1. Introduction
Graph product structure theory describes complicated graphs as subgraphs of strong productsFootnote ^{1} of simpler building blocks. The building blocks typically have bounded treewidth, which is the standard measure of how similar a graph is to a tree. Examples of graph classes that can be described this way include planar graphs [Reference Dujmović, Joret, Micek, Morin, Ueckerdt and Wood29, Reference Ueckerdt, Wood and Yi73], graphs of bounded Euler genus [Reference Distel, Hickingbotham, Huynh and Wood23, Reference Dujmović, Joret, Micek, Morin, Ueckerdt and Wood29], graphs excluding a fixed minor [Reference Dujmović, Joret, Micek, Morin, Ueckerdt and Wood29], and various nonminorclosed classes [Reference Dujmović, Morin and Wood31, Reference Hickingbotham and Wood41]. These results have been the key to solving several open problems regarding queue layouts [Reference Dujmović, Joret, Micek, Morin, Ueckerdt and Wood29], nonrepetitive colouring [Reference Dujmović, Esperet, Joret, Walczak and Wood28], $p$ centered colouring [Reference Döcebski, Felsner, Micek and Schröder25], adjacency labelling [Reference Dujmović, Esperet, Gavoille, Joret, Micek and Morin27, Reference Esperet, Joret and Morin36], twinwidth [Reference Bekos, Da Lozzo, Hliněný and Kaufmann3, Reference Édouard Bonnet and Wood11], and comparable box dimension [Reference Dvorák, Gonçalves, Lahiri, Tan and Ueckerdt33].
This paper shows that graph product structure theory can even be used to describe graphs of bounded treewidth in terms of simpler graphs. Here the building blocks are graphs of smaller treewidth and complete graphs of bounded size. For example, a classical theorem by the referee of [Reference Ding and Oporowski19] can be interpreted as saying that every graph $G$ of treewidth $k$ and maximum degree $\Delta$ is containedFootnote ^{2} in $T \boxtimes K_{O(k\Delta )}$ for some tree $T$ .
This result motivates the following definition. The underlying treewidth of a graph class $\mathcal{G}$ is the minimum $c \in{\mathbb{N}}_0$ such that, for some function $f$ , for every graph $G \in \mathcal{G}$ there is a graph $H$ with $\textrm{tw}(H) \leqslant c$ such that $G$ is contained in $H \boxtimes K_{f(\textrm{tw}(G))}$ . If there is no such $c$ , then $\mathcal{G}$ has unbounded underlying treewidth. We call $f$ the treewidthbinding function. For example, the abovementioned result in [Reference Ding and Oporowski19] says that any graph class with bounded degree has underlying treewidth at most $1$ with treewidthbinding function $O(k)$ .
This paper introduces disjointed coverings of graphs and shows that they are intimately related to underlying treewidth; see Section 3. Indeed, we show that disjointed coverings characterise the underlying treewidth of any graph class (Theorem 11). The remainder of the paper uses disjointed coverings to determine the underlying treewidth of several graph classes of interest, with a small treewidthbinding function as a secondary goal.
Minorclosed classes: We prove that every planar graph of treewidth $k$ is contained in $H \boxtimes K_{O(k^2)}$ where $H$ is a graph of treewidth $3$ . Moreover, this bound on the treewidth of $H$ is best possible. Thus the class of planar graphs has underlying treewidth $3$ (Theorem 21). We prove the following generalisations of this result: the class of graphs embeddable on any fixed surface has underlying treewidth $3$ (Theorem 22); the class of $K_{t}$ minorfree graphs has underlying treewidth $t2$ (Theorem 18); and for $t \geqslant \max \{s,3\}$ the class of $K_{s,t}$ minorfree graphs has underlying treewidth $s$ (Theorem 19). In all these results, the treewidthbinding function is $O(k^2)$ for fixed $s$ and $t$ .
Monotone classes: We characterise the monotone graph classes with bounded underlying treewidth. We show that a monotone graph class $\mathcal{G}$ has bounded underlying treewidth if and only if $\mathcal{G}$ excludes some fixed topological minor (Theorem 28). In particular, we show that for $t \geqslant 5$ the class of $K_t$ topologicalminorfree graphs has underlying treewidth $t$ (Theorem 25). The characterisation for monotone classes has immediate consequences. For example, it implies that the class of $1$ planar graphs has unbounded underlying treewidth. On the other hand, for any $k \in{\mathbb{N}}$ , we show that the class of outer $k$ planar graphs has underlying treewidth $2$ (Theorem 46), which generalises the wellknown fact that outerplanar graphs have treewidth $2$ .
We use our result for disjointed coverings to characterise the graphs $H$ for which the class of $H$ free graphs has bounded underlying treewidth. In particular, the class of $H$ free graphs has bounded underlying treewidth if and only if every component of $H$ is a subdivided star (Theorem 29). For specific graphs $H$ , including paths and disjoint unions of paths, we precisely determine the underlying treewidth of the class of $H$ free graphs.
Hereditary classes: We characterise the graphs $H$ for which the class of graphs with no induced subgraph isomorphic to $H$ has bounded underlying treewidth. The answer is precisely when every component of $H$ is a star, in which case the underlying treewidth is at most $2$ . Moreover, we characterise the graphs $H$ for which the class of graphs with no induced subgraph isomorphic to $H$ has underlying treewidth $0$ , $1$ or $2$ (Theorem 38).
Universal graphs: A graph $U$ is universal for a graph class $\mathcal{G}$ if $U \in \mathcal{G}$ and $U$ contains every graph in $\mathcal{G}$ . This definition is only interesting when considering infinite graphs. For each $k \in{\mathbb{N}}$ there is a universal graph $\mathcal{T}_k$ for the class of countable graphs of treewidth $k$ . Huynh, Mohar, Šámal, Thomassen, and Wood [Reference Huynh, Mohar, Šámal, Thomassen and Wood42] gave an explicit construction for $\mathcal{T}_k$ , and showed how product structure theorems for finite graphs lead to universal graphs. Their results imply that for any hereditary class $\mathcal{G}$ of countable graphs, if the class of finite graphs in $\mathcal{G}$ has underlying treewidth $c$ with treewidthbinding function $f$ , then every graph in $\mathcal{G}$ of treewidth at most $k$ is contained in $\mathcal{T}_c \boxtimes K_{f(k)}$ . This result is applicable to all the classes above. For example, every countable $K_t$ minorfree graph of treewidth $k$ is contained in $\mathcal{T}_{t2} \boxtimes K_{O(k^2)}$ .
The definition of underlying treewidth suggests an underlying version of any graph parameter. An extended version of this paper [Reference Campbell12] explores this idea, focusing on underlying chromatic number. It also includes details of some straightforward proofs omitted from this version.
The rest of this paper is organised as follows. Section 2 introduces some standard structural graph theory notions that will be useful. Section 3 presents disjointed coverings, our main technical tool that characterises underlying treewidth. Section 4 defines two graphs that provide lower bounds on the underlying treewidth of many graph classes. Sections 5–8 address the underlying treewidth of graph classes defined by excluded minors, topological minors, subgraphs, and induced subgraphs, respectively. Finally, Section 9 considers graphs defined by their drawings.
2. Preliminaries
2.1. Basic definitions
See [Reference Diestel17] for graphtheoretic definitions not given here. We consider simple, finite, undirected graphs $G$ with vertexset $V(G)$ and edgeset $E(G)$ . A graph class is a collection of graphs closed under isomorphism. A graph class is hereditary if it is closed under taking induced subgraphs. A graph class is monotone if it is closed under taking subgraphs. 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 $G$ is $H$ minorfree if $H$ is not a minor of $G$ . A graph class $\mathcal{G}$ is minorclosed if every minor of each graph in $\mathcal{G}$ is also in $\mathcal{G}$ .
The class of planar graphs is minorclosed. More generally, the class of graphs embeddable on a given surface (that is, a closed compact $2$ manifold) is minorclosed. The Euler genus of a surface with $h$ handles and $c$ crosscaps is $2h+c$ . The Euler genus of a graph $G$ is the minimum $g \in{\mathbb{N}}_0$ such that there is an embedding of $G$ in a surface of Euler genus $g$ ; see [Reference Mohar and Thomassen56] for more about graph embeddings in surfaces. A graph is linklessly embeddable if it has an embedding in $\mathbb{R}^3$ with no two linked cycles; see [Reference Robertson, Seymour and Thomas66] for a survey and precise definitions. The class of linklessly embeddable graphs is also minorclosed.
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 $P_{vw}$ has $t$ internal vertices, then $\tilde{G}$ is the $t$ subdivision of $G$ . If each $P_{vw}$ has at most $t$ internal vertices, then $\tilde{G}$ is a $(\leqslant t)$ subdivision of $G$ . A graph $H$ is a topological minor of $G$ if a subgraph of $G$ is isomorphic to a subdivision of $H$ . A graph $G$ is $H$ topologicalminorfree if $H$ is not a topological minor of $G$ .
A clique in a graph is a set of pairwise adjacent vertices. Let $\omega (G)$ be the size of the largest clique in a graph $G$ . An independent set in a graph is a set of pairwise nonadjacent vertices. Let $\alpha (G)$ be the size of the largest independent set in a graph $G$ . Let $\chi (G)$ be the chromatic number of $G$ . Note that $V(G) \leqslant \chi (G)\alpha (G)$ . A graph $G$ is $d$ degenerate if every nonempty subgraph of $G$ has a vertex of degree at most $d$ . A greedy algorithm shows that $\chi (G) \leqslant d+1$ for every $d$ degenerate graph $G$ .
Let $P_n$ be the $n$ vertex path. For a graph $G$ and $\ell \in{\mathbb{N}}$ , let $\ell \,G$ be the union of $\ell$ vertexdisjoint copies of $G$ . Let $\widehat{G}$ be the graph obtained from $G$ by adding one dominant vertex.
Let ${\mathbb{N}} \,:\!=\, \{1,2,\dots \}$ and ${\mathbb{N}}_0 \,:\!=\, \{0,1,\dots \}$ . All logarithms in this paper are binary.
2.2. Treedecompositions
For a tree $T$ , a $T$ decomposition of a graph $G$ is a collection $\mathcal{W} = (W_x \,:\, x \in V(T))$ of subsets of $V(G)$ indexed by the nodes of $T$ such that (i) for every edge $vw \in E(G)$ , there exists a node $x \in V(T)$ with $v,w \in W_x$ ; and (ii) for every vertex $v \in V(G)$ , the set $\{ x \in V(T) \,:\, v \in W_x \}$ induces a (connected) subtree of $T$ . Each set $W_x$ in $\mathcal{W}$ is called a bag. The width of $\mathcal{W}$ is $\max \{ W_x \,:\, x \in V(T) \}1$ . A treedecomposition is a $T$ decomposition for any tree $T$ . The treewidth $\textrm{tw}(G)$ of a graph $G$ is the minimum width of a treedecomposition of $G$ . Treewidth is the standard measure of how 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 [Reference Bodlaender6, Reference Harvey and Wood40, Reference Reed62] for surveys.
We use the following wellknown facts about treewidth. Every (topological) minor $H$ of a graph $G$ satisfies $\textrm{tw}(H) \leqslant \textrm{tw}(G)$ . In every treedecomposition of a graph $G$ , each clique of $G$ appears in some bag. Thus $\textrm{tw}(G) \geqslant \omega (G)1$ and $\textrm{tw}(K_n) = n1$ . If $\{v_1,\dots,v_k\}$ is a clique in a graph $G_1$ and $\{w_1,\dots,w_k\}$ is a clique in a graph $G_2$ , and $G$ is the graph obtained from the disjoint union of $G_1$ and $G_2$ by identifying $v_i$ and $w_i$ for each $i \in \{1,\dots,k\}$ , then $\textrm{tw}(G) = \max \{\textrm{tw}(G_1),\textrm{tw}(G_2)\}$ . For any graph $G$ , we have $\textrm{tw}(\widehat{G}) = \textrm{tw}(G) + 1$ and $\textrm{tw}(\ell \, G) = \textrm{tw}(G)$ for any $\ell \in{\mathbb{N}}$ , implying $\textrm{tw}(\widehat{\ell \, G}) = \textrm{tw}(G)+1$ . Finally, every graph $G$ is $\textrm{tw}(G)$ degenerate, implying $\chi (G) \leqslant \textrm{tw}(G)+1$ .
2.3. Partitions
To describe our main results in Section 1, it is convenient to use the language of graph products. However, to prove our results, it is convenient to work with the equivalent notion of graph partitions, which we now introduce.
For graphs $G$ and $H$ , an $H$ partition of $G$ is a partition $(V_x \,:\, x\in V(H))$ of $V(G)$ indexed by the nodes of $H$ , such that for every edge $vw$ of $G$ , if $v \in V_x$ and $w \in V_y$ , then $x = y$ or $xy \in E(H)$ . We say that $H$ is the quotient of such a partition. The width of such an $H$ partition is $\max \{ V_x \,:\, x \in V(H)\}$ . For $c \in{\mathbb{N}}_0$ , an $H$ partition where $\textrm{tw}(H) \leqslant c$ is called a $c$ treepartition. The $c$ treepartitionwidth of a graph $G$ , denoted $\textrm{tpw}_c(G)$ , is the minimum width of a $c$ treepartition of $G$ .
It follows from the definitions that a graph $G$ has an $H$ partition of width at most $\ell$ if and only if $G$ is contained in $H \boxtimes K_\ell$ . Thus, $\textrm{tpw}_c(G)$ equals the minimum $\ell \in{\mathbb{N}}_0$ such that $G$ is contained in $H \boxtimes K_{\ell }$ for some graph $H$ with $\textrm{tw}(H) \leqslant c$ . Hence, the underlying treewidth of a graph class $\mathcal{G}$ equals the minimum $c \in{\mathbb{N}}_0$ such that, for some function $f$ , every graph $G \in \mathcal{G}$ has $c$ treepartitionwidth at most $f(\textrm{tw}(G))$ . We henceforth use this as our working definition of underlying treewidth.
If a graph $G$ has an $H$ partition for some graph $H$ of treewidth $c$ , then we may assume that $H$ is edgemaximal of treewidth $c$ . So $H$ is a $c$ tree (which justifies the ‘ $c$ treepartition’ terminology). Such graphs $H$ are chordal. Chordal partitions are well studied with several applications [Reference Huynh, Mohar, Šámal, Thomassen and Wood42, Reference Reed and Seymour63, Reference Scott, Seymour and Wood68, Reference van den Heuvel, de Mendez, Quiroz, Rabinovich and Siebertz74, Reference van den Heuvel and Wood75]. For example, van den Heuvel and Wood [Reference van den Heuvel and Wood75] proved that every $K_t$ minorfree graph has a $(t2)$ treepartition in which each part induces a connected subgraph with maximum degree at most $t2$ (amongst other properties). Our results give chordal partitions with boundedsize parts (for graphs of bounded treewidth).
Before continuing, we review work on the $c = 1$ case. A treepartition is a $T$ partition for some tree $T$ . The treepartitionwidth Footnote ^{3} of $G$ , denoted by $\textrm{tpw}(G)$ , is the minimum width of a treepartition of $G$ . Thus $\textrm{tpw}(G) = \textrm{tpw}_1(G)$ , which equals the minimum $\ell \in{\mathbb{N}}_0$ for which $G$ is contained in $T \boxtimes K_\ell$ for some tree $T$ . Treepartitions were independently introduced by Seese [Reference Seese69] and Halin [Reference Halin39], and have since been widely investigated [Reference Bodlaender7, Reference Bodlaender and Engelfriet8, Reference Ding and Oporowski19, Reference Ding and Oporowski20, Reference Distel and Wood24, Reference Edenbrandt34, Reference Wood76, Reference Wood77]. Applications of treepartitions include graph drawing [Reference Carmi, Dujmović, Morin and Wood13, Reference Di Giacomo, Liotta and Meijer16, Reference Dujmović, Morin and Wood30, Reference Dujmović, Suderman and Wood32, Reference Wood and Telle80], nonrepetitive graph colouring [Reference Barát and Wood2], clustered graph colouring [Reference Alon, Ding, Oporowski and Vertigan1], monadic secondorder logic [Reference Kuske and Lohrey51], network emulations [Reference Bodlaender4, Reference Bodlaender5, Reference Bodlaender and van Leeuwen9, Reference Fishburn and Finkel37], size Ramsey number [Reference Draganić, Kaufmann, Correia, Petrova and Steiner26, Reference Kamcev, Liebenau, Wood and Yepremyan43], statistical learning theory [Reference Zhang and Amini81], and the edgeErdősPósa property [Reference Chatzidimitriou, Raymond, Sau and Thilikos14, Reference Giannopoulou, Kwon, Raymond and Thilikos38, Reference Raymond and Thilikos60]. Planarpartitions and other more general structures have also been studied [Reference Diestel and Kühn18, Reference Ding, Oporowski, Sanders and Vertigan21, Reference Ding, Oporowski, Sanders and Vertigan22, Reference Reed and Seymour63, Reference Wood and Telle80].
Bounded treepartitionwidth implies bounded treewidth, as noted by Seese [Reference Seese69]. This fact easily generalises for $c$ treepartitionwidth; see [Reference Campbell12] for a proof.
Observation 1. For every graph $G$ and $c\in{\mathbb{N}}_0$ , we have $\textrm{tw}(G) \leqslant (c+1)\textrm{tpw}_c(G)1$ .
Of course, $\textrm{tw}(T) = \textrm{tpw}(T) = 1$ for every tree $T$ . But in general, $\textrm{tpw}(G)$ can be much larger than $\textrm{tw}(G)$ . For example, fan graphs on $n$ vertices have treewidth $2$ and treepartitionwidth $\Omega (\sqrt{n})$ ; see Lemma 12 below. On the other hand, the referee of [Reference Ding and Oporowski19] showed that if the maximum degree and treewidth are both bounded, then so is the treepartitionwidth, which is one of the most useful results about treepartitions.
Lemma 2 ([Reference Ding and Oporowski19]). For $k,\Delta \in{\mathbb{N}}$ , every graph of treewidth less than $k$ and maximum degree at most $\Delta$ has treepartitionwidth at most $24k\Delta$ .
This bound is best possible up to the multiplicative constant [Reference Wood77]. Note that bounded maximum degree is not necessary for bounded treepartitionwidth (for example, stars). Ding and Oporowski [Reference Ding and Oporowski20] characterised graph classes with bounded treepartitionwidth in terms of excluded topological minors. We give an alternative characterisation, which says that graph classes with bounded treepartitionwidth are exactly those that have bounded treewidth and satisfy a further ‘disjointedness’ condition. Furthermore, this result naturally generalises for $c$ treepartitionwidth and thus for underlying treewidth.
3. Disjointed coverings
This section introduces disjointed coverings and shows that they can be used to characterise bounded $c$ treepartitionwidth and underlying treewidth (Theorem 11). On a high level, disjointed coverings are simply a weakening of $c$ treepartitions. As such, they are often easier to construct than $c$ treepartitions. This is important since disjointed coverings can in fact be used to construct $c$ treepartitions (Lemma 8).
Here is the intuition behind disjointed coverings. An important property of any $c$ tree $G$ is that for any set $S$ of $c+1$ vertices and any component $X$ of $GS$ , there is a set $Q$ of at most $c$ vertices in $X$ such that no component of $XQ$ is adjacent to all of $S$ . Given a $c$ treepartition of a graph, an analogous property holds for the parts of the partition. Weakening this property slightly and allowing the parts of the partition to overlap leads to the following definition of disjointed coverings.
An $\ell$ covering of a graph $G$ is a set $\beta \subseteq 2^{V(G)}$ such that $B \leqslant \ell$ for every $B \in \beta$ , and $\cup \{B \,:\, B\in \beta \} = V(G)$ .Footnote ^{4} If $B_1 \cap B_2 = \varnothing$ for all distinct $B_1,B_2 \in \beta$ , then $\beta$ is an $\ell$ partition. As illustrated in Figure 1, an $\ell$ covering $\beta$ of a graph $G$ is $(c,d)$ disjointed if for every $c$ tuple $(B_1,\dots,B_c) \in \beta ^c$ and every component $X$ of $G(B_1\cup \dots \cup B_c)$ there exists $Q \subseteq V(X)$ with $Q \leqslant d$ such that for each component $Y$ of $XQ$ , for some $i \in \{1,\dots,c\}$ we have $V(Y)\cap N_G\big(B^{\prime}_i\big) = \varnothing$ , where $B^{\prime}_i \,:\!=\, B_i\setminus (B_1 \cup \dotsb \cup B_{i1})$ . Note that we can take $Q = \varnothing$ if some $B^{\prime}_i = \varnothing$ , since $N_G(\varnothing ) = \varnothing$ .
Let $\beta$ be an $\ell$ covering of a graph $G$ . For $t \in{\mathbb{N}}$ , let $\beta [t]$ $\,:\!=\, \{\bigcup \mathcal{B} \,:\, \mathcal{B}\subseteq \beta, \mathcal{B}\leqslant t\}$ . So $\beta [t]$ is a $t\ell$ covering of $G$ . For a function $f \,:\,{\mathbb{N}} \to \mathbb{R}^+$ we say that $\beta$ is $(c,f)$ disjointed if $\beta [t]$ is $(c,f(t))$ disjointed for every $t \in{\mathbb{N}}$ .
While $(c,d)$ disjointed coverings are conceptually simpler than $(c,f)$ disjointed coverings, we show they are roughly equivalent (Theorem 4). Moreover, $(c,f)$ disjointed coverings are essential for the main proof (Lemma 8) and give better bounds on the $c$ treepartitionwidth, leading to smaller treewidthbinding functions when determining the underlying treewidth of several graph classes of interest (for $K_t$ minorfree graphs for example).
Note that we often consider the singleton partition $\beta \,:\!=\, \{\{v\} \,:\, v \in V(G)\}$ of a graph $G$ , which is $(c,f)$ disjointed if and only if, for every $t \in{\mathbb{N}}$ , every $t$ partition of $G$ is $(c,f(t))$ disjointed.
This section characterises $c$ treepartitionwidth in terms of $(c,d)$ disjointed coverings (or partitions) and $(c,f)$ disjointed coverings (or partitions). The following observation deals with the $c = 0$ case.
Observation 3. The following are equivalent for any graph $G$ and $d \in{\mathbb{N}}$ :

(a) $G$ has a $(0,d)$ disjointed covering;

(b) every covering of $G$ is $(0,d)$ disjointed;

(c) each component of $G$ has at most $d$ vertices;

(d) $G$ has $0$ treepartitionwidth at most $d$ .
Observation 3 implies that a graph class $\mathcal{G}$ has underlying treewidth $0$ if and only if there is a function $f$ such that every component of every graph $G \in \mathcal{G}$ has at most $f(\textrm{tw}(G))$ vertices.
We prove the following characterisation of bounded $c$ treepartitionwidth (which is new even in the $c = 1$ case).
Theorem 4. For fixed $c \in{\mathbb{N}}_0$ , the following are equivalent for a graph class $\mathcal{G}$ with bounded treewidth:

(a) $\mathcal{G}$ has bounded $c$ treepartitionwidth;

(b) for some $d,\ell \in{\mathbb{N}}$ , every graph in $\mathcal{G}$ has a $(c,d)$ disjointed $\ell$ partition;

(c) for some $d,\ell \in{\mathbb{N}}$ , every graph in $\mathcal{G}$ has a $(c,d)$ disjointed $\ell$ covering;

(d) for some $\ell \in{\mathbb{N}}$ and function $f$ , every graph in $\mathcal{G}$ has a $(c,f)$ disjointed $\ell$ partition;

(e) for some $\ell \in{\mathbb{N}}$ and function $f$ , every graph in $\mathcal{G}$ has a $(c,f)$ disjointed $\ell$ covering.
Proof. Observation 3 handles the $c = 0$ case. Now assume that $c \geqslant 1$ . Lemma 6 below says that (a) implies (b). Since every $\ell$ partition is an $\ell$ covering, (b) implies (c), and (d) implies (e). Lemma 5 below says that (c) implies (d). Finally, Lemma 8 below says that (e) implies (a).
By definition, every $(c,f)$ disjointed $\ell$ covering is $(c,f(1))$ disjointed. The next lemma gives a qualitative converse to this.
Lemma 5. Let $\ell,c,d\in{\mathbb{N}}$ , and let $\beta$ be a $(c,d)$ disjointed $\ell$ covering of a graph $G$ . Then $\beta$ is $(c,f)$ disjointed, where $f(t) \,:\!=\, d t^c$ for each $t \in{\mathbb{N}}$ .
Proof. Fix $t \in{\mathbb{N}}$ . Let $B_1,\dots, B_c \in \beta [t]$ . Let $X$ be a component of $G(B_1\cup \dots \cup B_c)$ . For each $i\in \{1,\dots,c\}$ , let $\mathcal{B}_i$ be a set of at most $t$ elements of $\beta$ whose union is $B_i$ . Let $\mathcal{F} \,:\!=\, \mathcal{B}_1 \times \dots \times \mathcal{B}_c$ , and for each $y = (A_1, \dots, A_c) \in \mathcal{F}$ , define $Q_y$ as follows. Let $X_y$ the component of $G  (A_1 \cup \dots \cup A_c)$ containing $X$ . Since $\beta$ is $(c,d)$ disjointed, there exists $Q_y \subseteq V(X_y)$ of size at most $d$ such that for every component $Y$ of $X_yQ_y$ there is some $i \in \{1, \dots, c\}$ such that $V(Y) \cap N_G(A_i \setminus (A_1 \cup \dots \cup A_{i1})) = \varnothing$ . Now let $Q \,:\!=\, \bigcup _{y \in \mathcal{F}} Q_y$ , and note that $Q\leqslant d\mathcal{F} \leqslant dt^c$ .
Suppose for contradiction that for some component $Y$ of $X  Q$ and each $i \in \{1,\dots, c\}$ , there is a vertex $b_i \in N_G(Y) \cap B^{\prime}_i$ , where $B^{\prime}_i \,:\!=\, B_i \setminus (B_1 \cup \dots \cup B_{i1})$ . Let $y = (A_1, \dots, A_c) \in \mathcal{F}$ be such that $(b_1, \dots, b_c) \in A_1 \times \dots \times A_c$ , and consider that component $Y^{\prime}$ of $X_y  Q_y$ containing $Y$ . By the definition of $Q_y$ , there is some $i \in \{1,\dots, c\}$ such that $Y^{\prime}$ contains no neighbour of a vertex in $A_i \setminus (A_1 \cup \dots \cup A_{i1})$ . In particular, all neighbours of vertices of $Y$ are either vertices of $Y^{\prime}$ or neighbours of vertices of $Y^{\prime}$ , so $b_i$ is not a neighbour of any vertex of $Y$ , a contradiction.
Now we prove that having a $(c,d)$ disjointed partition is necessary for bounded $c$ treepartitionwidth.
Lemma 6. For all $c,\ell \in{\mathbb{N}}_0$ , every graph $G$ with $c$ treepartitionwidth $\ell$ has a $(c,c\ell )$ disjointed $\ell$ partition.
Proof. By assumption, $G$ has an $H$ partition $\beta = (V_h \,:\, h \in V(H))$ where $H$ is a graph of treewidth at most $c$ and $V_h \leqslant \ell$ for all $h$ . We first show that the singleton partition of $H$ is $(c,c)$ disjointed. Let $v_1, \dotsc, v_c \in V(H)$ and let $X$ be a component of $H\{v_1,\dots, v_c\}$ . Let $(W_x \,:\, x \in V(T))$ be a treedecomposition of $H$ where $W_x \leqslant c + 1$ for all $x \in V(T)$ . We may assume that $W_x \neq W_y$ whenever $x \neq y$ . For each $i \in \{1,\dots,c\}$ , let $T_i$ be the subtree of $T$ induced by $\{x \in V(T) \,:\, v_i \in W_x\}$ .
First suppose that $V(T_i) \cap V(T_j) = \varnothing$ for some $i, j \in \{1,\dots,c\}$ . Let $z \in V(T_i)$ be the closest node (in $T$ ) to $T_j$ . Let $Q \,:\!=\, W_z \cap X$ . Note that $Q \subseteq W_z \setminus \{v_i\}$ so $Q \leqslant c$ . Any path from $v_i$ to $v_j$ in $H$ passes through $W_z$ , so each component of $X  Q$ is disjoint from $N_H(v_i)$ or $N_H(v_j)$ .
Now assume that $V(T_i) \cap V(T_j) \neq \varnothing$ for all $i, j\in \{1,\dots, c\}$ . Let $T_X$ be the subgraph of $T$ induced by $\{x \in V(T) \,:\, V(X) \cap W_x \neq \varnothing \}$ . Since $X$ is connected, $T_X$ is a subtree of $T$ . Suppose that $V(T_i) \cap V(T_X) = \varnothing$ for some $i$ . Since $N_H(v_i) \subseteq \bigcup (W_x \,:\, x \in V(T_i))$ , it follows that $N_H(v_i) \cap V(X) = \varnothing$ and so we may take $Q \,:\!=\, \varnothing$ in this case. Now assume that $V(T_i) \cap V(T_X) \neq \varnothing$ for all $i \in \{1,\dots,c\}$ . By the Helly property, $\tilde{T} \,:\!=\, T_1 \cap \dotsb \cap T_c \cap T_X$ is a nonempty subtree of $T$ . For $x \in V(\tilde{T})$ , we have $W_x \leqslant c + 1$ and so $W_x = \{v_1, \dotsc, v_c, u\}$ for some $u \in V(X)$ . First suppose that $V(\tilde{T}) \geqslant 2$ . Then there are adjacent $x, y \in V(\tilde{T})$ with $W_x = \{v_1, \dotsc, v_c, u\}$ and $W_y = \{v_1, \dotsc, v_c, v\}$ for $u, v \in V(X)$ . Since $W_x \neq W_y$ , we have $u \neq v$ and thus there is no $(u,v)$ path in $H  \{v_1, \dotsc, v_c\}$ , contradicting the connectedness of $X$ . Hence $\tilde{T}$ consists of a single vertex $z$ ; thus $W_z = \{v_1, \dotsc, v_c, u\}$ for some $u \in V(X)$ . Let $Q \,:\!=\, \{u\}$ and consider a component $Y$ of $X  Q$ . Let $T_{Y}$ be the subtree of $T$ induced by $\{y\in V(T) \,:\, V(Y) \cap W_y \neq \varnothing \}$ . Since $T_{Y}$ is connected and does not contain $z$ , it is disjoint from some $T_i$ . As above, $N_H(v_i) \cap V(Y) = \varnothing$ , as required.
We have shown that the singleton partition of $H$ is $(c,c)$ disjointed. Now focus on $G$ . By assumption, $\beta$ is an $\ell$ partition of $G$ . Let $V_{v_1}, \dotsc, V_{v_c}$ be parts in $\beta$ , and let $X$ be a component of $G  (V_{v_1} \cup \dotsb \cup V_{v_c})$ . Then $X \subseteq \bigcup \{V_h \,:\, h \in X^{\prime}\}$ where $X^{\prime}$ is a component of $H  \{v_1, \dotsc, v_c\}$ . Since $H$ is $(c,c)$ disjointed, there exists $Q^{\prime} \subseteq V(X^{\prime})$ of size at most $c$ such that each component $X^{\prime}  Q^{\prime}$ is disjoint from some $N_H(v_i)$ . Let $Q \,:\!=\, \bigcup \{V_h \,:\, h \in Q^{\prime}\}$ , which has size at most $c\ell$ . Each component of $X  Q$ is disjoint from some $N_G(V_{v_i})$ .
Note that $(c, f)$ disjointedness is preserved when restricting to a subgraph.
Lemma 7. If $\beta$ is a $(c,f)$ disjointed $\ell$ covering of a graph $G$ , then for every subgraph $\tilde{G}$ of $G$ , the restriction $\tilde{\beta } \,:\!=\, \{ B \cap V(\tilde{G}) \,:\, B \in \beta \}$ is a $(c,f)$ disjointed $\ell$ covering of $\tilde{G}$ .
Proof. Fix $t \in{\mathbb{N}}$ . Let $\tilde{B}_1, \dots, \tilde{B}_c \in \tilde{\beta }[t]$ and let $\tilde{X}$ be a component of $\tilde{G}\big(\tilde{B}_1\cup \dots \cup \tilde{B}_c\big)$ . For each $i \in \{1,\dots, c\}$ , there is a subset $S_i \subseteq \beta$ of size at most $t$ such that $\tilde{B}_i = \bigcup _{B \in S_i} (B\cap V(\tilde{G}))$ . Let $(B_1,\dots,B_c)\,:\!=\, (\bigcup S_1,\dots,\bigcup S_c)$ , and let $\beta^{\prime\prime}$ be the $t\ell$ covering of $G$ given by $\beta \cup \{B_1,\dots,B_c\}$ . Let $X$ be the component of ${G(B_1\cup \dots \cup B_c})$ which contains $\tilde{X}$ , and for each $i \in \{1,\dots, c\}$ let $B^{\prime}_i\,:\!=\, B_i\setminus (B_1\cup \dots \cup B_{i1})$ . Since $\beta$ is $(c,f)$ disjointed, there is a subset $Q$ of $V(X)$ of size at most $f(t)$ such that each component of $XQ$ disjoint from $N_G\big(B^{\prime}_i\big)$ for some $i \in \{1,\dots,c\}$ . Let $\tilde{Q}\,:\!=\, Q\cap V(\tilde{X})$ , and note that $\tilde{Q}\leqslant Q\leqslant f(t)$ . Each component of $\tilde{X}\tilde{Q}$ is contained in a component of $X  Q$ , and hence is disjoint from $N_{\tilde{G}}\big(\tilde{B}_i\setminus \big(\tilde{B}_1\cup \dots \cup \tilde{B}_{i1}\big)\big)\subseteq N_G\big(B^{\prime}_i\big)$ for some $i \in \{1,\dots, c\}$ . Hence $\tilde{\beta }$ is $(c,f)$ disjointed.
The next lemma lies at the heart of the paper.
Lemma 8. Let $k,c,\ell \in{\mathbb{N}}$ and $f \,:\,{\mathbb{N}} \to \mathbb{R}^+$ . For any graph $G$ , if $\textrm{tw}(G)\lt k$ and $G$ has a $(c,f)$ disjointed $\ell$ covering, then $G$ has $c$ treepartitionwidth $\textrm{tpw}_c(G) \leqslant \max \{ 12\ell k, 2 c \ell f(12 k)\}$ .
We prove Lemma 8 via the following induction hypothesis.
Lemma 9. Let $k,c,\ell \in{\mathbb{N}}$ and let $f \,:\,{\mathbb{N}} \to \mathbb{R}^+$ . Let $G$ be a graph of treewidth less than $k$ and let $\beta \subseteq 2^{V(G)}$ be a $(c,f)$ disjointed $\ell$ covering of $G$ . Let $S_1, \dotsc, S_{c  1}, R \subseteq V(G)$ , where $S_i \in \beta [12k]$ for each $i \in \{1, \dotsc, c  1\}$ and $4 k\leqslant R \leqslant f(12 k)$ . Then there exists a $c$ treepartition $(V_x \,:\, x \in V(H))$ of $G$ of width at most $W \,:\!=\, \max \{ 12\ell k, 2 c \ell f(12 k) \}$ , and there exists a $c$ clique $\{x_1,\dots,x_{c1},y\}$ of $H$ such that $V_{x_i} = S_i\setminus (S_1\cup \dots \cup S_{i1})$ for each $i \in \{1,\dots,c1\}$ , and $R \setminus (S_1 \cup \dots \cup S_{c  1}) \subseteq V_y$ with $V_y \leqslant 2 \ell (R  2 k)$ .
Proof. We proceed by induction on $V(G)$ . Let $S \,:\!=\, S_1 \cup \dotsb \cup S_{c  1}$ .
Case 0. $V(G) = R \cup S$ : Let $H$ be the complete graph on vertices $x_1, \dotsc, x_{c  1}, y$ . Let $V_{x_i} \,:\!=\, S_i \setminus (S_1 \cup \dotsb \cup S_{i  1})$ for each $i$ and let $V_y \,:\!=\, R$ . Then $(V_x \,:\, x \in V(H))$ is a $c$ treepartition of $G$ with width at most $W$ and $V_y = R \leqslant 2(R  2k) \leqslant 2 \ell (R  2k)$ . From now on assume that $G  (R \cup S)$ is nonempty.
Case 1. $4 k \leqslant R\leqslant 12 k$ : Since $\beta$ is an $\ell$ covering, and $R \leqslant 12k$ , we can pick $S_c \in \beta [12k]$ such that $R \subseteq S_c$ and $S_c \leqslant \ell R \leqslant 2 \ell (R  2k)$ .
Let $G_1, \dotsc, G_a$ be the connected components of $G  (S \cup S_c)$ . For each $i \in \{1,\dots,c\}$ , let $S^{\prime}_i \,:\!=\, S_i \setminus (S_1 \cup \dotsb \cup S_{i  1})$ . To complete this case, we first prove the following.
Claim 1. For each $j \in \{1,\dots,a\}$ , the subgraph $G[ V(G_j) \cup S \cup S_c ]$ has a $c$ treepartition $\big(V_h^j \,:\, h \in V(H_j)\big)$ of width at most $W$ such that there is a $c$ clique $K = \{x_1, \dotsc, x_c\}$ in $H_j$ , where $S^{\prime}_i = V^j_{x_i}$ for each $i \in \{1,\dots,c\}$ .
Proof. If $V(G_j) \lt 4 k$ , then take $H_j$ to be the complete graph on vertices $x_1, \dotsc, x_{c}, z$ with the partition $V^j_{x_i} \,:\!=\, S^{\prime}_i$ for $i \in \{1,\dots,c\}$ and $V^j_{z} \,:\!=\, V(G_j)$ . Then this gives us the desired $c$ treepartition of $G[V(G_j) \cup S \cup S_c]$ . So assume $V(G_j) \geqslant 4k$ . Note that $\beta [12k]$ is a $(c,f(12 k))$ disjointed $12k\ell$ covering containing $S_1, \dots, S_c$ , so there is a subset $Q^{\prime}_j \subseteq V(G_j)$ of size at most $f(12 k)$ and there is a partition $\{A_1, \dotsc, A_c\}$ of $V\big(G_j  Q^{\prime}_j\big)$ such that each $A_i$ is a union of vertex sets of components of $G_j  Q^{\prime}_j$ that do not intersect $N_{G}\big(S^{\prime}_i\big)$ . Let $Q_j$ be a set such that $Q^{\prime}_j \subseteq Q_j \subseteq V(G_j)$ and $4k \leqslant Q_j \leqslant f(12 k)$ . As illustrated in Figure 2, consider the subgraph
By Lemma 7, the restriction of $\beta$ to $V(F_i)$ is a $(c,f)$ disjointed $\ell$ covering of $F_i$ . Apply induction to $F_i$ with the sets $S_1,\dotsc,S_{i  1},S_{i + 1} \setminus S^{\prime}_i,\dotsc,S_c\setminus S^{\prime}_i$ in place of the sets $S_1,\dotsc,S_{c1}$ and the set $Q_{j}$ in the place of $R$ . For each $i \in \{1,\dotsc,c\}$ , this gives a graph $L_i$ of treewidth at most $c$ containing a $c$ clique $\{x_1, \dotsc, x_{i1}, x_{i+1},\dotsc, x_c, y\}$ such that $F_i$ has an $L_i$ partition $\big( V^{j,i}_x \,:\, x \in V(L_i) \big)$ with $V^{j,i}_{x_m} = S^{\prime}_m$ for all $m \in \{1,\dots, i1,i+1,\dots, c\}$ and $Q_j \setminus \big((S\cup S_c) \setminus S^{\prime}_i)\big) \subseteq V_y^{j,i}$ where
Let $L_i^+$ be $L_i$ together with a vertex $x_i$ adjacent to the clique $\{x_1, \dotsc, x_{i  1}, x_{i + 1}, \dotsc, x_c, y\}$ . So $\textrm{tw}(L^+_i)\leqslant c$ . Set $V^{j,i}_{x_i} \,:\!=\, S^{\prime}_i$ . Then $L^+_i$ contains the $(c + 1)$ clique $K^+ \,:\!=\, \{x_1, \dotsc, x_c, y\}$ and $\big(V^{j,i}_h \,:\, h \in V(L^+_i)\big)$ is an $L^+_i$ partition of $G[A_i \cup Q_j \cup S \cup S_c]$ . Now we may assume that $V(L^+_1), \dotsc, V(L^+_c)$ pairwise intersect in exactly the clique $K^+$ . Let $H_j \,:\!=\, L_1^+\cup \dots \cup L_c^+$ . Since each $L_i^+$ has treewidth at most $c$ , so does $H_j$ . For $x \notin K^+$ , set $V^j_x \,:\!=\, V^{j,i}_x$ for the unique $i$ for which $x \in V(L_i^+)$ , for $i \in \{1, \dotsc, c \}$ set $V^j_{x_i} \,:\!=\, S^{\prime}_i$ , and set $V^j_y \,:\!=\, \bigcup _{i \in \{1, \dotsc, c\}} V^{j,i}_y$ . Since $V^{j}_y \leqslant 2c \ell f(12 k)$ , the partition $\big( V^j_x \,:\, x \in V(H_j) \big)$ has width at most $W$ . Setting $K \,:\!=\, \{ x_1 \dotsc, x_c \}$ , the claim follows.
We may assume that $V(H_1), \dotsc, V(H_a)$ pairwise intersect in exactly the clique $K$ . Let $H \,:\!=\,{H_1\cup \dotsc \cup H_a}$ . For $x \in V(H)$ , setting $V_x \,:\!=\, V^j_x$ if $x \in V(H_j)$ is well defined, and yields an $H$ partition $(V_h \,:\, h \in V(H))$ of $G$ . Since each $H_i$ has treewidth at most $c$ , so does $H$ . Let $y \,:\!=\, x_c$ . Then $R \setminus S \subseteq S^{\prime}_c = V_{y}$ , and, as noted at the start of this case, $V_{y} = S^{\prime}_c \leqslant S_c \leqslant 2 \ell (R  2k)$ . Hence, the width of this partition is at most $W$ , as required.
Case 2. $12 k \lt R \leqslant f(12 k)$ : Since $\textrm{tw}(G) \lt k$ , by the separator lemma of Robertson and Seymour [Reference Robertson and Seymour65, (2.6)], there is a partition $(A, B, C)$ of $V(G)$ with no edges between $A$ and $B$ , where $C \leqslant k$ and $A \cap R, B \cap R \leqslant \tfrac{2}{3} R \setminus C$ . Let $G_1 \,:\!=\, G[A \cup C]$ and $G_2 \,:\!=\, G[B \cup C]$ . Let $R_1 \,:\!=\, (R \cap A) \cup C$ and $R_2 \,:\!=\, (R \cap B) \cup C$ . Since $R \geqslant 12 k$ ,
Hence, $4 k \leqslant R_1 \leqslant f(12k)$ and similarly $4 k \leqslant R_2 \leqslant f(12k)$ . Also $V(G)  V(G_1) = B \geqslant R_2  C \geqslant 4k  k \gt 0$ , so $V(G_1) \lt V(G)$ and likewise for $G_2$ . Fix $j \in \{1, 2\}$ . Let $\beta _j$ be the restriction of $\beta$ to $V(G_j)$ . By Lemma 7, $\beta _j$ is a $(c, f)$ disjointed $\ell$ covering of $G_j$ . Let $S^j_i \,:\!=\, S_i \cap V(G_j)$ for each $i \in \{1,\dots,c1\}$ ; note that each set $S^j_i$ is a union of at most $12k$ elements of $\beta _j$ .
Apply induction to $G_j$ with $S_i^j$ in place of $S_i$ and $R_j$ in place of