Polynomial expansion and sublinear separators

TL;DR

This paper proves that classes of graphs with sublinear separators have minors with bounded average degree, confirming a conjecture and linking separator properties to minor structure.

Contribution

It establishes a relationship between sublinear separators and the average degree of depth-$k$ minors in graph classes, confirming a conjecture of Dvořák and Norin.

Findings

Graphs with sublinear separators have minors with bounded average degree.

The bound depends on the depth of the minor and the separator size.

The result confirms a previously conjectured relationship in graph theory.

Abstract

Let $\mathcal{C}$ be a class of graphs that is closed under taking subgraphs. We prove that if for some fixed $0<\delta\le 1$, every $n$-vertex graph of $\mathcal{C}$ has a balanced separator of order $O(n^{1-\delta})$, then any depth-$k$ minor (i.e. minor obtained by contracting disjoint subgraphs of radius at most $k$) of a graph in $\mathcal{C}$ has average degree $O\big((k \text{ polylog }k)^{1/\delta}\big)$. This confirms a conjecture of Dvo\v{r}\'ak and Norin.

Equations

$$c_1\cdot (r+1{)}^{1/\delta }(\frac{1}{\delta }\log (r+3){)}^{c_2/\delta }\ge \max \left\{c_1(\log 3{)}^{c_2},c_1\exp (\frac{c_2}{\delta }\log \frac{\log 3}{\delta })\right\},$$

$$d(F)\ge \max \left\{{10}^8,\exp \left(4\cdot \frac{\beta +3}{\delta }\log (2\cdot \frac{\beta +3}{\delta })\right)\right\}$$

$$|N(S)|\ge \frac{1}{2^8\log |V(H)|(\log \log |V(H)|{)}^2}|S|\ge \frac{1}{2^8(\log |V(H)|{)}^3}|S|.$$

$$|A\cap B|\ge \frac{1}{2^8(\log |V(H)|{)}^3}|A\setminus B|.$$

$$\frac{1}{3}|V(H)|\le |A\cap B|(1+2^8(\log |V(H)|{)}^3).$$

$$\mathrm{tw}(H)\ge \frac{|V(H)|}{3\cdot 2^8(\log |V(H)|{)}^3+3}-1\ge \frac{|V(H)|}{2^{10}(\log |V(H)|{)}^3},$$

$$\mathrm{tw}(H^{\prime })\ge \frac{\alpha \mathrm{tw}(H)}{(\log \mathrm{tw}(H){)}^{\beta }}\ge \frac{\alpha |V(H)|}{2^{10}(\log |V(H)|{)}^{\beta +3}},$$

$$\mathrm{tw}(G^{\prime })\ge \mathrm{tw}(H^{\prime })\ge \frac{\alpha |V(H)|}{2^{10}(\log |V(H)|{)}^{\beta +3}}.$$

$$\mathrm{tw}(G^{\prime })\le 105C|V(G^{\prime }){|}^{1-\delta }\le 2^{7}C|V(G^{\prime }){|}^{1-\delta }.$$

$$\frac{\alpha |V(H)|}{2^{10}(\log |V(H)|{)}^{\beta +3}}$$

$$\displaystyle\leq 2^{7}C\,\big{(}(3r+1)|V(H)|\big{)}^{1-\delta}\text{, and thus}$$

$$\frac{|V(H){|}^{\delta }}{(\log |V(H)|{)}^{\beta +3}}$$

$$\le \frac{2^{17}C}{\alpha }(3r+1)\text{, and}$$

$$|V(H)|$$

$$\log |V(H)|\le \frac{1}{\delta }\log \left(\frac{2^{17}C}{\alpha }(3r+1)\right)+\frac{\beta +3}{\delta }\log \log |V(H)|.$$

$$\frac{\log |V(H)|}{\log \log |V(H)|}$$

$$\log |V(H)|$$

$$|V(H)|\le {\left(\frac{2^{17}C}{\alpha }(3r+1){\left(\frac{2}{\delta }\log \left(\frac{2^{17}C}{\alpha }(3r+1)\right)\right)}^{\beta +3}\right)}^{1/\delta }\le \frac{255}{256}{c}_1(r+1{)}^{1/\delta }(\frac{1}{\delta }\log (r+3){)}^{c_2/\delta },$$

Full text

The first author was supported by ANR Projects STINT (anr-13-bs02-0007) and GATO (anr-16-ce40-0009-01) and LabEx PERSYVAL-Lab (anr-11-labx-0025-01). The second author was supported by the Polish National Science Centre grant PRELUDIUM DEC-2013/11/N/ST6/02706.

1. Introduction

For an integer $r\geq 0$, a depth-$r$ minor of a graph $G$ is a subgraph of a graph that can be obtained from $G$ by contracting pairwise vertex-disjoint subgraphs of radius at most $r$. Let $d(G)$ denote the average degree of a graph $G=(V,E)$, i.e. $d(G)=2|E|/|V|$. For some function $f$, we say that a class $\mathcal{C}$ of graphs has expansion bounded by $f$ if for any graph $G\in\mathcal{C}$ and any integer $r$, any depth-$r$ minor of $G$ has average degree at most $f(r)$. We say that a class has bounded expansion if it has expansion bounded by some function $f$, and polynomial expansion if $f$ can be taken to be a polynomial.

Classes of bounded expansion play a central role in the study of sparse graphs [7]. From an algorithmic point of view, a very useful property of theses classes is that when their expansion is not too large (say subexponential), graphs in the class have sublinear separators. A separator in a graph $G=(V,E)$ is a pair of subsets $(A,B)$ of vertices of $G$ such that $A\cup B=V$ and no edge of $G$ has one endpoint in $A\setminus B$ and the other in $B\setminus A$. The separator $(A,B)$ is said to be balanced if both $|A\setminus B|$ and $|B\setminus A|$ contain at most $\tfrac{2}{3}|V|$ vertices. The order of the separator $(A,B)$ is $|A\cap B|$.

A class $\mathcal{C}$ of graphs is monotone if for any graph $G\in\mathcal{C}$, any subgraph of $G$ is in $\mathcal{C}$. Dvořák and Norin [5] observed that the following can be deduced from a result of Plotkin, Rao, and Smith [8].

Theorem 1** ([5]).**

Let $\mathcal{C}$ be a monotone class of graphs with expansion bounded by $r\mapsto c(r+1)^{1/4\delta-1}$, for some constant $c>0$ and $0<\delta\leq 1$. Then there is a constant $C$ such that every $n$-vertex graph of $\mathcal{C}$ has a balanced separator of order $Cn^{1-\delta}$.

Dvořák and Norin [5] also proved the following partial converse.

Theorem 2** ([5]).**

Let $\mathcal{C}$ be a monotone class of graphs such that for some fixed constants $C>0$ and $0<\delta\leq 1$, every $n$-vertex graph of $\mathcal{C}$ has a balanced separator of order $Cn^{1-\delta}$. Then the expansion of $\mathcal{C}$ is bounded by some function $f(r)=O(r^{5/\delta^{2}})$.

They conjectured that the exponent $5/\delta^{2}$ of the polynomial expansion in Theorem 2 could be improved to match (asymptotically) that of Theorem 1.

Conjecture \theconjecture ([5]).

There exists a real $c>0$ such that the following holds. Let $\mathcal{C}$ be a monotone class of graphs such that for some fixed constants $C>0$ and $0<\delta\leq 1$, every $n$-vertex graph of $\mathcal{C}$ has a balanced separator of order $Cn^{1-\delta}$. Then the expansion of $\mathcal{C}$ is bounded by some function $f(r)=O(r^{c/\delta})$.

In this short note, we prove this conjecture.

Theorem 3**.**

For any $C>0$ and $0<\delta\leq 1$, if a monotone class $\mathcal{C}$ has the property that every $n$-vertex graph in $\mathcal{C}$ has a balanced separator of order at most $Cn^{1-\delta}$, then $\mathcal{C}$ has expansion bounded by the function $f:r\mapsto c_{1}\cdot(r+1)^{1/\delta}(\tfrac{1}{\delta}\log(r+3))^{c_{2}/\delta}$, for some constants $c_{1}$ and $c_{2}$ depending only on $\mathcal{C}$.

In particular Section 1 holds for any real number $c>1$. The proof of Theorem 3 is given in the next section, and we conclude with some open problems in Section 3.

2. Proof of Theorem 3

We need the following results. The first is a classical connection between balanced separators and tree-width (see [5]).

Lemma \thelemma.

Any graph $G$ has a balanced separator of order at most $\operatorname{tw}(G)+1$.

Dvořák and Norin [4] proved that the following partial converse holds.

Theorem 4** ([4]).**

If every subgraph of $G$ has a balanced separator of order at most $k$, then $G$ has tree-width at most $105k$.

Note that in our proof of Theorem 3 we could also use the weaker (and easier) result of [1] that under the same hypothesis, $G$ has tree-width at most $1+k\log|V(G)|$, but the computation is somewhat less cumbersome if we use Theorem 4 instead.

For a set $S$ of vertices in a graph $G$, we let $N(S)$ denote the set of vertices not in $S$ with at least one neighbor in $S$. We will use the following result of Shapira and Sudakov [9].

Theorem 5** ([9]).**

Any graph $G$ contains a subgraph $H$ of average degree $d(H)\geq\tfrac{255}{256}d(G)$ such that for any set $S$ of at most $n/2$ vertices of $H$ (where $n=|V(H)|$), $|N(S)|\geq\tfrac{1}{2^{8}\log n(\log\log n)^{2}}|S|$.

In fact, we will only need a much weaker version, where the vertex-expansion is of order $\Omega\left(\tfrac{1}{\operatorname{polylog}n}\right)$ instead of $\Omega\left(\tfrac{1}{\log n(\log\log n)^{2}}\right)$.

Finally, we need a result of Chekuri and Chuzhoy [2] on bounded-degree subgraphs of large tree-width in a graph of large tree-width.

Theorem 6** ([2]).**

There are constants $\alpha,\beta$ such that for any integer $k\geq 2$, any graph $G$ of tree-width at least $k$ contains a subgraph $H$ of tree-width at least $\alpha k/(\log k)^{\beta}$ and maximum degree 3.

Let us remark that instead of Theorem 6, our proof of Theorem 3 could rely on an earlier result of Chekuri and Chuzhoy [3] which, under the same assumptions, merely guarantees the existence of a subgraph of $G$ of treewidth $\Omega(k/(\log k)^{6})$ and maximum degree $O((\log k)^{3})$.

We are now ready to prove our main result.

Proof of Theorem 3. Let $G$ be a graph of $\mathcal{C}$ and let $F$ be a depth-$r$ minor of $G$. Our goal is to prove that $d(F)\leq c_{1}\cdot(r+1)^{1/\delta}(\tfrac{1}{\delta}\log(r+3))^{c_{2}/\delta}$, for some constants $c_{1}$ and $c_{2}$ depending only on $C$. Note that for any $r\geq 0$ and $0<\delta\leq 1$,

[TABLE]

so we can assume without loss of generality that

[TABLE]

by choosing appropriate values of $c_{1},c_{2}$. By Theorem 5, $F$ has a subgraph $H$ of average degree $d(H)\geq\tfrac{255}{256}d(F)$ such that for any set $S$ of at most $|V(H)|/2$ vertices of $H$,

[TABLE]

It follows from Section 2 that $H$ contains a balanced separator $(A,B)$ with $|A\cap B|\leq\operatorname{tw}(H)+1$. As $A\setminus B$ and $B\setminus A$ are disjoint, one of them contains at most half of the vertices. We may assume without loss of generality that $|A\setminus B|\leq|V(H)|/2$. As $N(A\setminus B)\subseteq A\cap B$, we get

[TABLE]

Since $(A,B)$ is balanced, $|A\setminus B|+|A\cap B|\geq\tfrac{1}{3}|V(H)|$ and so

[TABLE]

Given that $|A\cap B|\leq\operatorname{tw}(H)+1$, we deduce

[TABLE]

using that $|V(H)|\geq d(H)\geq\tfrac{255}{256}\cdot 10^{8}$.

By Theorem 6, $H$ has a subgraph $H^{\prime}$ of maximum degree 3 such that

[TABLE]

since $\operatorname{tw}(H)\leq|V(H)|$. Note that $H^{\prime}$ is a subgraph of $H$ (and $F$) and therefore also a depth-$r$ minor of $G$. In $G$, $H^{\prime}$ corresponds to a subgraph $G^{\prime}$ (before contraction of the subgraphs of radius $r$) with $|V(G^{\prime})|\leq(3r+1)|V(H^{\prime})|\leq(3r+1)|V(H)|$. Indeed, since $H^{\prime}$ has maximum degree 3, each subgraph of radius at most $r$ in $G^{\prime}$ whose contraction corresponds to a vertex of $H^{\prime}$ contains at most $3r+1$ vertices. Since $H^{\prime}$ is a minor of $G^{\prime}$, we have

[TABLE]

Since $\mathcal{C}$ is monotone, every subgraph of $G^{\prime}$ is in $\mathcal{C}$ and thus has a balanced separator of order at most $C|V(G^{\prime})|^{1-\delta}$. Hence, by Theorem 4,

[TABLE]

We just obtained lower and upper bounds on $\operatorname{tw}(G^{\prime})$. Putting them together, we obtain:

[TABLE]

It follows that

[TABLE]

Since the function $n\mapsto\tfrac{\log n}{\log\log n}$ is increasing for $n\geq 16$, a direct consequence of our initial assumption that $|V(H)|\geq\exp\left(4\cdot\tfrac{\beta+3}{\delta}\log(2\cdot\tfrac{\beta+3}{\delta})\right)$ is that

[TABLE]

We conclude that

[TABLE]

for some constants $c_{1},c_{2}$ depending only on $C$ and the constants $\alpha,\beta$ of Theorem 6. Recall that $d(F)\leq\tfrac{256}{255}d(H)$. Since $d(H)\leq|V(H)|$, we obtain $d(F)\leq c_{1}\cdot(r+1)^{1/\delta}(\tfrac{1}{\delta}\log(r+3))^{c_{2}/\delta}$, as desired. This concludes the proof of Theorem 3. $\Box$

3. Open problems

A natural problem is to determine the infimum real $c>0$, such that if a monotone class $\mathcal{C}$ has the property that every $n$-vertex graph in $\mathcal{C}$ has a balanced separator of order $O(n^{1-\delta})$, then $\mathcal{C}$ has expansion bounded by some function $r\mapsto O(r^{c/\delta})$. Theorem 3 implies that $c\leq 1$. On the other hand, it directly follows from Theorem 1 that $c\leq\tfrac{1}{4+\epsilon}$ would imply that if any $n$-vertex graph in $\mathcal{C}$ has a balanced separator of order $O(n^{1-\delta})$, then any $n$-vertex graph in $\mathcal{C}$ has a balanced separator of order $O(n^{1-(1+\epsilon/4)\delta})$. Therefore, Theorem 1 implies that $c\geq\tfrac{1}{4}$ (moreover, the proof of Theorem 1 in [5] can be slightly optimized to show that $c\geq\tfrac{1}{2}$). A good candidate to prove a better lower bound for $c$ would be the family of all finite subgraphs of the infinite $d$-dimensional grid. The $n$-vertex graphs in this class have balanced separators of order $O(n^{1-1/d})$ (see [6]), and it might be the case that they have expansion $\Omega(r^{cd})$ for some $c>\tfrac{1}{2}$.

One way to measure the sparsity of a class of graphs is via its expansion (as defined in Section 1). Another way (which turns out to be equivalent) is via its generalized coloring parameters. Given a linear order $L$ on the vertices of a graph $G$, and an integer $r$, we say that a vertex $v$ of $G$ is strongly $r$-reachable from a vertex $u$ (with respect to $L$) if $v\leq_{L}u$, and there is a path $P$ of length at most $r$ between $u$ and $v$, such that $u<_{L}w$ for any internal vertex $w$ of $P$. If we only require that $v$ is the minimum of the vertices of $P$ (with respect to $L$), we say that $v$ is weakly $r$-reachable from $u$. The strong $r$-coloring number $\text{col}_{r}(G)$ of $G$ is the minimum integer $k$ such that there is a linear order $L$ on the vertices of $G$ such that for any vertex $u$ of $G$, at most $k$ vertices are strongly $r$-reachable from $u$ (with respect to $L$). By replacing strongly by weakly in the previous definition, we obtain the weak $r$-coloring number $\text{wcol}_{r}(G)$ of $G$. Note that for any graph $G$ and any integer $r$, $\text{col}_{r}(G)\leq\text{wcol}_{r}(G)$. For more on these parameters and their connections with the expansion of graph classes, the reader is referred to [7].

As we have seen before, it follows from [5] that a monotone class of graphs has polynomial expansion if and only if, for some fixed $0<\delta\leq 1$, each $n$-vertex graph in the class has a balanced separator of order $O(n^{1-\delta})$. Joret and Wood asked whether this is also equivalent to having weak and strong $r$-coloring numbers bounded by a polynomial function of $r$.

Problem \theprob (Joret and Wood, 2017).

Assume that $\mathcal{C}$ is a monotone class of graphs. Are the following statements equivalent?

1. (1)

$\mathcal{C}$* has polynomial expansion.* 2. (2)

There exists a constant $c$, such that for every $r$, every graph in $\mathcal{C}$ has strong $r$-coloring number at most $O(r^{c})$. 3. (3)

There exists a constant $c$, such that for every $r$, every graph in $\mathcal{C}$ has weak $r$-coloring number at most $O(r^{c})$.

Note that clearly (3) implies (2). It was known that (3) implies (1) (this is a consequence of Lemma 7.11 in [7]), and Norin recently made the following observation, which shows that (2) implies (1).

Observation \theobservation (Norin, 2017).

Every depth-$r$ minor of a graph $G$ has average degree at most $2\,\mbox{\rm{col}}_{4r}(G)$.

Proof.

Let $L$ be a linear order on the vertices of $G$, such that for any vertex $v$ of $G$, at most $\text{col}_{r}(G)$ vertices are strongly $r$-reachable from $v$ (with respect to $L$). Let $H$ be a depth-$r$ minor of a graph $G$. For any vertex $u$ of $H$, let $S_{u}$ be a subgraph of $G$ of radius at most $r$, such that the $S_{u}$’s are vertex-disjoint and for any edge $uv$ of $H$, there is an edge in $G$ between a vertex of $S_{u}$ and a vertex of $S_{v}$. It is enough to prove that there is a linear order $L^{\prime}$ on the vertices of $H$ such that any vertex $u$ of $H$, at most $\text{col}_{4r}(G)$ vertices of $H$ are strongly 1-reachable from $u$.

We construct $L^{\prime}$ from $L$ as follows: for $u,v$ in $H$, we set $u<_{L^{\prime}}v$ if and only if, with respect to $L$, the smallest vertex of $S_{u}$ precedes the smallest vertex of $S_{v}$. This clearly defines a linear order on the vertices of $H$. Consider a vertex $u$ of $H$ and let $x$ be the smallest vertex of $S_{u}$ (with respect to $L$). Let $v$ be a neighbor of $u$ in $H$ with $v<_{L^{\prime}}u$ (i.e. $v$ is strongly 1-reachable from $u$ in $H$). Let $t\in S_{u}$ and $z\in S_{v}$ be such that $tz$ is an edge of $G$. Observe that there is a path $P_{u}$ from $x$ to $t$ in $S_{u}$ (and $x$ is the smallest vertex in this path with respect to $L$), and a path $P_{v}$ from $z$ to $y$ in $S_{v}$. Let $w$ be the first vertex of $P_{v}$ such that $w<_{L}x$ (note that possibly $w=z$). The concatenation of $P_{u}$, $zt$, and the subpath of $P_{v}$ between $z$ and $w$ has length at most $4r$ and thus shows that $w$ is strongly $4r$-reachable from $x$ in $G$. Hence, at most $\text{col}_{4r}(G)$ vertices of $H$ are strongly 1-reachable from $u$ in $H$ with respect to $L^{\prime}$, as desired. ∎

Acknowledgements

We thank Zdeněk Dvořák for the discussion about [5], Gwenaël Joret and David Wood for allowing us to mention Section 3, and Sergey Norin for allowing us to mention Section 3 and its proof.