Graphs of Linear Growth have Bounded Treewidth

The growth of a (possibly infinite) graph¹¹1We consider undirected graphs $G$ with vertex-set ${V(G)}$ and edge-set ${E(G)}$. For integers ${m,n\in\mathbb{Z}}$, let ${[m,n]:=\{z\in\mathbb{Z}\colon m\leqslant z\leqslant n\}}$ and ${[n]:=[1,n]}$. Let $\mathbb{N}$ be the set of positive integers. Let $\log$ be the natural logarithm $\log_{\_}e$. $G$ is the function ${f_{\_}G\colon\mathbb{N}\to\mathbb{N}\cup\{\infty\}}$ where ${f_{\_}G(r)}$ is the supremum of ${|V(H)|}$ taken over all subgraphs $H$ of $G$ with radius at most $r$. Growth in graphs is an important topic in group theory [36, 31, 19, 20, 16, 40, 17], where growth of a finitely generated group is defined through the growth of the corresponding Cayley graphs. Growth of graphs also appears in metric geometry [28], algebraic graph theory [39, 25, 14, 15, 23, 24], and in models of random infinite planar graphs [11, 1]. A graph class $\mathcal{G}$ has linear/quadratic/polynomial/exponential growth if ${\sup\{f_{\_}G(r)\colon G\in\mathcal{G}\}}$ is bounded from above and below by a linear/quadratic/polynomial/exponential function of $r$.

This paper focuses on graph classes with linear growth. Linear growth has previously been studied in the context of infinite vertex-transitive graphs [39, 25, 14, 15, 23, 24]. Notably, Imrich and Seifter [24] characterised when an infinite vertex-transitive graph has linear growth in terms of its automorphism group. We take a more structural and less algebraic approach, and prove that graph classes with linear growth have a tree-like structure.

To formalise this result, we need the following definition. A tree-decomposition of a graph $G$ is a collection ${(B_{\_}x\subseteq V(G)\colon x\in V(T))}$ of subsets of ${V(G)}$ (called bags) indexed by the nodes of a tree $T$, such that:

• •

  for every edge ${uv\in E(G)}$, some bag $B_{\_}x$ contains both $u$ and $v$, and

• •

  for every vertex ${v\in V(G)}$, the set ${\{x\in V(T)\colon v\in B_{\_}x\}}$ induces a non-empty subtree of $T$.

The width of a tree-decomposition is the size of the largest bag, minus $1$. The treewidth of a graph $G$, denoted by $\operatorname{tw}(G)$, is the smallest integer $w$ for which there is a tree-decomposition of $G$ of width $w$, or $\infty$ if no such $w$ exists. Treewidth can be thought of as measuring how structurally similar a graph is to a tree. Indeed, a connected graph has treewidth $1$ if and only if it is a tree. Treewidth is of fundamental importance in structural and algorithmic graph theory; see [32, 21, 3] for surveys.

Our main result shows that graphs with linear growth have bounded treewidth.

thmMain For any ${c\geqslant 1}$, every graph $G$ with growth ${f_{\_}G(r)\leqslant cr}$ has treewidth at most ${49c^{2}+30c}$.

It suffices to prove Section 1 for finite graphs, since for ${k\in\mathbb{N}}$, an infinite graph has treewidth at most $k$ if and only if every finite subgraph has treewidth at most $k$ (see [37, 38]).

Section 1 is proved in Section 2, where we also prove an ${\Omega(c\log c)}$ lower bound on the treewidth in Section 1. Section 3 considers graphs with linear growth in proper minor-closed graph classes. In this case, we improve the upper bound on the treewidth in Section 1 to ${O(c)}$. Section 4 explores the product structure of graphs with linear growth. Combining Section 1 with results from the literature, we show that graphs with linear growth are subgraphs of bounded ‘blow-ups’ of trees with bounded maximum degree, which is a qualitative strengthening of Section 1. This section also presents two conjectures about the product structure of graphs with linear and polynomial growth. Finally, Section 5 studies the growth of subdivisions of graphs. We show that a finite graph with bounded treewidth and bounded maximum degree has a subdivision with linear growth. We also show that for any superlinear function $f$ with ${f(r)\geqslant 1+\Delta r}$, every finite graph with maximum degree $\Delta$ (regardless of its treewidth) has a subdivision with growth bounded above by $f$. These results show that, for instance, in Section 1, “treewidth” cannot be replaced by “pathwidth”, while “${cr}$” cannot be replaced by “${cr^{1+\varepsilon}}$”.

Graphs with bounded treewidth have many attractive properties, and Section 1 implies that all such properties hold for graphs of linear growth. To conclude this introduction, we give one such example. A $k$-stack layout of a graph $G$ is a pair ${(\leqslant,\varphi)}$ where $\leqslant$ is a linear ordering on $V(G)$ and ${\varphi\colon E(G)\to[k]}$ is a function such that ${\varphi(ux)\neq\varphi(vy)}$ for any two edges ${ux,vy\in E(G)}$ with ${u<v<x<y}$. The stack-number of a (possibly infinite) graph $G$ is the minimum integer ${k\geqslant 0}$ such that there exists a $k$-stack layout of $G$, or $\infty$ if no such $k$ exists. This topic is widely studied; see [9, 8, 2, 26, 42, 43] for example.

Eppstein et al. [12] recently showed that ${P_{\_}n\boxtimes P_{\_}n\boxtimes P_{\_}n}$, which has growth ${(2r+1)^{3}}$, has unbounded stack-number (as ${n\to\infty}$). Motivated by this discovery, they asked whether graphs of quadratic or of linear growth have bounded stack-number. Ganley and Heath [13] showed that every finite graph with treewidth $k$ has stack-number at most ${k+1}$. Section 1 thus implies a positive answer to the second part of this question.

As before, it suffices to prove Theorem 1 for finite graphs, since a standard compactness argument shows that for ${k\in\mathbb{N}}$, an infinite graph has stack-number at most $k$ if and only if every finite subgraph has stack-number at most $k$ (see Appendix A).

For the remainder of the paper, we assume that every graph is finite.

This section proves our main result, Section 1, as well as a lower bound for the growth of the class of graphs of tree-width at most $c$, Lemma 6.

The key tool we use is that of balanced separations. A separation of a graph $G$ is a pair ${(A,B)}$ of subsets of ${V(G)}$ such that ${A\cup B=V(G)}$ and no edge of $G$ has one end in ${A\setminus B}$ and the other in ${B\setminus A}$. The order of the separation ${(A,B)}$ is ${|A\cap B|}$. For ${\alpha\in[\frac{2}{3},1)}$, a separation ${(A,B)}$ of a graph on $n$ vertices is $\alpha$-balanced if ${|A|\leqslant\alpha n}$ and ${|B|\leqslant\alpha n}$. The $\alpha$-separation number $\operatorname{sep}_{\_}{\alpha}(G)$ of a graph $G$ is the smallest integer $s$ such that every subgraph of $G$ has an $\alpha$-balanced separation of order at most $s$.

Robertson and Seymour [33] showed that ${\operatorname{sep}_{\_}{2/3}(G)\leqslant\operatorname{tw}(G)+1}$. Dvořák and Norin [10] established the following converse.

The next two lemmas are folklore. The first one enables us to work in the more general setting of $\alpha$-balanced separation.

The next lemma is the heart of this paper.

We are now ready to prove our main theorem which we restate for convenience.

*

We conclude this section by showing that the function $49c^{2}+30c$ in Section 1 cannot be replaced by any function in ${o(c\log c)}$. For a vertex $v$ in a graph $G$, the $r$-ball at $v$ is the set $B_{\_}r(v)$${\;:=\{w\in V(G)\colon\operatorname{dist}_{\_}G(v,w)\leqslant r\}}$.

thmLowerBound If $g$ is any function such that for any $c\geqslant 1$, every graph $G$ of growth $f_{\_}G(r)\leqslant cr$ has treewidth at most $g(c)$, then $g(c)\in\Omega(c\log c)$.

This section studies growth in proper minor-closed graph classes. A graph $H$ is a minor of a graph $G$ if $H$ is isomorphic to a graph obtained from a subgraph of $G$ by contracting edges. A graph class $\mathcal{G}$ is minor-closed if for every $G\in\mathcal{G}$ every minor of $G$ is also in $\mathcal{G}$. A minor-closed class $\mathcal{G}$ is proper if some graph is not in $\mathcal{G}$. A graph parameter $\lambda$ is minor-monotone if $\lambda(H)\leqslant\lambda(G)$ whenever $H$ is a minor of $G$.

Grid graphs are the key examples here. For $n\in\mathbb{N}$, the $n\times n$ grid is the graph with vertex set $\{(v_{\_}1,v_{\_}2):v_{\_}1,v_{\_}2\in[n]\}$ where $(v_{\_}1,v_{\_}2)$ and $(u_{\_}1,u_{\_}2)$ are adjacent if $v_{\_}1=u_{\_}1$ and $|v_{\_}2-u_{\_}2|=1$, or $v_{\_}2=u_{\_}2$ and $|v_{\_}1-u_{\_}1|=1$. This graph has treewidth $n$ (see [21]), and is a canonical example of a graph with large treewidth in the sense that every graph $G$ with sufficiently large treewidth contains the $n\times n$ grid as a minor [34]. Since treewidth is minor-monotone, Section 1 implies that any graph $G$ with growth $f_{\_}G(r)\leqslant cr$ cannot contain the $n\times n$ grid as a minor, where $n=\lceil 49c^{2}+30c\rceil$. We prove this directly with $n=\lceil 2c\rceil$.

thmGrowthGridMinor For any ${c\geqslant 1}$, every graph $G$ with growth ${f_{\_}G(r)\leqslant cr}$ does not contain the $\lceil 2c\rceil\times\lceil 2c\rceil$ grid as a minor.

Demaine and Hajiaghayi [4] showed that for any fixed graph $H$, every $H$-minor-free graph $G$ with treewidth $k$ contains an ${\Omega(k)\times\Omega(k)}$ grid as a minor (see [27] for explicit bounds). In this case, Section 3 implies the following improvement on Section 1.

In the case of planar graphs, Robertson et al. [35] showed that every planar graph containing no ${n\times n}$-grid minor has treewidth at most $6n-5$. Section 3 thus implies:

Recall that Lemma 6 provides an ${\Omega(c\log c)}$ lower bound on the treewidth of graphs $G$ with growth ${f_{\_}G(r)\leqslant cr}$. Thus to conclude the ${O(c)}$ upper bounds in Corollaries 7 and 8, it is essential to make some assumption such as excluding a fixed minor.

Much of the research on the growth of finite graphs has centred around polynomial growth. In this setting, Krauthgamer and Lee [28] showed that every graph $G$ of growth $f_{\_}G(r)\leqslant r^{d}$ (for $r\geqslant 2$) is isomorphic to a subgraph of the strong product of $O(d\log d)$ sufficiently long paths. Here the strong product $G\boxtimes H$ of graphs $G$ and $H$ is the graph with vertex-set $V(G)\times V(H)$ with an edge between two vertices $(v,w)$ and $(v^{\prime},w^{\prime})$ if $vv^{\prime}\in E(G)$ and $w=w^{\prime}$ or $ww^{\prime}\in E(H)$, or $v=v^{\prime}$ and $ww^{\prime}\in E(H)$. Note that $G\boxtimes K_{\_}t$ is simply the graph obtained from $G$ by replacing each vertex of $G$ by a copy of $K_{\_}t$ and replacing each edge of $G$ by $K_{\_}{t,t}$ between the corresponding copies of $K_{\_}t$, sometimes called a blow-up of $G$. The following result, due to a referee of [6] and refined in [7, 41], allows us to describe graphs of linear growth as subgraphs of blow-ups of trees.

A graph $G$ with growth $f_{\_}G(r)\leqslant cr$ has maximum degree at most $c-1$. Thus the following result²²2It follows from a result of Kuske and Lohrey [29] (or Huynh et al. [22] in the countable case) that Theorem 10 also holds for infinite graphs. is a consequence of Sections 1 and 9.

The graph $T\boxtimes K_{\_}{\lfloor 882c^{3}\rfloor}$ preserves the boundedness of the treewidth of $G$. However, the growth of $T\boxtimes K_{\_}{\lfloor 882c^{3}\rfloor}$ is at least the growth of $T$ which can be exponential, for example if $T$ is a complete binary tree. This leads us to conjecture the following rough characterisation of graphs of linear growth.

This conjecture (if true) would approximately characterise graphs of linear growth in the sense that every subgraph $H$ of $T\boxtimes K_{\_}{g(c)}$ has growth $f_{\_}H(r)\leqslant g(c)h(c)r\in O(r)$.

More generally, for graphs of polynomial growth, we conjecture the following rough characterisation.

Again, this conjecture (if true) would approximately characterise graphs of degree-$d$ polynomial growth in the sense that if $H$ is a subgraph of $T_{\_}1\boxtimes\cdots\boxtimes T_{\_}d\boxtimes K_{\_}{g(c,d)}$, then $f_{\_}H(r)\leqslant g(c,d)\,(h(c,d)r)^{d}\in O(r^{d})$.

This section considers the growth of subdivisions of graphs. A graph $\tilde{G}$ is a subdivision of a graph $G$ if $\tilde{G}$ can be obtained from $G$ by replacing each edge ${vw}$ by a path $P_{\_}{vw}$ with endpoints $v$ and $w$ (internally disjoint from the rest of $\tilde{G}$). If each of these paths has the same length, then $\tilde{G}$ is said to be uniform.

Theorem 14 below shows that every graph with bounded degree and bounded treewidth has a subdivision with linear growth. By Lemma 9, we can obtain Theorem 14 from the following result.

The following theorem is a direct consequence of Lemmas 9 and 13.

Note that the growth of any subdivision $\tilde{G}$ of a graph $G$ depends on the maximum degree $\Delta$ of $G$ (since ${f_{\_}{\tilde{G}}(1)\geqslant\Delta+1}$). We now show that every graph $G$ with bounded treewidth is a minor of a graph $G^{\prime}$ with linear growth where the growth of $G^{\prime}$ does not depend on the maximum degree of $G$. Markov and Shi [30] proved that every graph $G$ is a minor of some graph $G^{\prime}$ with maximum degree $3$ and treewidth at most ${\operatorname{tw}(G)+1}$. Theorem 14 applied to $G^{\prime}$ gives the following result.