Related papers: Relations between global forcing number and maximu…

The \emph{matching preclusion number} of a graph $G$, denoted by $\mpo(G)$, is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. In this paper, we first give some…

A graph is called weakly perfect if its vertex chromatic number equals its clique number. Let $R$ be a ring and $I(R)^*$ be the set of all left proper non-trivial ideals of $R$. The intersection graph of ideals of $R$, denoted by $G(R)$, is…

In this paper we compare the brushing number of a graph with the zero-forcing number of its line graph. We prove that the zero-forcing number of the line graph is an upper bound for the brushing number by constructing a brush configuration…

Let $\alpha(G)$ denote the cardinality of a maximum independent set, while $\mu(G)$ be the size of a maximum matching in $G=\left( V,E\right) $. It is known that if $\alpha(G)+\mu(G)=\left\vert V\right\vert $, then $G$ is a…

A bisection of a graph is a bipartition of its vertex set such that the two resulting parts differ in size by at most 1, and its size is the number of edges that connect vertices in the two parts. The perfect matching condition and…

The restricted hypercube-like graphs, variants of the hypercube, were proposed as desired interconnection networks of parallel systems. The matching preclusion number of a graph is the minimum number of edges whose deletion results in the…

Let $G$ be a graph of order $n$ and let $k\in\{1,\ldots,n-1\}$. The $k$-token graph $F_k(G)$ of $G$, is the graph whose vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$ whenever their symmetric difference…

A zero forcing set is a set $S$ of vertices of a graph $G$, called forced vertices of $G$, which are able to force the entire graph by applying the following process iteratively: At any particular instance of time, if any forced vertex has…

Let ${\mathcal G}$ be an infinite family of connected graphs and let $k$ be a positive integer. We say that $k$ is ${\it forcing}$ for ${\mathcal G}$ if for all $G \in {\mathcal G}$ but finitely many, the following holds. Any…

We provide a short proof of a conjecture of Davila and Kenter concerning a lower bound on the zero forcing number $Z(G)$ of a graph $G$. More specifically, we show that $Z(G)\geq (g-2)(\delta-2)+2$ for every graph $G$ of girth $g$ at least…

For a connected graph $G$, let $\mu(G)$ denote the distance spectral radius of $G$. A matching in a graph $G$ is a set of disjoint edges of $G$. The maximum size of a matching in $G$ is called the matching number of $G$, denoted by…

Given a graph $G$, a subset $S \subseteq V(G)$ is \textit{cycle convex}, if for any vertex $v \in V(G) \setminus S$, the induced subgraph, $G[S \cup \{v\}]$ cannot form a cycle containing the vertex $v$. The \textit{exchange number} of $G$,…

Let $G$ be a simple connected graph with vertex set $V(G)$ and edge set $E(G)$. A $k$-matching of a graph $G$ is a function $f:E(G)\rightarrow \{0,1,\ldots, k\}$ satisfying $\sum_{e \in E_G(v)} f(e) \leq k$ for every vertex $v \in V(G)$,…

A total labeling of a graph $G = (V, E)$ is said to be local total antimagic if it is a bijection $f: V\cup E \to\{1,\ldots ,|V|+|E|\}$ such that adjacent vertices, adjacent edges, and incident vertex and edge have distinct induced weights…

Let $ G $ be a graph. A subset $S \subseteq V(G) $ is called a total dominating set if every vertex of $G$ is adjacent to at least one vertex of $S$. The total domination number, $\gamma_{t}$($G$), is the minimum cardinality of a total…

A vertex subset $S$ of a graph $G$ is a dominating set if every vertex of $G$ either belongs to $S$ or is adjacent to a vertex of $S$. The cardinality of a smallest dominating set is called the dominating number of $G$ and is denoted by…

We introduce a new variant of zero forcing - signed zero forcing. The classical zero forcing number provides an upper bound on the maximum nullity of a matrix with a given graph (i.e. zero-nonzero pattern). Our new variant provides an…

Given a graph $G$, the number of its vertices is represented by $n(G)$, while the number of its edges is denoted as $m(G)$. An independent set in a graph is a set of vertices where no two vertices are adjacent to each other and the size of…

A fractional matching of a graph $G$ is a function $f$ giving each edge a number in $[0,1]$ such that $\sum_{e\in\Gamma(v)}f(e)\leq1$ for each vertex $v\in V(G)$, where $\Gamma(v)$ is the set of edges incident to $v$. The fractional…

Zero forcing number has recently become an interesting graph parameter studied in its own right since its introduction by the "AIM Minimum Rank -- Special Graphs Work Group", whereas metric dimension is a well-known graph parameter. We…