The Grid-Minor Theorem Revisited

The seminal Graph Minors series of Robertson and Seymour is the foundation of modern structural graph theory. In this work, treewidth is a central concept that measures how similar a given graph is to a tree. A key theorem of Robertson and Seymour [38] states that a minor-closed graph class $\mathcal{G}$ has bounded treewidth if and only if some planar graph is not in $\mathcal{G}$. In particular, for every planar graph $X$, every $X$-minor-free graph has treewidth at most some function $g(X)$. This result is often called the Grid-Minor Theorem since it suffices to prove it when $X$ is a planar grid. The asymptotics of $g$ have been substantially improved since the original work. Most significantly, Chekuri and Chuzhoy [4] showed that $g$ can be chosen to be polynomial in $|V(X)|$. The current best bound is $g(X)\in\widetilde{\mathcal{O}}(|V(X)|^{9})$, which follows from a result of Chuzhoy and Tan [5]. Dependence on $|V(X)|$ is unavoidable, since the complete graph on $|V(X)|-1$ vertices is $X$-minor-free, but has treewidth $|V(X)|-2$. Our goal is to prove a qualitative strengthening of the Grid-Minor Theorem via graph product structure theory.

Graph product structure theory describes graphs in complicated classes as subgraphs of strong products¹¹1The strong product $G_{1}\boxtimes G_{2}$ of two graphs $G_{1}$ and $G_{2}$ is the graph with vertex set $V(G_{1}\boxtimes G_{2}):=V(G_{1})\times V(G_{2})$ and that includes the edge with endpoints $(v,x)$ and $(w,y)$ if and only if $vw\in E(G_{1})$ and $x=y$; $v=w$ and $xy\in E(G_{2})$; or $vw\in E(G_{1})$ and $xy\in E(G_{2})$. of simpler graphs. For example, the Planar Graph Product Structure Theorem by Dujmović, Joret, Micek, Morin, Ueckerdt, and Wood [11] says that for every planar graph $G$ there is a graph $H$ of treewidth at most $3$ and a path $P$ such that $G\mathrel{\ooalign{\raise 0.75348pt\hbox{$\subset$}\cr\raise-3.87495pt\hbox{\scalebox{0.9}{$\sim$}}\cr}}H\boxtimes P\boxtimes K_{3}$. Here, $G_{1}\mathrel{\ooalign{\raise 0.75348pt\hbox{$\subset$}\cr\raise-3.87495pt\hbox{\scalebox{0.9}{$\sim$}}\cr}}G_{2}$ means that $G_{1}$ is isomorphic to a subgraph of $G_{2}$.

Inspired by this viewpoint, we prove the following product structure extension of the Grid-Minor Theorem. Note that $H\boxtimes K_{c}$ is the graph obtained from $H$ by ‘blowing-up’ each vertex of $H$ by a complete graph $K_{c}$. Let $\operatorname{tw}(G)$ denote the treewidth of a graph $G$, and let $\operatorname{td}(G)$ denote the treedepth of $G$ (both defined in Section˜2).

The point of Theorem 1 is that the treewidth of $H$ only depends on the treedepth of $X$, not on $|V(X)|$ ²²2Arbitrarily large graphs can have bounded treedepth, such as edgeless graphs (treedepth 1) or stars (treedepth 2).. The described product structure of $G$ is a more refined description of $G$ compared to the output of the Grid-Minor Theorem since $\operatorname{tw}(H\boxtimes K_{c})\leqslant(\operatorname{tw}(H)+1)c-1$. This refinement is useful because various graph parameters can be bounded on $H\boxtimes K_{c}$ by a fast-growing function of $\operatorname{tw}(H)$ times a slow-growing (usually linear) function of $c$; this includes queue-number [11], nonrepetitive chromatic number [12], and others [15, 2]. As concrete applications of Theorem˜1, we use it to improve bounds for weak coloring numbers and $p$-centered colorings of $X$-minor-free graphs.

The Grid-Minor Theorem relates to treewidth in the same way as the Excluded-Tree-Minor Theorem by Robertson and Seymour [37] relates to pathwidth. The latter says that a minor-closed class $\mathcal{G}$ has bounded pathwidth if and only if some tree is not in $\mathcal{G}$. In particular, there is a function $g$ such that for every tree $X$, every $X$-minor-free graph has pathwidth at most $g(|V(X)|)$. The following product structure version of this result was proved by Dujmović, Hickingbotham, Joret, Micek, Morin, and Wood [13]: there exists a function $f$ such that for every tree $X$, there exists a positive integer $c$ such that for every $X$-minor-free graph $G$, there exists a graph $H$ of pathwidth at most $f(\operatorname{td}(X))$ such that $G\mathrel{\ooalign{\raise 0.75348pt\hbox{$\subset$}\cr\raise-3.87495pt\hbox{\scalebox{0.9}{$\sim$}}\cr}}H\boxtimes K_{c}$. (In fact, they prove a stronger statement in which the pathwidth of $H$ is bounded by $2h-1$, where $h$ is the radius of $X$.)

We actually prove the following result for an arbitrary excluded minor, which combined with the Grid-Minor Theorem immediately implies Theorem˜1.

Theorem˜2 was inspired by and can be restated in terms of the parameter ‘underlying treewidth’ introduced by Campbell, Clinch, Distel, Gollin, Hendrey, Hickingbotham, Huynh, Illingworth, Tamitegama, Tan, and Wood [3]. They defined the underlying treewidth of a graph class $\mathcal{G}$, denoted by $\operatorname{utw}(\mathcal{G})$, to be the smallest integer such that, for some function $f$, for every graph $G\in\mathcal{G}$ there is a graph $H$ of treewidth at most $\operatorname{utw}(\mathcal{G})$ such that $G\mathrel{\ooalign{\raise 0.75348pt\hbox{$\subset$}\cr\raise-3.87495pt\hbox{\scalebox{0.9}{$\sim$}}\cr}}H\boxtimes K_{f(\operatorname{tw}(G))}$. Here, $f$ is called the treewidth binding function. Campbell et al. [3] showed that the underlying treewidth of the class of planar graphs equals 3, and the same holds for any fixed surface. More generally, let $\mathcal{G}_{X}$ be the class of graphs excluding a given graph $X$ as a minor. Campbell et al. [3] showed that $\operatorname{utw}(\mathcal{G}_{K_{t}})=t-2$ and $\operatorname{utw}(\mathcal{G}_{K_{s,t}})=s$ (for $t\geqslant\max\{s,3\})$. In these results, the treewidth binding function is quadratic. Illingworth, Scott and Wood [23] reproved these results with a linear treewidth binding function.

Determining the underlying treewidth of the class of $X$-minor-free graphs, for an arbitrary graph $X$, was one of the main problems left unsolved by Campbell et al. [3] and Illingworth, Scott and Wood [23]. Theorem˜2 together with a well-known lower bound construction given in Section˜3 shows that $\operatorname{utw}(\mathcal{G}_{X})$ and $\operatorname{td}(X)$ are tied:

This shows that treedepth is the right parameter to consider in Theorems˜1 and 2. Moreover, in the upper bound the treewidth binding function is linear.

Application #1. Weak Coloring Numbers. Weak coloring numbers are a family of graph parameters studied extensively in structural and algorithmic graph theory. See the book by Nešetřil and Ossona de Mendez [31], or the recent lecture notes by Pilipczuk, Pilipczuk, and Siebertz [34] for more information on this topic. For the algorithmic side, see Dvořák [14] and also Theorem 5.2 in [34], which contains a polynomial-time approximation algorithm for $r$-dominating set, with approximation ratio bounded by a function of a weak coloring number of the input graph. We now quickly introduce the definition. The length of a path is the number of its edges. For two vertices $u$ and $v$ in a graph $G$, a $u$–$v$ path is a path in $G$ with endpoints $u$ and $v$. Let $G$ be a graph and let $\sigma$ be an ordering of the vertices of $G$. For an integer $r\geqslant 0$ and two vertices $u$ and $v$ of $G$, we say that $u$ is weakly $r$-reachable from $v$ in $\sigma$, if there exists a $u$–$v$ path of length at most $r$ such that for every vertex $w$ on the path, $u\leqslant_{\sigma}w$. The set of vertices that are weakly $r$-reachable from a vertex $v$ in $\sigma$ is denoted by $\operatorname{WReach}_{r}[G,\sigma,v]$. The $r$-th weak coloring number of $G$, denoted by $\operatorname{wcol}_{r}(G)$, is defined as

where $\sigma$ ranges over the set of all vertex orderings of $G$. Several papers give bounds for weak coloring numbers of graphs in a given sparse class. For example, if $G$ is planar, then $\operatorname{wcol}_{r}(G)=\mathcal{O}(r^{3})$; if $G$ has no $K_{t}$-minor, then $\operatorname{wcol}_{r}(G)=\mathcal{O}(r^{t-1})$ as proved by van den Heuvel, Ossona de Mendez, Quiroz, Rabinovich, and Siebertz [21]; and if $\operatorname{tw}(G)\leqslant t$, then $\operatorname{wcol}_{r}(G)\leqslant\binom{r+t}{t}$ and this bound is tight as proved by Grohe, Kreutzer, Rabinovich, Siebertz, and Stavropoulos [20].

Fix a planar graph $X$. What is known about $\operatorname{wcol}_{r}(G)$ when $X$ is not a minor of $G$? Since $G$ is $K_{|V(X)|}$-minor-free, we have $\operatorname{wcol}_{r}(G)=\mathcal{O}(r^{|V(X)|-1})$. However, thanks to Theorem˜1, there exists a graph $H$ with $\operatorname{tw}(H)\leqslant f(\operatorname{td}(X))=\mathcal{O}(2^{\operatorname{td}(X)})$ and $c$ depending only on $X$ such that $G\mathrel{\ooalign{\raise 0.75348pt\hbox{$\subset$}\cr\raise-3.87495pt\hbox{\scalebox{0.9}{$\sim$}}\cr}}H\boxtimes K_{c}$. Therefore,

Indeed, the first inequality follows from the monotonicity of $\operatorname{wcol}_{r}$ and the second inequality is an easy property³³3 Let $\sigma_{H}$ be an ordering of $V(H)$ such that $|\operatorname{WReach}_{r}[H,\sigma,x]|\leqslant\operatorname{wcol}_{r}(H)$ for every $x\in V(H)$. Consider $\sigma$ an ordering of $V(H\boxtimes K_{c})$ such that for every $(x,u),(x^{\prime},u^{\prime})\in V(H\boxtimes K_{x})$ if $x<_{\sigma_{H}}x^{\prime}$, then $(x,u)<_{\sigma}(x^{\prime},u^{\prime})$. It is easy to see that $\operatorname{WReach}_{r}[H\boxtimes K_{c},\sigma,(x,u)]\subseteq\{(y,v)\in V(H\boxtimes K_{c})\mid y\in\operatorname{WReach}_{r}[H,\sigma_{H},x]\}$, and so $|\operatorname{WReach}_{r}[H\boxtimes K_{c},\sigma,u]|\leqslant c\cdot\operatorname{wcol}_{r}(H)$. of $\operatorname{wcol}_{r}$. The obtained upper bound on $\operatorname{wcol}_{r}(G)$ is polynomial in $r$, where the exponent depends only on $\operatorname{td}(X)$ and not on $|V(X)|$.

Again, the point of Theorem˜3 is that the degree of the polynomial in $r$ bounding $\operatorname{wcol}_{r}(G)$ depends only on $\operatorname{td}(X)$ and not on $|V(X)|$. In the previous best bound for weak colouring number of $X$-minor free graphs, the degree of the polynomial in $r$ depended on the vertex-cover⁴⁴4The vertex-cover number $\tau(G)$ of a graph $G$ is the size of a smallest set $S\subseteq V(G)$ such that every edge of $G$ has at least one endpoint in $S$. number $\tau(X)$. In particular, it follows from a result by van den Heuvel and Wood [41, Proposition 28] regarding the weak colouring number of $K_{s,t}^{\star}$-minor-free graphs that $\operatorname{wcol}_{r}(G)\leqslant c\cdot r^{\tau(X)+1}$ for every $X$-minor-free graph $G$ and integer $r\geqslant 1$. Theorem˜3 is qualitatively stronger since $\operatorname{td}(X)\leqslant\tau(X)+1$ and there are graphs $X$ with $\operatorname{td}(X)=3$ and arbitrarily large $\tau(X)$.

The proof of the theorem relies on the same decomposition lemma as the proof of Theorem˜1. The ordering of the vertices witnessing the bound on $\operatorname{wcol}_{r}$ in Theorem˜3 is built via chordal partitions—a powerful proof technique originally developed in the 1980s in the context of the cops and robber game [1] that was rediscovered and used in [21] to bound weak coloring numbers, and has subsequently found several other applications in structural graph theory.

Application #2. Product Structure for Apex-Minor-Free Graphs. As already mentioned, Dujmović et al. [11] proved that every planar graph is isomorphic to a subgraph of $H\boxtimes P\boxtimes K_{3}$ for some graph $H$ with treewidth at most 3 and for some path $P$. This result has been the key ingredient in the solution of several open problems on planar graphs [11, 12, 17, 8]. Building on this work, Distel, Hickingbotham, Huynh, and Wood [7] proved that every graph of Euler genus $g$ is isomorphic to a subgraph of $H\boxtimes P\boxtimes K_{\max\{2g,3\}}$ for some graph $H$ with treewidth at most 3 and for some path $P$. More generally, Dujmović et al. [11] characterized the graphs $X$ for which there exist integers $t$ and $c$ such that every $X$-minor-free graph is isomorphic to a subgraph of $H\boxtimes P\boxtimes K_{c}$ where $\operatorname{tw}(H)\leqslant t$ and $P$ is a path. The answer is precisely the apex graphs. Here a graph $X$ is apex if $V(X)=\emptyset$ or $X-u$ is planar for some vertex $u$ of $X$. The following natural problem arises: for a given apex graph $X$, what is the minimum integer $t(X)$ such that, for some integer $c$, every $X$-minor-free graph is isomorphic to a subgraph of $H\boxtimes P\boxtimes K_{c}$ where $\operatorname{tw}(H)\leqslant t(X)$ and $P$ is a path? Illingworth, Scott and Wood [23] showed that $t(X)\leqslant\tau(X)$. We show, via an application of Theorem˜2, that $t$ is tied to treedepth. In particular,

The proof of this result is presented in Section˜8.

Application #3. $p$-Centered Colorings. Theorem˜1 can also be used to improve bounds for $p$-centered chromatic numbers of graphs excluding a fixed minor. For an integer $p\geqslant 1$, a vertex coloring $\phi$ of a graph $G$ is $p$-centered if for every connected subgraph $H$ of $G$ either $\phi$ uses more than $p$ colors in $H$ or there is a color that appears exactly once in $H$. The $p$-centered chromatic number of $G$, denoted by $\chi_{p}(G)$, is the least number of colors in a $p$-centered coloring of $G$. Centered colourings are important since they characterize graph classes of bounded expansion [31], and are a central tool for designing parameterized algorithms in classes of bounded expansion [35, 36]; see [15] for an overview.

If $\operatorname{tw}(G)\leqslant t$ then $\chi_{p}(G)\leqslant\binom{p+t}{t}$, and this bound is again tight [35, 15]. If $X$ is a planar graph and $G$ is $X$-minor-free, then by Theorem˜1, there exists a graph $H$ with $\operatorname{tw}(H)\leqslant f(\operatorname{td}(X))$ and $c$ depending only on $X$ such that $G\mathrel{\ooalign{\raise 0.75348pt\hbox{$\subset$}\cr\raise-3.87495pt\hbox{\scalebox{0.9}{$\sim$}}\cr}}H\boxtimes K_{c}$. Therefore,

where the first inequality follows from the monotonicity of $\chi_{p}$ and the second inequality is an easy property of $p$-centered colorings (Lemma 8 in [15]). The obtained upper bound on $\chi_{p}(G)$ is polynomial in $p$, where the exponent depends only on $\operatorname{td}(X)$. Similarly, for an apex graph $X$, we use (2) to show that for some $c=c(X)$, every $X$-minor-free graph $G$ satisfies $\chi_{p}(G)\leqslant c\cdot p^{f(\operatorname{td}(X))}$.

Outline. Section˜2 gives all the necessary definitions, as well as some preliminary results about tree-decompositions. Section˜3 proves the lower bounds in (1) and (2). Section˜4 provides a decomposition lemma (Corollary˜13) for graphs avoiding an ‘attached model’ of a fixed graph, which is a key ingredient in the results that follow. This part of the argument, in particular Lemma˜12, is inspired by a result of Kawarabayashi [24] on rooted minors, which in turn is inspired by results of Robertson and Seymour [39]. Section˜5 contains the proof of Theorem˜2. Section˜6 contains the proof of Theorem˜3, which relies on chordal partitions (see Lemma 19 and Figure 5), and a variant of the Helly property for $K_{t}$-minor-free graphs that is of independent interest (see Lemma˜21). Section 7 contains the proof of Lemma˜21. This part builds on the work of Pilipczuk and Siebertz [35] for bounded genus graphs, and on the Graph Minor Structure Theorem of Robertson and Seymour [40] for $K_{t}$-minor-free graphs. Section˜8 presents a product structure decomposition for graphs excluding an apex graph of small treedepth as a minor. As a consequence, we obtain better bounds for the $p$-centered chromatic number of such graphs. Section˜9 concludes with four questions that we find relevant and exciting for future work.

For a positive integer $k$, we use the notation $[k]=\{1,\ldots,k\}$, and when $k=0$ let $[k]=\emptyset$. The empty graph is the graph with no vertices. All graphs considered in this paper are finite and may be empty.

Let $G$ be a graph. Recall that the length of a path $P$, denoted by $\operatorname{len}(P)$, is the number of edges of $P$. The distance between two vertices $u$ and $v$ in $G$, denoted by $\mathrm{dist}_{G}(u,v)$, is the minimal length of a path with endpoints $u,v$ in $G$, if such a path exists, and $+\infty$ otherwise. A path $P$ is a geodesic in $G$ if it is a shortest path between its endpoints in $G$.

Let $u$ be a vertex of $G$. The neighborhood of $u$ in $G$, denoted by $N_{G}(u)$, is the set $\{v\in V(G)\mid uv\in E(G)\}$. For every set $X$ of vertices of $G$, let $N_{G}(X)=\bigcup_{u\in X}N_{G}(u)-X$. For every integer $r\geqslant 1$, we denote by $N_{G}^{r}[u]=\{v\in V(G)\mid\mathrm{dist}_{G}(u,v)\leqslant r\}$. We omit $G$ in the subscripts when it is clear from the context.

A rooted forest is a disjoint union of rooted trees. The vertex-height of a rooted forest $F$ is the maximum number of vertices on a path from a root to a leaf in $F$. For two vertices $u$, $v$ in a rooted forest $F$, we say that $u$ is a descendant of $v$ in $F$ if $v$ lies on the path from a root to $u$ in $F$. The closure of $F$ is the graph with vertex set $V(F)$ and edge set $\{vw\mid v\text{ is a strict descendant of $w$ in $F$}\}$. The treedepth of a graph $G$, denoted by $\operatorname{td}(G)$, is $0$ if $G$ is empty, and otherwise is the minimum vertex-height of a rooted forest $F$ with $V(F)=V(G)$ such that $G$ is a subgraph of the closure of $F$.

Consider the following family of graphs $\{U_{h,d}\}$. For every positive integer $d$, define $U_{0,d}$ to be the empty graph. For all positive integers $h$ and $d$, define $U_{h,d}$ to be the closure of the disjoint union of $d$ complete $d$-ary trees of vertex-height $h$. Observe that $U_{h,d}$ has treedepth $h$. Moreover, this family of graphs is universal for graphs of bounded treedepth: For every graph $X$ of treedepth at most $h$, there exists $d$ such that $X\mathrel{\ooalign{\raise 0.75348pt\hbox{$\subset$}\cr\raise-3.87495pt\hbox{\scalebox{0.9}{$\sim$}}\cr}}U_{h,d}$. Thus, every $X$-minor-free graph is $U_{h,d}$-minor-free, and to prove Theorem˜2 it suffices to do so for $X=U_{h,d}$.

A tree-decomposition $\mathcal{W}$ of a graph $G$ is a pair $(T,(W_{x}\mid x\in V(T)))$, where $T$ is a tree and the sets $W_{x}$ for each $x\in V(T)$ are subsets of $V(G)$ called bags satisfying:

1. (1)

   for each edge $uv\in E(G)$ there is a bag containing both $u$ and $v$, and

2. (2)

   for each vertex $v\in V(G)$ the set of vertices $x\in V(T)$ with $v\in W_{x}$ induces a non-empty subtree of $T$.

The width of $\mathcal{W}$ is $\max\{|W_{x}|-1\mid x\in V(T)\}$, and its adhesion is $\max\{|W_{x}\cap W_{y}|\mid xy\in E(T)\}$. The treewidth of a graph $G$, denoted $\operatorname{tw}(G)$, is the minimum width of a tree-decomposition of $G$.

A clique in a graph is a set of pairwise adjacent vertices. Given two graphs $G_{1},G_{2}$, a clique $K^{1}$ in $G_{1}$, a clique $K^{2}$ in $G_{2}$, a function $f:K^{2}\to K^{1}$, the clique-sum of $G_{1}$ and $G_{2}$ according to $f$ is the graph $G$ obtained from the disjoint union of $G_{1}$ and $G_{2}$ by identifying $x$ with $f(x)$ for every $x\in K^{2}$. Note that $f$ does not have to be injective. It is well known that $\operatorname{tw}(G)\leqslant\max\{\operatorname{tw}(G_{1}),\operatorname{tw}(G_{2})\}$.

Given two graphs $G$ and $H$, an $H$-partition of $G$ is a partition $(V_{x}\mid x\in V(H))$ of $V(G)$ with possibly empty parts such that for all distinct $x,y\in V(H)$, if there is an edge between $V_{x}$ and $V_{y}$ in $G$, then $xy\in E(H)$. The width of such an $H$-partition is $\max\{|V_{x}|\colon x\in V(H)\}$.

A partition of a graph $G$ is a family $\mathcal{P}$ of induced subgraphs of $G$ such that every vertex in $G$ is in the vertex set of exactly one member of $\mathcal{P}$. Given a partition $\mathcal{P}$ of $G$, define $G/\mathcal{P}$ to be the graph with vertex set $\mathcal{P}$ and edge set all the pairs $P,Q\in\mathcal{P}$ such that there is an edge between $V(P)$ and $V(Q)$ in $G$.

A layering of a graph $G$ is a sequence $(L_{0},L_{1},\ldots)$ of disjoint subsets of $V(G)$ whose union is $V(G)$ and such that for every edge $uv$ of $G$ there is a non-negative integer $i$ such that $u,v\in L_{i}\cup L_{i+1}$. A layered partition of $G$ is a pair $(\mathcal{P},\mathcal{L})$ where $\mathcal{P}$ is a vertex partition of $G$ and $\mathcal{L}$ is a layering of $G$.

We finish these preliminaries with three simple statements on tree-decompositions.

The following technical lemma encapsulates a step in the main proof. In this lemma, we “capture” a given set of vertices $Y$ with a superset $X$ such that $X$ is not too large and each component of $G-X$ has a bounded number of neighbors in $X$.

This section explores connections between our results and improper graph colorings, which lead to the lower bound on $\operatorname{utw}(\mathcal{G}_{X})$ in (1) and the lower bound on $t(X)$ in (2).

A graph $G$ is $k$-colorable with defect $d$ if each vertex can be assigned one of $k$ colors such that each monochromatic subgraph has maximum degree at most $d$. A graph $G$ is $k$-colorable with clustering $c$ if each vertex can be assigned one of $k$ colors such that each monochromatic connected subgraph has at most $c$ vertices. The defective chromatic number of a graph class $\mathcal{G}$ is the minimum integer $k$ such that for some integer $d$, every graph in $\mathcal{G}$ is $k$-colorable with defect $d$. Similarly, the clustered chromatic number of a graph class $\mathcal{G}$ is the minimum integer $k$ such that for some integer $c$, every graph in $\mathcal{G}$ is $k$-colorable with clustering $c$. These topics have been widely studied in recent years; see [9, 41, 33, 32, 28, 27, 29, 25, 42, 16, 26] for example. Clustered coloring is closely related to the results in this paper, since a graph $G$ is $k$-colorable with clustering $c$ if and only if $G\mathrel{\ooalign{\raise 0.75348pt\hbox{$\subset$}\cr\raise-3.87495pt\hbox{\scalebox{0.9}{$\sim$}}\cr}}H\boxtimes K_{c}$ for some graph $H$ with $\chi(H)\leqslant k$. Our results are stronger in that they replace the condition $\chi(H)\leqslant k$ by the qualitatively stronger statement that $\operatorname{tw}(H)\leqslant k$ (since $\chi(H)\leqslant\operatorname{tw}(H)+1$). Of course, this is only possible when $G$ itself has bounded treewidth.

The treedepth of $X$ is the right parameter to consider when studying the defective or clustered chromatic number of the class of $X$-minor-free graphs. Fix any connected⁵⁵5These results hold for possibly disconnected $X$, but with treedepth replaced by a variant parameter called connected treedepth, which differs from treedepth by at most 1. graph $X$ with treedepth $h$. Ossona de Mendez, Oum and Wood [33, Proposition 6.6.] proved that the defective chromatic number of the class of $X$-minor-free graphs is at least $h-1$, and conjectured that equality holds. Norin, Scott, Seymour and Wood [32] proved a relaxation of this conjecture with an exponential bound, and in the stronger setting of clustered coloring. In particular, they showed that every $X$-minor-free graph is $(2^{h+1}-4)$-colorable with clustering $c(X)$. The proof of Norin et al. [32] went via treewidth. In particular, they showed that every $X$-minor-free graph with treewidth $t$ is $(2^{h}-2)$-colorable with clustering $ct$ where $c=c(X)$; that is, $G\mathrel{\ooalign{\raise 0.75348pt\hbox{$\subset$}\cr\raise-3.87495pt\hbox{\scalebox{0.9}{$\sim$}}\cr}}H\boxtimes K_{ct}$ for some graph $H$ with $\chi(H)\leqslant 2^{h}-2$. Theorem˜2 provides a qualitative strengthening of this result by showing that $G\mathrel{\ooalign{\raise 0.75348pt\hbox{$\subset$}\cr\raise-3.87495pt\hbox{\scalebox{0.9}{$\sim$}}\cr}}H\boxtimes K_{ct}$ for some graph $H$ with $\operatorname{tw}(H)\leqslant 2^{h+1}$ where $c=c(X)$. Liu [25] recently established the original conjecture of Ossona de Mendez et al. [33], which also implies that the clustered chromatic number of $X$-minor-free graphs is at most $3h-3$, by a result of Liu and Oum [26, Theorem 1.5].

For the sake of completeness, we now adapt the argument of Ossona de Mendez et al. [33] to conclude the lower bound in (1) on underlying treewidth, and the lower bound in (2) related to the product structure of apex-minor-free graphs. We start with the following well-known statement (see [3, Lemma 12] for a similar result).

The next lemma proves the lower bound in (1).

The next result, which is an adaptation of Theorem 19 in [11], proves the lower bound in (2).

Let $G$ and $H$ be graphs. Then $H$ is a minor of $G$ if a graph isomorphic to $H$ can be obtained from $G$ by deleting edges, deleting vertices and contracting edges. If $H$ is not a minor of $G$, then $G$ is $H$-minor-free. A model of $H$ in $G$ is a family $(B_{x}\mid x\in V(H))$ of pairwise disjoint subsets of $V(G)$ such that:

1. (1)

   for every $x\in V(H)$, the subgraph induced by $B_{x}$ is non-empty and connected.

2. (2)

   for every edge $xy\in E(H)$, there is an edge between $B_{x}$ and $B_{y}$ in $G$.

The sets $B_{x}$ for $x\in V(H)$ are called the branch sets of the model. Note that $H$ is a minor of $G$ if and only if there is an $H$-model in $G$.

The join of graphs $G_{1}$ and $G_{2}$, denoted by $G_{1}\oplus G_{2}$, is the graph obtained from the disjoint union of $G_{1}$ and $G_{2}$ by adding all edges between vertices in $G_{1}$ and vertices in $G_{2}$. Similarly, given a set $U$ and a graph $G$ with $V(G)\cap U=\emptyset$, denote by $U\oplus G$ the graph with vertex set $U\cup V(G)$ and edge set $E(G)\cup\{uv\mid u\in U,v\in V(G)\}\cup\{uu^{\prime}\mid u,u^{\prime}\in U,u\neq u^{\prime}\}$.

Let $G$ and $H$ be graphs. Let $a$ and $k$ be integers with $a\geqslant k\geqslant 0$, and let $R_{1},\ldots,R_{k}$ be pairwise disjoint subsets of $V(G)$. A model $(B_{v}\mid v\in V(K_{a}\oplus H))$ of $K_{a}\oplus H$ in $G-\bigcup_{i=1}^{k}R_{i}$ is $\{R_{1},\dots,R_{k}\}$-attached in $G$ if there are $k$ distinct vertices $v_{1},\ldots,v_{k}$ of $K_{a}$ such that $B_{v_{i}}$ contains a neighbor of a vertex in $R_{i}$ in $G$ for each $i\in[k]$. If $R=\{r_{1},\dots,r_{k}\}\subseteq V(G)$ is a set of $k$ vertices, then we say that a model of $K_{a}\oplus H$ in $G$ is $R$-attached in $G$ if it is $\{\{r_{1}\},\dots,\{r_{k}\}\}$-attached in $G$.