Induced subdivisions and bounded expansion

TL;DR

This paper proves that certain graph classes excluding specific induced subgraphs have bounded expansion, providing new characterizations of such classes and strengthening previous results in graph theory.

Contribution

It establishes that classes of graphs excluding K_s, K_{s,s}, or subdivisions of H as induced subgraphs have bounded expansion, enhancing understanding of graph class properties.

Findings

Graph classes excluding specific induced subgraphs have bounded expansion.

Provides new characterizations of bounded expansion and nowhere-dense classes.

Strengthens previous results by Kuhn and Osthus.

Abstract

We prove that for every graph H and for every integer s, the class of graphs that do not contain K_s, K_{s,s}, or any subdivision of H as an induced subgraph has bounded expansion; this strengthens a result of Kuhn and Osthus. The argument also gives another characterization of graph classes with bounded expansion and of nowhere-dense graph classes.

Equations

$$g(k,n)=f(0,n)+\sum _{t=1}^k{d}_{r(t,n),t,s(n)}=f(0,n)+\sum _{t=1}^{k}O((f(t,n)f(0,n)t{)}^{12}),$$

Full text

For a non-negative integer $k$, a $(\leq\!k)$-subdivision of a graph $H$ is a graph obtained from $H$ by subdividing each of its edges by at most $k$ vertices (not necessarily the same number on each edge). For a graph $G$, let $\widetilde{\nabla}_{k}(G)$ denote the maximum average degree of a graph $H$ whose $(\leq\!k)$-subdivision appears as a subgraph in $G$; in particular, $\widetilde{\nabla}_{0}(G)$ is the maximum average degree of a subgraph of $G$. We say that a class of graphs $\mathcal{G}$ has bounded expansion if there exists a function $f:\mathbf{Z}_{0}^{+}\to\mathbf{Z}_{0}^{+}$ such that for every $G\in\mathcal{G}$ and every non-negative integer $k$, $\widetilde{\nabla}_{k}(G)\leq f(k)$. We say that $\mathcal{G}$ is nowhere-dense if there exists a function $h:\mathbf{Z}_{0}^{+}\to\mathbf{Z}_{0}^{+}$ such that for every $G\in\mathcal{G}$ and every non-negative integer $k$, $G$ does not contain a $(\leq\!k)$-subdivision of $K_{h(k)}$ as a subgraph.

There are many equivalent definitions of nowhere-dense graph classes and classes of graphs with bounded expansion [7]. For example, to clarify the relationship to bounded expansion classes, it is also possible to define nowhere dense classes in the terms of the density of graphs whose bounded depth subdivisions appear in the graphs of the class, as follows. A function $g:\mathbf{Z}_{0}^{+}\to\mathbf{Z}_{0}^{+}$ is subpolynomial if $\lim_{n\to\infty}\frac{\log g(n)}{\log n}=0$, and a function $f:\mathbf{Z}_{0}^{+}\times\mathbf{Z}_{0}^{+}\to\mathbf{Z}_{0}^{+}$ is subpolynomial in the second argument if for every non-negative integer $k$, the function $f(k,\cdot)$ is subpolynomial. A class $\mathcal{G}$ is nowhere-dense if and only if there exists a function $f:\mathbf{Z}_{0}^{+}\times\mathbf{Z}_{0}^{+}\to\mathbf{Z}_{0}^{+}$ subpolynomial in the second argument such that for every non-negative integer $k$ and graph $G\in\mathcal{G}$, each subgraph $G^{\prime}$ of $G$ satisfies $\widetilde{\nabla}_{k}(G^{\prime})\leq f(k,|V(G^{\prime})|)$.

The notions of bounded expansion and nowhere-denseness formalize in a robust way the concept of sparseness of a graph class. Such graph classes have a number of important algorithmic and structural properties, including fixed-parameter tractability of any problem expressible in first-order logic when restricted to a class with bounded expansion [2] or a nowhere-dense class [3] and existence of low tree-depth colorings [6]. Also importantly, many naturally defined graph classes have bounded expansion, including proper minor closed classes, classes with bounded maximum degree, and more generally proper classes closed on topological minors, graphs drawn in a fixed surface with a bounded number of crossings on each edge, all graph classes with strongly sublinear separators, and many others. We refer the reader to the book of Nešetřil and Ossona de Mendez [7] for a more thorough treatment of the subject.

Most of the mentioned examples of graph classes with bounded expansion are closed on taking subgraphs. Graph classes that are only closed on induced subgraphs, and in particular graph classes characterized by forbidden induced minors or induced subdivisions, has been less studied in the context. The major issue is that such classes typically contain arbitrarily large cliques or bicliques (balanced complete bipartite graphs $K_{s,s}$), which have unbounded minimum degree. However, Kühn and Osthus [5] shown that at least with regards to the maximum average degree $\widetilde{\nabla}_{0}$, this is the only obstruction.

Note that if $G$ contains a large biclique as a subgraph, applying Ramsey’s theorem to each part of the biclique gives either a large clique or a large biclique as an induced subgraph. Hence, the previous theorem can be reformulated as follows.

In this note, we show that the result of Kühn and Osthus [5] can be easily extended to prove that such graph classes actually have bounded expansion.

Theorem 3 is a consequence of the following new characterization of graph classes with bounded expansion. For a graph $G$, let $\widetilde{\nabla}^{\circ}_{k}(G)$ denote the maximum of average degrees of graphs $H$ whose $(\leq\!k)$-subdivision appears as an induced subgraph in $G$. The $k$-subdivision of a graph $H$ is the graph obtained from $H$ by subdividing each edge exactly $k$ times. Let $\widetilde{\nabla}^{e}_{k}(G)$ denote denote the maximum of average degrees of graphs $H$ whose $k$-subdivision appears as an induced subgraph in $G$.

• (a)

$\mathcal{G}$* has bounded expansion.*

• (b)

There exists a function $f:\mathbf{Z}_{0}^{+}\to\mathbf{Z}_{0}^{+}$ such that for every $G\in\mathcal{G}$ and every non-negative integer $k$, $\widetilde{\nabla}^{\circ}_{k}(G)\leq f(k)$.

• (c)

There exists a function $f:\mathbf{Z}_{0}^{+}\to\mathbf{Z}_{0}^{+}$ such that for every $G\in\mathcal{G}$ and every non-negative integer $k$, $\widetilde{\nabla}^{e}_{k}(G)\leq f(k)$.

The same argument gives a characterization of nowhere-dense graph classes.

• (a)

$\mathcal{G}$* is nowhere-dense.*

• (b)

There exists a function $f:\mathbf{Z}_{0}^{+}\times\mathbf{Z}_{0}^{+}\to\mathbf{Z}_{0}^{+}$ subpolynomial in the second argument such that for every non-negative integer $k$ and graph $G\in\mathcal{G}$, every induced subgraph $G^{\prime}$ of $G$ satisfies $\widetilde{\nabla}^{\circ}_{k}(G^{\prime})\leq f(k,|V(G^{\prime})|)$.

• (c)

There exists a function $f:\mathbf{Z}_{0}^{+}\times\mathbf{Z}_{0}^{+}\to\mathbf{Z}_{0}^{+}$ subpolynomial in the second argument such that for every non-negative integer $k$ and graph $G\in\mathcal{G}$, every induced subgraph $G^{\prime}$ of $G$ satisfies $\widetilde{\nabla}^{e}_{k}(G^{\prime})\leq f(k,|V(G^{\prime})|)$.

1 The proofs

Let $G$ be a bipartite graph with bipartition $(A,B)$. A hat over this bipartition is a $3$-vertex path in $G$ with endpoints in $B$ (and the midpoint in $A$). We say that a set $\mathcal{P}$ of hats is uncrowded if any two hats in $\mathcal{P}$ join distinct pair of vertices and have distinct midpoints. We say that it is induced if the midpoint of each hat has exactly two neighbors in $B$, i.e., $\bigcup\mathcal{P}$ is an induced subgraph of $G$.

Kühn and Osthus [5] proved the following.

Lemma 6** (Kühn and Osthus [5, Lemma 18]).**

Let $r$ be a positive integer and let $G$ be a bipartite graph with bipartition $(A,B)$, such that each vertex of $A$ has degree at most $4r$. If $G$ contains an uncrowded set of at least $\tfrac{r^{11}}{2^{8}}|B|$ hats, then $G$ has an induced subgraph $G^{\prime}$ with bipartition $(A^{\prime},B^{\prime})=(A\cap V(G^{\prime}),B\cap V(G^{\prime}))$ such that $B^{\prime}\neq\emptyset$ and the set of all $3$-vertex paths in $G^{\prime}$ with midpoints in $A^{\prime}$ forms an induced uncrowded set of at least $\tfrac{r^{9}}{2^{15}}|B^{\prime}|$ hats over $(A^{\prime},B^{\prime})$.

We also need another lemma of Kühn and Osthus [5] (the branch vertices of a subdivision of a graph $H$ are the vertices of the subdivision corresponding to the original vertices of $H$).

Lemma 7** (Kühn and Osthus [5, Lemma 20]).**

Let $r\geq 2^{25}$ be an integer. Let $(A,B)$ be a partition of vertices of a graph $G$ such that $A$ is an independent set of $G$, $\chi(G[B])\leq r$ and $\widetilde{\nabla}_{0}(G[B])\leq r^{3}$. Let $G^{\prime}$ be the spanning bipartite subgraph of $G$ containing exactly the edges of $G$ with one end in $A$ and the other end in $B$. If $G^{\prime}$ contains an induced uncrowded set of at least $\tfrac{r^{9}}{2^{15}}|B|$ hats over $(A,B)$, then $G$ contains an induced subgraph $G^{\prime\prime}$ such that $G^{\prime\prime}$ is the $1$-subdivision of a graph of average degree at least $r$, with all branch vertices contained in $B$.

Combining these lemmas with Theorem 1 gives the following result.

Lemma 8**.**

For all integers $r,k,s\geq 1$, there exists an integer $d_{r,k,s}=O((rsk)^{12})$ such that for every graph $G$, if $\widetilde{\nabla}_{0}(G)\leq s$ and $\widetilde{\nabla}^{e}_{k}(G)<r$, then $\widetilde{\nabla}_{k}(G)<\widetilde{\nabla}_{k-1}(G)+d_{r,k,s}$.

Proof.

Without loss of generality, we can assume that $r\geq\max(2^{25},s+1,sk/2)$. Let $d_{r,k,s}=\tfrac{r^{11}(sk+1)}{2^{6}}$.

Suppose for a contradiction that $\widetilde{\nabla}_{k}(G)\geq\widetilde{\nabla}_{k-1}(G)+d_{r,k,s}$. Let $H$ be a graph of average degree at least $\widetilde{\nabla}_{k-1}(G)+d_{r,k,s}$ whose $(\leq\!k)$-subdivision appears as a subgraph of $G$. That is, there exists a function $\varphi$ that assigns to vertices of $H$ distinct vertices of $G$ and to edges of $H$ paths of length at most $k+1$ in $G$, such that for every $uv\in E(H)$, the path $\varphi(uv)$ has endpoints $\varphi(u)$ and $\varphi(v)$ and its internal vertices do not belong to $\varphi(V(H))$, and for distinct edges $e_{1},e_{2}\in E(H)$, the paths $\varphi(e_{1})$ and $\varphi(e_{2})$ do not intersect except possibly in their endpoints. Without loss of generality, we can assume that $\varphi(e)$ is an induced path in $G$ for every $e\in E(H)$. Let $B=\varphi(V(H))$. Let $H_{1}$ be the subgraph of $H$ consisting of the edges such that the path $\varphi(e)$ has length exactly $k+1$. Since $H$ has average degree at least $\widetilde{\nabla}_{k-1}(G)+d_{r,k,s}$ and a $(\leq\!k-1)$-subdivision of $H-E(H_{1})$ is a subgraph of $G$, we conclude that the graph $H_{1}$ has average degree at least $d_{r,k,s}$.

Let $G_{1}$ be an auxiliary graph with vertex set $E(H_{1})$, such that distinct $e_{1},e_{2}\in E(H_{1})$ are adjacent in $G_{1}$ if and only if there exists an edge of $G$ with one end in an internal vertex of $\varphi(e_{1})$ and the other end in an internal vertex of $\varphi(e_{2})$. Let $C$ be any subset of $E(H_{1})=V(G_{1})$, and let $D$ be the union of internal vertices of the paths $\varphi(e)$ for $e\in C$; we have $|D|=k|C|$, and $|E(G_{1}[C])|\leq|E(G[D])|\leq\tfrac{\widetilde{\nabla}_{0}(G)}{2}|D|\leq\tfrac{sk}{2}|C|$. Hence, $\widetilde{\nabla}_{0}(G_{1})\leq sk$, and in particular $\chi(G_{1})\leq sk+1$. Hence, $G_{1}$ contains an independent set $S$ of size at least $\tfrac{|E(H_{1})|}{sk+1}$. Let $H_{2}$ be the spanning subgraph of $H_{1}$ with edge set $S$, and note that the average degree of $H_{2}$ is at least $\tfrac{d_{r,k,s}}{sk+1}=\tfrac{r^{11}}{2^{6}}$. By the choice of $S$, for any distinct $e_{1},e_{2}\in E(H_{2})$, there are no edges between the internal vertices of $\varphi(e_{1})$ and $\varphi(e_{2})$ in $G$.

Let $A_{2}=E(H_{2})$ and let $G_{2}$ be an auxiliary bipartite graph with bipartition $(A_{2},B)$, where $ev$ with $e\in E(H_{2})$ and $v\in B$ is an edge of $G_{2}$ if and only if there exists an edge of $G$ between an internal vertex of $\varphi(e)$ and $v$. Let $D_{2}$ be the union of internal vertices of the paths $\varphi(e)$ for $e\in A_{2}$. Since the average degree of $H_{2}$ is greater than $2$, we have $|B|<|A_{2}|\leq|D_{2}|$. Note that $|E(G_{2})|\leq|E(G[D_{2}\cup B])|\leq\tfrac{\widetilde{\nabla}_{0}(G)}{2}|D_{2}\cup B|\leq s|D_{2}|\leq sk|A_{2}|\leq 2r|A_{2}|$. Hence, less than half of the vertices of $A_{2}$ have degree more than $4r$ in $G_{2}$. Let $A_{2}^{\prime}\subseteq A_{2}$ consist of vertices of $A_{2}$ whose degree in $G_{2}$ is at most $4r$, and let $G^{\prime}_{2}$ be the induced subgraph of $G_{2}$ with bipartition $(A^{\prime}_{2},B)$. We have $|A_{2}^{\prime}|\geq|A_{2}|/2=|E(H_{2})|/2\geq\tfrac{r^{11}}{2^{8}}|B|$. Let $\mathcal{P}$ be the set of paths $uev$ in $G_{2}$, where $e\in A_{2}^{\prime}$ and $u$ and $v$ are the endpoints of $\varphi(e)$; then $\mathcal{P}$ is an uncrowded set of at least $\tfrac{r^{11}}{2^{8}}|B|$ hats over $(A^{\prime}_{2},B)$. Let $G_{3}$ with bipartition $(A_{3},B_{3})$ be the induced subgraph of $G^{\prime}_{2}$ obtained by applying Lemma 6, and let $H_{3}$ be the subgraph of $H_{2}$ with vertex set $\varphi^{-1}(B_{3})$ and edge set $A_{3}$. We have $|E(H_{3})|=|A_{3}|\geq\tfrac{r^{9}}{2^{15}}|B_{3}|$, and for each $e\in E(H_{3})$, the internal vertices of $\varphi(e)$ have no neighbors in $B_{3}$ other than the endpoints of $\varphi(e)$.

Finally, we consider an auxiliary graph $G_{4}=G_{3}\cup G[B_{3}]$. Note that $\widetilde{\nabla}_{0}(G_{3}[B_{3}])=\widetilde{\nabla}_{0}(G[B_{3}])\leq s\leq r^{3}$ and $\chi(G_{3}[B_{3}])\leq\widetilde{\nabla}_{0}(G[B_{3}])+1\leq s+1\leq r$. Let $G_{5}$ be the induced bipartite subgraph of $G_{4}$ obtained by applying Lemma 7, with bipartition $(A_{5},B_{5})=(A_{3}\cap V(G_{5}),B_{3}\cap V(G_{5}))$. Then $\varphi$ restricted to $A_{5}\cup B_{5}$ shows that $G$ contains the $k$-subdivision of a graph of average degree at least $r$ as an induced subgraph, and thus $\widetilde{\nabla}^{e}_{k}(G)\geq r$; this is a contradiction. ∎

The new characterizations of classes with bounded expansion and nowhere-dense classes now readily follow.

Proof of Theorem 4.

For every graph $G$ and a non-negative integer $k$, we have $\widetilde{\nabla}_{k}(G)\geq\widetilde{\nabla}^{\circ}_{k}(G)\geq\widetilde{\nabla}^{e}_{k}(G)$, and thus $\text{(a)}\Rightarrow\text{(b)}\Rightarrow\text{(c)}$.

Suppose now that $f:\mathbf{Z}_{0}^{+}\to\mathbf{Z}_{0}^{+}$ is a function such that for every $G\in\mathcal{G}$ and every non-negative integer $k$, $\widetilde{\nabla}^{e}_{k}(G)\leq f(k)$. Let $g(0)=f(0)$ and for every positive integer $k$, let $d_{r,k,s}$ be the constant from Lemma 8, where $r=f(k)+1$ and $s=f(0)$, and let $g(k)=g(k-1)+d_{r,k,s}$. Consider a graph $G\in\mathcal{G}$. By induction, we will show that $\widetilde{\nabla}_{k}(G)\leq g(k)$ for every non-negative integer $k$. For $k=0$, we have $\widetilde{\nabla}_{0}(G)=\widetilde{\nabla}^{e}_{0}(G)\leq f(0)=g(0)$; hence, we can assume that $k\geq 1$ and that $\widetilde{\nabla}_{k-1}(G)\leq g(k-1)$. However, then Lemma 8 implies $\widetilde{\nabla}_{k}(G)<\widetilde{\nabla}_{k-1}(G)+d_{f(k)+1,k,f(0)}\leq g(k-1)+d_{f(k)+1,k,f(0)}=g(k)$. We conclude that $\mathcal{G}$ has bounded expansion, and thus $\text{(c)}\Rightarrow\text{(a)}$. ∎

Proof of Theorem 5.

Again, the implications $\text{(a)}\Rightarrow\text{(b)}\Rightarrow\text{(c)}$ are trivial.

Suppose now that $f:\mathbf{Z}_{0}^{+}\times\mathbf{Z}_{0}^{+}\to\mathbf{Z}_{0}^{+}$ is a function subpolynomial in the second argument such that for every non-negative integer $k$ and graph $G\in\mathcal{G}$, every induced subgraph $G^{\prime}$ of $G$ satisfies $\widetilde{\nabla}^{e}_{k}(G^{\prime})\leq f(k,|V(G^{\prime})|)$. We can without loss of generality assume that $f$ is non-decreasing in the second argument. Let us define $g(0,n)=f(0,n)$. For any positive integer $k$, let $d_{r(k,n),k,s(n)}$ be the constant from Lemma 8, where $r(k,n)=f(k,n)+1$ and $s(n)=f(0,n)$, and let $g(k,n)=g(k-1,n)+d_{r(k,n),k,s(n)}$. As in the proof of Theorem 4, induction on $k$ shows that $\widetilde{\nabla}_{k}(G^{\prime})\leq g(k,|V(G^{\prime})|)$ for every (induced) subgraph $G^{\prime}$ of a graph $G\in\mathcal{G}$. Furthermore, for any fixed $k$,

[TABLE]

and thus $g$ is subpolynomial in the second argument. It follows that $\mathcal{G}$ is nowhere-dense, and thus $\text{(c)}\Rightarrow\text{(a)}$. ∎

To derive Theorem 3, we also need the following result.

Theorem 9** (Komlós and Szemerédi [4], Bollobás and Thomason [1]).**

There exists $c>0$ such that for every positive integer $n$, every graph of average degree at least $cn^{2}$ contains a subdivision of $K_{n}$ as a subgraph.

Proof of Theorem 3.

Let $n=|V(H)|$, and let $c$ be the constant from Theorem 9. Let $d$ be the constant of Corollary 2 for $H$ and $s$. Let $f(0)=d$ and let $f(k)=cn^{2}$ for every $k\geq 1$.

Consider any graph $G\in\mathcal{G}$. By assumptions and Corollary 2, every induced subgraph of $G$ has average degree at most $d$, and thus $\widetilde{\nabla}^{e}_{0}(G)\leq f(0)$. Consider any positive integer $k$ and let $H_{1}$ be any graph whose $k$-subdivision appears in $G$ as an induced subgraph $H^{\prime}_{1}$. Since $H^{\prime}_{1}$ does not contain a subdivision of $H$ as an induced subgraph, and each edge of $H_{1}$ is subdivided at least once to obtain $H^{\prime}_{1}$, we conclude that $H_{1}$ does not contain a subdivision of $H$ as a subgraph. Consequently, $H_{1}$ does not contain a subdivision of $K_{n}$ as a subgraph, and the average degree of $H_{1}$ is less than $cn^{2}$ by Theorem 9. It follows that $\widetilde{\nabla}^{e}_{k}(G)<cn^{2}=f(k)$ for every positive integer $k$. Hence, $\mathcal{G}$ has bounded expansion by Theorem 4. ∎

Acknowledgments

I would like to thank Ken-ichi Kawarabayashi for fruitful discussions that motivated this result, and in particular for pointing my attention to the result of Kühn and Osthus.