater than 3 lies on the outer face: Essentially, replace the square grid by a hexagonal grid and join the ‘new’ vertices only to degree-2 vertices on the perimeter.

Suppose that $H$ is a graph with the property that every graph $G$ that does not contain $H$ as a topological minor satisfies $\beta(G)<s$ for some constant $s=s(H)$. Then $H$ is a topological minor of $H_{s}$ and therefore planar. Moreover, $H$ “inherits” a drawing in the plane in which all vertices of degree greater than 3 lie on the outer face.

The simplest case of a deep structure theorem for graphs excluding a fixed graph as a topological minor [15, Theorem 3] asserts a converse to this in a strong form.

Theorem 10** ([15]).**

Let $H$ be a graph drawn in the plane so that every vertex of degree greater than 3 lies on the outer face. Then there exists an $r=r(H)$ such that every graph that does not contain $H$ as a topological minor has a tree-decomposition in which every torso contains at most $r$ vertices of degree at least $r$.

It is now easy to characterize the graphs whose exclusion as a topological minor bounds the block number.

Corollary 11**.**

Let $H$ be a graph. The following are equivalent:

1. (i)

There is an integer $s=s(H)$ such that every graph $G$ that does not contain $H$ as a topological minor satisfies $\beta(G)<s$. 2. (ii)

$H$* can be drawn in the plane such that every vertex of degree greater than 3 lies on the outer face.*

Proof.

(i) $\rightarrow$ (ii): By assumption, the graph $H_{s}$ contains $H$ as a topological minor. The desired drawing of $H$ can then be obtained from the drawing of $H_{s}$.

(ii) $\rightarrow$ (i): By Theorem 10 and Theorem 1 (ii). ∎

Note that every graph that contains $H_{k}$ as a topological minor necessarily has a $k$-block. Theorem 10 thereby implies a qualitative version of Theorem 1 (i), but without explicit bounds.

Corollary 12**.**

Let $\mathcal{G}$ be a class of graphs. The following are equivalent:

1. (1)

There is a $k\in\mathbb{N}$ such that $\beta(G)\leq k$ for every $G\in\mathcal{G}$. 2. (2)

There is an $m\in\mathbb{N}$ such that no $G\in\mathcal{G}$ contains $H_{m}$ as a topological minor. 3. (3)

There is an $r\in\mathbb{N}$ such that every graph in $\mathcal{G}$ has a tree-decomposition in which every torso has at most $r$ vertices of degree at least $r$.

4.2 The unbounded case

We now turn to the case where $\mathcal{G}$ is a class of graphs excluding some fixed graph as a topological minor for which there exists no upper bound on the block number of graphs in $\mathcal{G}$. If $\mathcal{G}$ is closed under taking topological minors, then by Corollary 12 this implies $H_{k}\in\mathcal{G}$ for all $k\in\mathbb{N}$. Since $|H_{k}|\leq\beta(H_{k})^{3}$, the bound in Theorem 2 is optimal up to a constant factor.

Our aim now is to prove Theorem 2. In light of Lemma 5, it clearly suffices to show the following.

Theorem 13**.**

Let $\mathcal{G}$ be a class of graphs excluding some fixed graph as a topological minor. There exists a constant $c=c(\mathcal{G})$ such that every $G\in\mathcal{G}$ containing a $k$-fan-set $X$ has at least $c|X|k^{2}$ vertices.

This immediately yields the following strengthening of Theorem 2.

Corollary 14**.**

Let $\mathcal{G}$ be a class of graphs excluding some fixed graph as a topological minor. Let $G\in\mathcal{G}$ and let $X$ be the set of all vertices of $G$ that lie in some $k$-block of $G$. Then $|X|\leq|G|/(ck^{2})$, where $c=c(\mathcal{G})$ is the constant from Theorem 13.

Proof.

By Lemma 5, every $k$-block of $G$ is a $k$-fan-set. It is easy to see that a union of $k$-fan-sets is again a $k$-fan-set. Since $X$ is the union of all $k$-blocks, it is therefore a $k$-fan-set. By Theorem 13 we have $|G|\geq c|X|k^{2}$ for $c=c(\mathcal{G})$. ∎

We now turn to the proof of Theorem 13 above. Excluding a topological minor ensures that our graph and all its topological minors are sparse. The following is well-known, see [10, Chapter 7].

Lemma 15**.**

Let $\mathcal{G}$ be a class of graphs excluding some fixed graph as a topological minor. There exist constants $\alpha,d>0$ such that every topological minor $G$ of a graph in $\mathcal{G}$ has at most $d|G|$ edges and an independent set of order at least $\alpha|G|$.

Proof of Theorem 13.

Let $G\in\mathcal{G}$ and $k\in\mathbb{N}$. To ease notation, we assume that $X\subseteq V(G)$ is a $(k+1)$-fan-set instead of just a $k$-fan-set. This only has an effect on the constant $c$.

For every $x\in X$ let $\mathcal{Q}_{x}$ be a $k$-fan from $x$ to $X\setminus\{x\}$. Taking subpaths, if necessary, we may assume that no $Q\in\mathcal{Q}_{x}$ has an internal vertex in $X$. We use initial segments of the paths in $\mathcal{Q}_{x}$ to construct a subdivision of a star with center $x$. Lemma 15 will enable us to find many disjoint such subgraphs.

We adopt an idea from [16]. For some integer $r$ that we are going to choose later, let $\mathcal{P}$ be a maximal set of internally disjoint $X$-paths of length at most $2r$ such that for any two $x,y\in X$ there is at most one path in $\mathcal{P}$ joining them. The paths in $\mathcal{P}$ will be used as barriers to separate the subdivided stars. Let $B:=X\cup\bigcup\mathcal{P}$.

For $x\in X$ and $Q\in\mathcal{Q}_{x}$, let $Q^{\prime}\subseteq Q$ be the maximal subpath of length at most $r$ with $Q^{\prime}\cap B=\{x\}$. If the length of $Q^{\prime}$ is less than $r$, then the next vertex along $Q$ lies in $B$. We say that this vertex stops the path $Q^{\prime}$. Define $\mathcal{Q}_{x}^{\prime}:=\{Q^{\prime}\colon Q\in\mathcal{Q}_{x}\}$ and $S_{x}:=\bigcup\mathcal{Q}_{x}^{\prime}$.

The paths of $\mathcal{P}$ provide us control on the overlap of the stars and allow us to separate them. Let $H=H(\mathcal{P})$ be the auxiliary graph with vertex-set $X$ where $xy\in E(H)$ if and only if some $P\in\mathcal{P}$ joins $x$ and $y$.

[TABLE]

Indeed, if $\bigcup\mathcal{Q}_{x}^{\prime}\cap\bigcup\mathcal{Q}_{y}^{\prime}\neq\emptyset$ then we can find a path $P$ of length at most $2r$ between $x$ and $y$ which is internally disjoint from all paths in $\mathcal{P}$. By maximality of $\mathcal{P}$, there must already be some $R\in\mathcal{P}$ joining $x$ and $y$. Similarly

[TABLE]

The graph $H$ is clearly a topological minor of $G$. It follows from Lemma 15 that $|\mathcal{P}|=|E(H)|\leq d|X|$ and that $H$ contains an independent set $Y\subseteq X$ with $|Y|\geq\alpha|X|$. By (2), the stars with centers in $Y$ are pairwise disjoint and $z\in Y$ does not stop any $Q^{\prime}\in\mathcal{Q}_{y}^{\prime}$ for $y\in Y$. We will show that, on average, many paths in $\mathcal{Q}_{y}^{\prime}$, $y\in Y$, have length $r$.

For $y\in Y$ let $q_{y}$ be the number of $Q^{\prime}\in\mathcal{Q}_{y}^{\prime}$ that were stopped. Extending each $Q^{\prime}\in\mathcal{Q}_{y}^{\prime}$ that was stopped by a single edge, we obtain a path $Q^{\prime\prime}$ from $y$ to the vertex $v\in B$ that stopped $Q^{\prime}$. Note that if $v$ stops some $Q^{\prime}\in\mathcal{Q}_{y}^{\prime}$, then $v\in B\setminus Y$. We therefore obtain a bipartite graph $J$ with $V(J)=Y\cup(B\setminus Y)$ as a topological minor of $G$, where $yv\in E(J)$ if and only if $v$ stops some $Q^{\prime}\in\mathcal{Q}_{y}^{\prime}$. It follows from Lemma 15 that

[TABLE]

Since the paths in $\mathcal{Q}_{y}$ intersect only in $y$, no vertex can stop more than one $Q^{\prime}\in\mathcal{Q}_{y}^{\prime}$. Therefore

[TABLE]

It follows that

[TABLE]

Setting $r:=\lfloor\frac{\alpha k}{4d^{2}}\rfloor$ yields the desired result. ∎

5 Admissibility and $k$-blocks

We now prove Theorem 3, which asserts that block number and $\infty$-admissibility are within a constant multiplicative factor. By Lemma 5, every $k$-block is a $k$-fan-set and so

[TABLE]

It thus only remains to show $\beta(G)\geq\lfloor(\rm adm_{\infty}(G)+1)/2\rfloor$. Lemma 9 provides a sufficient condition for a set of vertices to be $(<\!k)$-inseparable. Our proof is an adaptation of the proof of [4, Theorem 4.2], where it is shown that $\beta(G)\geq\lfloor\delta(G)/2\rfloor+1$. This is also a consequence of our result, since $V(G)$ itself is a $(\delta(G)+1)$-fan-set.

Proof of Theorem 3.

The inequality $\beta(G)\leq\rm adm_{\infty}(G)$ follows from Lemma 5.

Suppose now that $\rm adm_{\infty}(G)\geq 2k-1$ and let $X\subseteq V(G)$ be a $(2k-1)$-fan-set. We will show that $X$ contains a $(<\!k)$-inseparable set.

By Lemma 8 there exists a $k$-lean tree-decomposition $(T,\mathcal{V})$ of $G$. Let $S\subseteq T$ be a minimal subtree such that $X\subseteq\bigcup_{s\in S}V_{s}$. Let $t\in S$ be a leaf of $S$. If $S=\{t\}$, then $X\subseteq V_{t}$ and from every $x\in X$ there is a $(2k-1)$-fan to $V_{t}$. By Lemma 9, $X$ itself is already $(<\!k)$-inseparable.

Otherwise, let $t^{\prime}$ be the unique neighbor of $t$ in $S$ and let $W:=X\cap V_{t}\setminus V_{t^{\prime}}$. Note that $W\neq\emptyset$, for otherwise $S-t$ would violate the minimality of $S$. Let $w\in W$ arbitrary and let $\mathcal{Q}_{w}$ be a $(2k-1)$-fan from $w$ to $X$. Every $Q\in\mathcal{Q}_{w}$ whose endvertex is not in $W$ must meet $V_{t}\cap V_{t^{\prime}}$. Thus at most $|V_{t}\cap V_{t^{\prime}}|<k$ paths from $\mathcal{Q}_{w}$ have endvertices outside $W$. In particular, $|W|\geq k$. Furthermore by stopping every $Q\in\mathcal{Q}_{w}$ when it hits $V_{t}\cap V_{t^{\prime}}$ (if it does) we obtain a $(2k-1)$-fan from $w$ to $V_{t}$. By Lemma 9 the vertices of $W$ cannot be separated by deleting fewer than $k$ vertices. ∎

6 Tree-width and $k$-blocks

This section is devoted to the relation between tree-width and the occurrence of $k$-blocks in a minor. By considering random graphs, one can show that there are graphs $G_{n}$ on $n$ vertices with $2n$ edges and tree-width at least $\gamma n$ for some absolute constant $\gamma>0$ (see [17, Corollary 5.2]). By (1) we have $\beta(H)\leq 1+\sqrt{4n}$ for every $H\prec G_{n}$. Hence the bound in Theorem 4 is best possible up to constant factors.

We now show that every graph of tree-width at least $2(k^{2}-1)$ has a minor with block number at least $k$. In fact, this follows easily from a lemma in the proof of the Grid Minor Theorem given by Diestel, Jensen, Gorbunov and Thomassen [14]. To state their result, we need to introduce some terminology.

Let $G$ be a graph. Call a set $X$ of vertices externally $k$-linked in $G$ if for any $Y,Z\subseteq X$, $|Y|=|Z|\leq k$, there are $|Y|$ disjoint $X$-paths joining $Y$ and $Z$. A $k$-mesh of order $m$ is a separation $(A,B)$ with $|A\cap B|=m$ such that $A\cap B$ is externally $k$-linked in $G[B]-E(A\cap B)$ and there is a tree $T\subseteq G[A]$ with $\Delta(T)\leq 3$ such that every vertex of $A\cap B$ lies in $T$ and has degree at most 2 in $T$.

Lemma 16** ([14, Lemma 4] ).**

Let $G$ be a graph and $m\geq k\geq 1$ integers. If $\rm tw(G)\geq k+m-1$, then $G$ has a $k$-mesh of order $m$.

Lemma 17**.**

Let $p\geq 0,k\geq 2$ be integers and let $T$ be a tree with $\Delta(T)\leq 3$ and $X\subseteq V(T)$ a set of at least $(2p+1)(k-1)$ vertices of degree at most 2. Then there are disjoint subtrees $T_{1},\ldots,T_{p}\subseteq T$ such that $|T_{i}\cap X|\geq k$ for every $i\in[p]$.

Proof.

By induction on $p$. The case where $p\in\{0,1\}$ is trivial. In the inductive step, declare a leaf $r$ of $T$ as the root and thus introduce an order on $T$. Choose $t\in T$ maximal in the tree-order such that $\lfloor t\rfloor$, the subtree containing $t$ and all its descendants, contains at least $k$ vertices of $X$. Note that $t\neq r$, since $|X|>k$.

If $t\in X$, then $|\lfloor t\rfloor\cap X|=k$ because $t$ has only one successor $s$ and $|\lfloor s\rfloor\cap X|<k$. If $t\notin X$, then similarly $|\lfloor t\rfloor\cap X|\leq 2(k-1)$. Let $S:=T-\lfloor t\rfloor$ and note that $|S\cap X|\geq|X|-2(k-1)$. By the inductive hypothesis applied to $S$ and $S\cap X$ we find disjoint $S_{1},\ldots,S_{p-1}\subseteq S$ with $|S_{i}\cap X|\geq k$ for all $i\in[p-1]$. For $1\leq i<p$ let $T_{i}:=S_{i}$ and put $T_{p}:=\lfloor t\rfloor$. These subtrees of $T$ are as desired. ∎

We thus obtain the following more precise version of Theorem 4.

Theorem 18**.**

Let $G$ be a graph and $p\geq k\geq 2$ integers. If the tree-width of $G$ is at least $2(k-1)(p+1)$, then some minor of $G$ contains a $(<\!k)$-inseparable independent set of size $p$.

Proof.

Let $m:=\rm tw(G)-k+1$. By Lemma 16 above, $G$ has a $k$-mesh $(A,B)$ of order $m$. Let $T\subseteq G[A]$ be the tree guaranteed by the definition.

Since $m\geq(k-1)(2p+1)$, we can apply the lemma above to find disjoint subtrees $T_{1},\ldots,T_{p}\subseteq T$ such that each contains at least $k$ vertices of $A\cap B$.

Let $W:=(B\setminus A)\cup\bigcup_{i=1}^{p}V(T_{i})$ and obtain $H$ from $G[W]$ by deleting all edges between $T_{i}$ and $T_{j}$ for $i\neq j$. Given $1\leq i,j\leq p$, the graph $H$ contains $k$ disjoint paths between $T_{i}\cap(A\cap B)$ and $T_{j}\cap(A\cap B)$ with no internal vertices or edges in $A\cap B$, since $A\cap B$ is externally $k$-linked in $G[B]-E(A,B)$.

Contracting each $T_{i}$ to a single vertex thus yields the desired $(<\!k)$-inseparable independent set in a minor of $H$ and thus of $G$. ∎

Taking $p=k$ clearly yields Theorem 4.

7 Concluding remarks

From Theorem 4 and Lemma 5 we deduce the following.

Corollary 19**.**

Let $k\geq 1$ be an integer. Every graph of tree-width at least $2k^{2}-2$ has a minor with $\infty$-admissibility at least $k$.

Richerby and Thilikos [19] proved the existence of a function $g$ such that graphs of tree-width at least $g(k)$ have a minor with $\infty$-admissibility $\geq k$. In their proof, $g(k)$ is the minimum $N$ such that graphs of tree-width at least $N$ have the $k^{3/2}\times k^{3/2}$-grid as a minor. The existence of such an $N$ is the rather difficult Grid-Minor Theorem of Robertson and Seymour [21]. In comparison, our proof is short and simple: the only non-trivial step was a lemma from [14], whose proof is about a page long and in fact the first step in their proof of the Grid-Minor Theorem. Moreover, we have provided an explicit quadratic bound on $g(k)$, while even the existence of a polynomial bound upper bound for $N$ is a recent breakthrough-result of Chekuri and Chuzhoy [9].

Dvořák proved that for every $k$ there are integers $m$ and $d$ such that every graph with $\infty$-admissibility at most $k$ has a tree-decomposition in which every torso contains at most $m$ vertices of degree at least $d$ ([15, Corollary 5]). The proof is based on a deep structure theorem for graphs excluding a topological minor [15, Theorem 3] and does not yield explicit bounds for $m$ and $d$. Combining Theorem 1 with Lemma 5, we obtain a much simpler proof that avoids the use of advanced graph minor theory and moreover provides explicit values for the parameters involved.

Corollary 20**.**

Let $k\geq 1$ be an integer. If $G$ has $\infty$-admissibility at most $k$, then $G$ has a tree-decomposition of adhesion less than $k$ in which every torso contains at most $k$ vertices of degree at least $2k(k-1)$.

It seems challenging to obtain stronger estimates: What is the minimum $N=N(k)$ such that every graph without a $k$-block has a tree-decomposition in which every torso contains at most $N$ vertices of degree at least $N$? Can we always find a tree-decomposition in which every torso has a bounded number of vertices of degree at least $\alpha k$ for some constant $\alpha>0$?

Admissibility of graphs has primarily been studied with a length-restriction imposed. We call the maximum length of a path in a fan $\mathcal{Q}$ the radius of $\mathcal{Q}$. A $(k,r)$-fan-set is a set $X\subseteq V(G)$ such that from every $x\in X$ there is a $k$-fan of radius at most $r$ to $X$. The $r$-admissibility $\rm adm_{r}(G)$ is the maximum $k$ for which $G$ has a $(k,r)$-fan-set. In particular

[TABLE]

Note that for every integer $r\geq 1$ trivially

[TABLE]

since a fan cannot contain $|G|/(r+1)$ paths of length $>r$.

Grohe et al showed in [16] that for every class of graphs $\mathcal{G}$ excluding a topological minor we have $\rm adm_{r}(G)=\mathcal{O}(r)$ for every $G\in\mathcal{G}$. Taking (4) into account we obtain the trivial estimate $\rm adm_{\infty}(G)=\mathcal{O}(\sqrt{|G|})$ for $G\in\mathcal{G}$, which also follows from a simple edge-count and Lemma 15. On the other hand, Theorem 13 shows that $\rm adm_{r}(G)=\mathcal{O}(\sqrt[3]{|G|})$ for every $r$. Hence for values of $r$ which are large with respect to $|G|$, namely for $r\geq K\sqrt[3]{|G|}$ for some constant $K>0$, our result is a substantial improvement of the estimate of Grohe et al.

Let $\mathcal{G}$ be class of graphs excluding a topological minor. For $n,r\in\mathbb{N}$ let

[TABLE]

We know by Theorem 13 that $F(n,r)=\mathcal{O}(\sqrt[3]{n})$ for all $r\in\mathbb{N}$, while Grohe et al [16] showed $F(n,r)=\mathcal{O}(r)$ for all $n\in\mathbb{N}$. It appears to be an interesting problem to try to obtain a unified bound.

Acknowledgements

I would like to thank the people of the graph-theory group at the University of Hamburg, in particular Joshua Erde who pointed out that $k$-lean tree-decompositions might help in a proof of Theorem 3.