Related papers: A Characterization of Claw-Free Graphs using Zero …

The forcing number of a perfect matching $M$ in a graph $G$ is the smallest number of edges inside $M$ that can not be contained in other perfect matchings. The anti-forcing number of $M$ is the smallest number of edges outside $M$ whose…

Call a graph $G$ zero-forcing for a finite abelian group $\mathcal{G}$ if for every $\ell : V(G) \to \mathcal{G}$ there is a connected $A \subseteq V(G)$ with $\sum_{a \in A} \ell(a) = 0$. The problem we pose here is to characterise the…

The anti-forcing number of a perfect matching $M$ of a graph $G$ is the minimal number of edges not in $M$ whose removal to make $M$ as a unique perfect matching of the resulting graph. The set of anti-forcing numbers of all perfect…

For any graphs $G$ and $H$, we say that a bound is of Vizing-type if $\gamma(G\square H)\geq c \gamma(G)\gamma(H)$ for some constant $c$. We show several bounds of Vizing-type for graphs $G$ with forbidden induced subgraphs. In particular,…

Let $G$ be a graph. We say that $G$ is perfectly divisible if for each induced subgraph $H$ of $G$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $\omega(H[B])<\omega(H)$. We use $P_t$ and $C_t$ to denote a path…

Given two graphs $G$ and $H$, we say that $G$ contains $H$ as an induced minor if a graph isomorphic to $H$ can be obtained from $G$ by a sequence of vertex deletions and edge contractions. We study the complexity of Graph Isomorphism on…

The \emph{zero forcing number}, $Z(G)$, of a graph $G$ is the minimum cardinality of a set $S$ of black vertices (whereas vertices in $V(G)-S$ are colored white) such that $V(G)$ is turned black after finitely many applications of "the…

A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight. For…

A hereditary class of graphs $\mathcal{G}$ is \emph{$\chi$-bounded} if there exists a function $f$ such that every graph $G \in \mathcal{G}$ satisfies $\chi(G) \leq f(\omega(G))$, where $\chi(G)$ and $\omega(G)$ are the chromatic number and…

Assume that $G$ is a graph with edge ideal $I(G)$ and matching number ${\rm match}(G)$. For every integer $s\geq 1$, we denote the $s$-th squarefree power of $I(G)$ by $I(G)^{[s]}$. It is shown that for every positive integer $s\leq {\rm…

We give short and simple proofs of the inequalities $B(G)\leq Z(L(G))$ and $Z(G)\leq Z(L(G))$ first established by Erzurumluo\u{g}lu, Meagher, and Pike, where $G$ is a graph without isolated vertices, $B(G)$ is the brushing number of $G$,…

We describe $Forb\{K_{1,3}, \overline {K_{1,3}}\}$, the class of graphs $G$ such that $G$ and its complement $ \overline{G}$ are claw-free. With few exceptions, it is made of graphs whose connected components consist of cycles of length at…

A function $f$ of a graph is called a complete graph invariant if the isomorphism of graphs $G$ and $H$ is equivalent to the equality $f(G)=f(H)$. If, in addition, $f(G)$ is a graph isomorphic to $G$, then $f$ is called a canonical form for…

For a graph $G$, let $\chi(G)$ and $\omega(G)$ respectively denote the chromatic number and clique number of $G$. We give an explicit structural description of ($P_5$,gem)-free graphs, and show that every such graph $G$ satisfies…

An $X$-TAR (token addition/removal) reconfiguration graph has as its vertices sets that satisfy some property $X$, with an edge between two sets if one is obtained from the other by adding or removing one element. This paper considers the…

A {\em fork} is a graph obtained from $K_{1,3}$ (usually called {\em claw}) by subdividing an edge once, an {\em antifork} is the complement graph of a fork, and a {\em co-cricket} is a union of $K_1$ and $K_4-e$. A graph is perfectly…

Zero forcing is an iterative graph coloring process where at each discrete time step, a colored vertex with a single uncolored neighbor forces that neighbor to become colored. The zero forcing number of a graph is the cardinality of the…

It has been conjectured that for every claw-free graph $G$ the choice number of $G$ is equal to its chromatic number. We focus on the special case of this conjecture where $G$ is perfect. Claw-free perfect graphs can be decomposed via…

A graph $H$ is said to be positive if the homomorphism density $t_H(G)$ is non-negative for all weighted graphs $G$. The positive graph conjecture proposes a characterisation of such graphs, saying that a graph is positive if and only if it…

Let $G$ be a bipartite graph, and let $H$ be a bipartite graph with a fixed bipartition $(B_H,W_H)$. We consider three different, natural ways of forbidding $H$ as an induced subgraph in $G$. First, $G$ is $H$-free if it does not contain…