Related papers: Clique Polynomials and Chordal Graphs

This is the second in a series of two papers dealing with $(2P_3,C_4,C_6)$-free graphs, or equivalently, $(2P_3,\text{even hole})$-free graphs. In this two-paper series, we give a full structural description of $(2P_3,C_4,C_6)$-free graphs…

Let $G$ be a finite simple graph and $B \subseteq V(G)$. We introduce the \emph{bivariate $B$-restricted clique polynomial} \[ C_B(G;x,y) = \sum_{\substack{K \subseteq V \\ K \text{ is a clique}}} x^{|K|} y^{|K \cap B|}, \] where the…

There is a class of graphs with well-covered dimension equal to the simplicial clique number that contains all chordal graphs and infinitely many other graphs. These graphs generalize a result by Brown and Nowakowski on the well-covered…

Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G, x)=\sum_{i=1}^n d(G,i) x^i$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. For two graphs $G$ and $H$, let $\mathcal{C} =…

A graph is even-hole-free if it has no induced even cycles of length 4 or more. A cap is a cycle of length at least 5 with exactly one chord and that chord creates a triangle with the cycle. In this paper, we consider (cap, even hole)-free…

Curve pseudo-visibility graphs generalize polygon and pseudo-polygon visibility graphs and form a hereditary class of graphs. We prove that every curve pseudo-visibility graph with clique number $\omega$ has chromatic number at most $3\cdot…

Clique separator decomposition introduced by Tarjan and Whitesides is one of the most important graph decompositions. A graph is an {\em atom} if it has no clique separator. A {\em hole} is a chordless cycle with at least five vertices, and…

This article studies real roots of the flow polynomial $F(G,\lambda)$ of a bridgeless graph $G$. For any integer $k\ge 0$, let $\xi_k$ be the supremum in $(1,2]$ such that $F(G,\lambda)$ has no real roots in $(1,\xi_k)$ for all graphs $G$…

We prove that if $\mathcal{C}$ is a hereditary class of graphs that is polynomially $\chi$-bounded, then the class of graphs that admit decompositions into pieces belonging to $\mathcal{C}$ along cuts of bounded rank is also polynomially…

It is well known that the coefficients of the matching polynomial are unimodal. Unimodality of the coefficients (or their absolute values) of other graph polynomials have been studied as well. One way to prove unimodality is to prove…

A clique covering of a graph $G$ is a set of cliques of $G$ such that any edge of $G$ is contained in one of these cliques, and the weight of a clique covering is the sum of the sizes of the cliques in it. The sigma clique cover number…

In this paper, we introduce and study families of squarefree monomial ideals called clique ideals and independence ideals that can be associated to a finite graph. A family of clique ideals with linear resolutions has been characterized.…

We initiate the study of graph classes of power-bounded clique-width, that is, graph classes for which there exist integers $k$ and $\ell$ such that the $k$-th powers of the graphs are of clique-width at most $\ell$. We give sufficient and…

For a directed graph $G$, and a linear order $\ll$ on the vertices of $G$, we define backedge graph $G^\ll$ to be the undirected graph on the same vertex set with edge $\{u,w\}$ in $G^\ll$ if and only if $(u,w)$ is an arc in $G$ and $w \ll…

A complete graph is the graph in which every two vertices are adjacent. For a graph $G=(V,E)$, the complete width of $G$ is the minimum $k$ such that there exist $k$ independent sets $\mathtt{N}_i\subseteq V$, $1\le i\le k$, such that the…

We study a very large family of graphs, the members of which comprise disjoint paths of cliques with extremal cliques identified. This broad characterisation naturally generalises those of various smaller families of graphs having…

Let $K_4^+$ be the 5-vertex graph obtained from $K_4$, the complete graph on four vertices, by subdividing one edge precisely once (i.e. by replacing one edge by a path on three vertices). We prove that if the chromatic number of some graph…

We introduce a class of pairs of graphs consisting of two cliques joined by an arbitrary number of edges. The members of a pair have the property that the clique-bridging edge-set of one graph is the complement of that of the other. We…

The boxicity of a graph G is defined as the minimum integer k such that G is an intersection graph of axis-parallel k-dimensional boxes. Chordal bipartite graphs are bipartite graphs that do not contain an induced cycle of length greater…

We show that for every two cycles $C,D$, there exists $c>0$ such that if $G$ is both $C$-free and $\overline{D}$-free then $G$ has a clique or stable set of size at least $|G|^c$. ("$H$-free" means with no induced subgraph isomorphic to…