Graphs of bounded chordality

For basic notions and notation not defined here we refer readers to [30]. In this paper we consider finite, undirected graphs with no loops or parallel edges. For a set $S$ we denote the power set of $S$ by $2^{S}$, and the set of all size-two elements of $2^{S}$ by $\binom{S}{2}$. Let $G$ be a graph. We call a subset of $V(G)$ a clique (respectively a stable set) of $G$ if it is a set of pairwise adjacent (respectively non-adjacent) vertices. A clique of size three is called a triangle. The clique number of $G$, denoted by $\omega(G)$, is the maximum size of a clique in $G$. For vertices $u,v\in V(G)$ a $(u,v)$-path in $G$ is a path which has as ends the vertices $u$ and $v$. A non-edge of $G$ is an element of the set $\binom{V(G)}{2}\setminus E(G)$. Given a graph $H$ we say that $G$ is $H$-free (respectively contains $H$) if it contains no (respectively contains an) induced subgraph isomorphic to $H$. For a set $X\subseteq V(G)$, we denote by $G[X]$ the subgraph of $G$ which is induced by $X$. A class of graphs is hereditary if it is closed under isomorphism and under taking induced subgraphs.

Let $G_{1},\ldots,G_{k}$ be graphs. Then, their intersection (respectively union), which we denote by $\cap_{i\in[k]}G_{i}$ (respectively $\cup_{i\in[k]}G_{i}$), is the graph $(\cap_{i\in[k]}V(G_{i}),\cap_{i\in[k]}E(G_{i}))$ (respectively $(\cup_{i\in[k]}V(G_{i}),\cup_{i\in[k]}E(G_{i}))$). Given graph classes $\mathcal{G}_{1},\ldots,\mathcal{G}_{k}$, we denote by $\mathcal{G}_{1}\overset{\mathfrak{g}}{\cap}\ldots\overset{\mathfrak{g}}{\cap}\mathcal{G}_{k}$ the class $\{G:\exists G_{i}\in\mathcal{G}_{i}\text{ such that }G=G_{1}\cap\ldots\cap G_{k}\}$, which we call the graph intersection of $\mathcal{G}_{1},\ldots,\mathcal{G}_{k}$. The graph union of $\mathcal{G}_{1},\ldots,\mathcal{G}_{k}$, which we denote by $\mathcal{G}_{1}\overset{\mathfrak{g}}{\cup}\ldots\overset{\mathfrak{g}}{\cup}\mathcal{G}_{k}$, is the class $\{G:\exists G_{i}\in\mathcal{G}_{i}\text{ such that }G=G_{1}\cup\ldots\cup G_{k}\}$.

Given a class of graphs $\mathcal{A}$ and a graph $G$, we follow Kratochvíl and Tuza [17], and define the intersection dimension of $G$ with respect to $\mathcal{A}$ to be the minimum integer $k$ such that $G\in\overset{\mathfrak{g}}{\bigcap}_{i\in[k]}\mathcal{A}$ if such a $k$ exists, and $+\infty$ otherwise. We remark that the intersection dimension of graphs with respect to graph classes has been also studied by by Cozzens and Roberts [6] under a different name: they called a graph property $P$ dimensional if for every graph $G$, the intersection dimension of $G$ with respect to the class $\mathcal{A}(P):=\{G:G\text{ has the property }P\}$ is finite. For a positive integer $n$, we denote by $K_{n}$ the complete graph on $n$ vertices, and by $K_{n}^{-}$ the graph we obtain from $K_{n}$ by deleting an edge. It is easy to observe that a graph property $P$ is dimensional if and only if for every for every positive integer $n$, both the graphs $K_{n}$ and $K_{n}^{-}$ have the property $P$.

A hole in a graph $G$ is an induced cycle of length at least four. A graph is chordal if it contains no holes, and we denote the class of chordal graphs by $\mathcal{C}$. Following McKee and Scheinerman [20] we call the intersection dimension of a graph $G$ with respect to $\mathcal{C}$ the chordality of $G$ and we denote it by $\mathsf{chor}(G)$. Since, for every positive integer $n$, both the graphs $K_{n}$ and $K_{n}^{-}$ are chordal, it follows that the chordality of every graph is finite (and upper bounded by the number of its non-edges). To the best of our knowledge, chordality was first studied by Cozzens and Roberts [6] under the name rigid circuit dimension.

Given a finite family of nonempty sets $\mathcal{S}$, the intersection graph of $\mathcal{S}$ is the graph which has as vertices the elements of $\mathcal{S}$ and two vertices are adjacent if and only if they have a non-empty intersection. Given a graph $G$ and a family $\mathcal{S}$ of subgraphs of $G$, the intersection graph of $\mathcal{S}$ is the intersection graph of the family $\{V(H):H\in\mathcal{S}\}$.

In the 1970s, Buneman [3], Gavril [12], and Walter [28, 29], proved independently that chordal graphs are exactly the intersection graphs of subtrees in trees. Let $H$ be a chordal graph. A tree $T$ is a representation tree of $H$ if there exists a function $\beta:V(T)\rightarrow 2^{V(H)}$ such that for every $v\in V(H)$, the subgraph $T[\beta^{-1}(v)]$ of $T$ is connected, and $H$ is isomorphic to the intersection graph of the family $\{\beta^{-1}(v):v\in V(H)\}$. In this case we call the pair $(T,\beta)$ a representation of $H$. By the aforementioned characterization of chordal graphs, it follows that every chordal graph has a representation. In Section 2, we prove a characterization of graphs of chordality at most $k$ which generalizes the above characterization of chordal graphs. We continue with some definitions before we state the main result of Section 2.

An interval graph is any graph which is isomorphic to the intersection graph of a family of intervals on the real line. We denote the hereditary class of interval graphs by $\mathcal{I}$. It is easy to see that the intersection graphs of subpaths in paths are exactly the interval graphs, and thus every interval graph is also chordal.

Let $G$ be a graph. A chordal completion (respectively interval completion) of $G$ is a supergraph of $G$ on the same vertex which is chordal (respectively interval). Since every complete graph is an interval graph, it follows that every graph has an interval and thus a chordal completion.

A tree-decomposition of $G$ is a representation $(T,\beta)$ of a chordal completion $H$ of $G$. Fix a chordal completion $H$ of $G$ and a representation $(T,\beta)$ of $H$. For every $t\in V(T)$, we call the set $\beta(t)$ the bag of $t$. It is easy to see that every bag is a clique of $H$ and that every clique of $H$ is contained in a bag of $T$. We say that $(T,\beta)$ is a complete tree-decomposition of $G$ if for every $t\in V(T)$, the set $\beta(t)$ is a clique of $G$. If $T$ is a path, then $H$ is an interval completion of $G$ and we call tree-decomposition $(T,\beta)$ a path-decomposition of $G$. It is easy to see that a graph has a complete tree-decomposition (respectively complete path-decomposition) if and only if it is chordal (respectively interval). The width of a tree-decomposition is the the clique number of the corresponding chordal completion minus one¹¹1The “minus one” in the definition of the width serves so that trees have tree-width one.. The tree-width (respectively path-width) of $G$, denoted by $\mathsf{tw}(G)$ (respectively $\mathsf{pw}(G)$) is the minimum width of a tree-decomposition (respectively path-decomposition) of $G$. That is, $\mathsf{tw}(G):=\min\{\omega(H)-1:H\text{ is a chordal completion of }G\}$, and $\mathsf{pw}(G):=\min\{\omega(I)-1:I\text{ is an interval completion of }G\}$. A tree-decomposition separates a non-edge $e$ if $e$ is a non-edge of the chordal completion which corresponds to this tree-decomposition. Let $\mathcal{T}$ be a family of tree-decompositions of $G$. We say that $\mathcal{T}$ is a non-edge-separating family of tree-decompositions if for every non-edge $e$ of $G$, there exists a tree-decomposition in $\mathcal{T}$ which separates $e$.

Below is the main result of Section 2, we postpone some definitions for Section 2.

In Section 3 we focus on the chromatic number of graphs of bounded chordality.

For a positive integer $k$ we denote by $[k]$ the set of integers $\{1,\ldots,k\}$. A $k$-coloring of $G$ is a function $f\colon V(G)\rightarrow[k]$ such that for every $i\in[k]$ we have that $f^{-1}(i)$ is a stable set. A graph is $k$-colorable if it admits a $k$-coloring, and the chromatic number of a graph $G$, denoted by $\chi(G)$, is the minimum integer $k$, for which $G$ is $k$-colorable.

It is immediate that for every graph $G$ we have $\chi(G)\geq\omega(G)$, and it is easy to see that there are graphs $G$ for which we have $\chi(G)>\omega(G)$ (for example odd cycles). Moreover, the gap between the chromatic number and the clique number can be arbitrarily large. Indeed, Tutte [7, 8] first proved in the 1940s that there exist triangle-free graphs of arbitrarily large chromatic number (for other such constructions see also [4, 21, 31]). Thus, in general, the chromatic number is not upper-bounded by a function of the clique number.

A graph $G$ is perfect if every induced subgraph $H$ of $G$ satisfies $\chi(H)=\omega(H)$. Berge [2] proved in 1960 that chordal graphs are perfect. What can we say for the connection between $\chi$ and $\omega$ for graphs of bounded chordality?

In his seminal paper “Problems from the world surrounding perfect graphs”, Gyárfás [15] introduced the $\chi$-bounded graph classes as “natural extensions of the world of perfect graphs”. We say that a hereditary class $\mathcal{A}$ is $\chi$-bounded if there exists a function $f\colon\mathbb{N}\rightarrow\mathbb{R}$ such that for every graph $G\in\mathcal{A}$, we have $\chi(G)\leq f(\omega(G))$. Such a function $f$ is called a $\chi$-bounding function for $\mathcal{A}$. For more on $\chi$-boundedness we refer the readers to the surveys of Scott and Seymour [26], and Scott [25]. The examples of triangle-free graphs of arbitrarily large chromatic number that we mention above imply that the class of all graphs is not $\chi$-bounded.

A natural direction of research on $\chi$-boundedness is to consider operations that we can apply among graphs of different classes in order to obtain new classes of graphs, and study (from the perspective of $\chi$-boundedness) graph classes which are obtained via this way from $\chi$-bounded classes.

Gyárfás [15, Section 5] considered graph intersections and graph unions of $\chi$-bounded graph classes from the perspective of $\chi$-boundedness. Graph unions of $\chi$-bounded graph classes are $\chi$-bounded²²2It is easy to observe that for any two graphs $G_{1}$ and $G_{2}$, we have $\chi(G_{1}\cup G_{2})\leq\chi(G_{1})\chi(G_{2})$ and that $\omega(G_{1}\cup G_{2})\geq\max\{\omega(G_{1}),\omega(G_{2})\}$. Thus, if for each $i\in[k]$ we have that $f_{i}$ is a $\chi$-bounding function for a class $\mathcal{G}_{i}$, then $f:=\prod_{i\in[k]}f_{i}$ is a $\chi$-bounding function for the class $\overset{\mathfrak{g}}{\bigcup}_{i\in[k]}\mathcal{G}_{i}$.. The situation with intersections of graphs is different. In an upcoming paper two of us, in joint work with Rimma Hämäläinen and Hidde Koerts, study further the interplay between graph intersections and $\chi$-boundedness.

Since interval graphs are chordal, it follows that they are perfect as well. Following [23], we define the boxicity of $G$ to be the minimum integer $k$ such that $G$ is isomorphic to the intersection graph of a family of axis-aligned boxes in $\mathbb{R}^{k}$. We denote the boxicity of a graph $G$ by $\mathsf{box}(G)$. It easy to see that the boxicity $k$ of a graph is equal to its intersection dimension with respect to the class of interval graphs.

In 1965, in his Ph.D. thesis [4] Burling introduced a sequence $\{B_{k}\}_{k\geq 1}$ of families of axis-aligned boxes in $\mathbb{R}^{3}$ such that for each $k$ the intersection graph of $B_{k}$ is triangle-free and has chromatic number at least $k$. Thus, for every $k\geq 3$ the class of all graphs of boxicity at most $k$, that is, the class $\overset{\mathfrak{g}}{\bigcap}_{i\in[k]}\mathcal{I}$, is not $\chi$-bounded. Hence, for every $k\geq 3$ the class of graphs of chordality at most $k$ is not $\chi$-bounded.

What about the class $\mathcal{C}\overset{\mathfrak{g}}{\cap}\mathcal{C}$? Asplund and Grünbaum [1], in one of the first results which provides an upper bound of the chromatic number in terms of the clique number for a class of graphs, proved in 1960 that every intersection graph of axis-aligned rectangles in the plane with clique number $\omega$ is $\mathcal{O}(\omega^{2})$-colorable. Hence the class $\mathcal{I}\overset{\mathfrak{g}}{\cap}\mathcal{I}$ is $\chi$-bounded (see also [5] for a better $\chi$-bounding function).

Since the class $\mathcal{I}\overset{\mathfrak{g}}{\cap}\mathcal{I}$ is $\chi$-bounded it is natural to ask whether any proper superclasses of this class are $\chi$-bounded as well. Gyárfás, asked the following question:

In Subsection 3.1 we discuss a result of Felsner, Joret, Micek, Trotter and Wiechert [10] which implies that Burling graphs are contained in $\mathcal{C}\overset{\mathfrak{g}}{\cap}\mathcal{I}$, and thus that the answer to Gyárfás’ question is negative.

In the rest of Section 3 we consider two families of subclasses of the class $\mathcal{C}\overset{\mathfrak{g}}{\cap}\mathcal{C}$, which we prove are $\chi$-bounded. In Subsection 3.2 we prove the following:

We remark that each of the classes which satisfies the assumptions of Theorem 1.3 is a proper superclass of $\mathcal{I}\overset{\mathfrak{g}}{\cap}\mathcal{I}$.

Let $u$ and $v$ be two vertices of a graph $G$. Then their distance, which we denote by $d(u,v)$, is the length of a shortest $(u,v)$-path in $G$. A rooted tree is a tree $T$ with one fixed vertex $r\in V(T)$ which we call the root of $T$. The height of a rooted tree $T$ with root $r$ is $h(T,r):=\max\{d(r,t):t\in V(T)\}$. The radius of a tree $T$, which we denote by $\mathsf{rad}(T)$, is the nonnegative integer $\min\{h(T,r):r\in V(T)\}$. In Subsection 3.3 we prove the following:

In Section 4, we consider the recognition problem for the class of graphs of chordality at most $k$. The $k$-Chordality Problem is the following: Given a graph $G$ as an input, decide whether or not $\mathsf{chor}(G)\leq k$. We prove the following:

Since chordal graphs can be recognized efficiently (see, for example, [11, 19]), the only open case in order to fully classify the complexity of the $k$-Chordality Problem, is the case $k=2$. In an upcoming paper, in joint work with Therese Biedl and Taite LaGrange, we prove that the $2$-Chordality Problem is $\NP$-complete as well.

In this section we prove Theorem 1.1 which provides different characterizations of graphs of bounded chordality. We also point out how our proof of Theorem 1.1 can be adapted (in a straight-forward way) to provided analogous characterizations for graphs of bounded boxicity.

The key notion that we use for the proof of Theorem 1.1 is that of the tree-median-dimension of a graph, introduced by Stavropoulos [27], which we prove is equivalent to chordality. We introduce the tree-median-dimension of a graph in Subsection 2.1. We first need some definitions.

Let $G$ be a graph. For $u,v\in V(G)$, a $(u,v)$-geodesic is a shortest $(u,v)$-path. We denote by $d(u,v)$ the distance of $u$ and $v$. We also denote by $I(u,v)$ the set of all vertices of $G$ which lie in a $(u,v)$-geodesic, that is, $I(u,v):=\{x\in V(G)\mid d(u,v)=d(u,x)+d(x,v)\}$. Given three distinct vertices $u,v,w\in V(G)$ we denote by $I(u,v,w)$ the set $I(u,v)\cap I(u,w)\cap I(v,w)$.

A graph $M$ is a median graph if it is connected and for every choice of three distinct vertices $u,v,w\in V(M)$, there exists a vertex $x$ with the property that $I(u,v,w)=\{x\}$. In this case, the vertex $x$ is called the median of $u,v,w$. For three distinct vertices $u,v,w\in V(G)$, we denote their median vertex by $\mathsf{median}(u,v,w)$. It is immediate that trees are median graphs.

Given two graphs $G$ and $H$, their Cartesian product is the graph $G\mathbin{\Box}H:=(V,E)$ where $V:=V(G)\times V(H)$ and $\{(v_{1},h_{1}),(v_{2},h_{2})\}\in E$ if and only if $v_{1}=v_{2}$ and $h_{1}h_{2}\in E(H)$, or $h_{1}=h_{2}$ and $v_{1}v_{2}\in E(G)$. A graph is $G$ isometrically embeddable into a graph $H$ if there exists a map $\phi:V(G)\rightarrow V(H)$ such that for every $u,v\in V(G)$ we have $d_{G}(u,v)=d_{H}(\phi(u),\phi(v))$. In this case we write $G\hookrightarrow H$ and we call the map $\phi$ an isometric embedding. The tree-dimension (respectively path-dimension) of a graph $G$, denoted by $\mathsf{td}(G)$ (respectively $\mathsf{pd}(G)$) is the minimum $k$ such that $G$ has an isometric embedding into the Cartesian product of $k$ trees (respectively paths) if such an embedding exists, and infinite otherwise.

For every positive integer $n$ the hypercube $Q_{n}$ is a graph isomorphic to the Cartesian product of $n$ copies of $K_{2}$. A partial cube is a graph which is isometrically embeddable into a hypercube. Median graphs form a proper subclass of partial cubes (see, for example, [22, Theorem 5.75]). Hence, both the tree-dimension and the path-dimension of every median graph are finite. We observe that for any graph $G$, we have $\mathsf{td}(G)\leq\mathsf{pd}(G)$.

We say that a set $S\subseteq V(G)$ is geodesically convex or simply convex if for every $u,v\in S$, we have $I(u,v)\subseteq S$. We remark that if $G$ is a connected graph and $S\subseteq V(G)$ is a convex set, then $G[S]$ is connected. A subgraph $H$ of $G$ is a convex subgraph if the set $V(H)$ is convex.

We study classes of graphs of bounded chordality from the perspective of $\chi$-boundedness.