Related papers: On the connected coalition number

The eccentric connectivity index of a connected graph $G$ is the sum over all vertices $v$ of the product $d_{G}(v) e_{G}(v)$, where $d_{G}(v)$ is the degree of $v$ in $G$ and $e_{G}(v)$ is the maximum distance between $v$ and any other…

Let $G=(V,E)$ be a simple graph. A set $S\subseteq V(G)$ is called an outer-connected dominating set (or ocd-set) of $G$, if $S$ is a dominating set of $G$ and either $S=V(G)$ or $V\backslash S$ is a connected graph. In this paper we…

Let $G$ be a (finite or infinite) group such that $G/Z(G)$ is not simple. The non-commuting, non-generating graph $\Xi(G)$ of $G$ has vertex set $G \setminus Z(G)$, with vertices $x$ and $y$ adjacent whenever $[x,y] \ne 1$ and $\langle x, y…

An identifying code in a graph $G$ is a dominating set $C$ such that the closed neighborhood of each vertex in $G$ has a distinct intersection with $C$. In 2008, Gravier et al. determined the minimum cardinality of an identifying code of…

For a graph $G=(V,E)$, a set $S\subseteq V$ is a dominating set if every vertex in $V-S$ has at least a neighbor in $S$. A dominating set $S$ is a global offensive alliance if for each vertex $v$ in $V-S$ at least half the vertices from the…

A graph $G$ of order $nv$ where $n\geq 2$ and $v\geq 2$ is said to be weakly $(n,v)$-clique-partitioned if its vertex set can be decomposed in a unique way into $n$ vertex-disjoint $v$-cliques. It is strongly $(n,v)$-clique-partitioned if…

For an edge-colored graph $G$, we call an edge-cut $M$ of $G$ monochromatic if the edges of $M$ are colored with the same color. The graph $G$ is called monochromatic disconnected if any two distinct vertices of $G$ are separated by a…

For a graph G=(V,E), the k-dominating graph of G, denoted by $D_{k}(G)$, has vertices corresponding to the dominating sets of G having cardinality at most k, where two vertices of $D_{k}(G)$ are adjacent if and only if the dominating set…

Counting dominating sets in a graph $G$ is closely related to the neighborhood complex of $G$. We exploit this relation to prove that the number of dominating sets $d(G)$ of a graph is determined by the number of complete bipartite…

A path in a vertex-colored graph is a {\it vertex-proper path} if any two internal adjacent vertices differ in color. A vertex-colored graph is {\it proper vertex $k$-connected} if any two vertices of the graph are connected by $k$ disjoint…

A set $S$ of vertices in a graph $G(V,E)$ is called a dominating set if every vertex $v\in V$ is either an element of $S$ or is adjacent to an element of $S$. A set $S$ of vertices in a graph $G(V,E)$ is called a total dominating set if…

A path in an edge-colored graph is called proper if no two consecutive edges of the path receive the same color. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to…

The double graph of a graph $G$ is defined as $\mathcal{D}[G]$ = $G \times T_2$, where \(T_2\) is the total graph with 2 vertices and $\times$ stands for the Kronecker product of graphs. In this paper, sufficient conditions for double…

A dominating set of a graph $G$ is a set $S \subseteq V(G)$ such that every vertex in $V(G) \setminus S$ has a neighbor in $S$, where two vertices are neighbors if they are adjacent. A secure dominating set of $G$ is a dominating set $S$ of…

A mixed graph $G$ is a graph that consists of both undirected and directed edges. An orientation of $G$ is formed by orienting all the undirected edges of $G$, i.e., converting each undirected edge $\{u,v\}$ into a directed edge that is…

For an edge-colored graph $G$, a set $F$ of edges of $G$ is called a \emph{proper cut} if $F$ is an edge-cut of $G$ and any pair of adjacent edges in $F$ are assigned by different colors. An edge-colored graph is \emph{proper disconnected}…

A set $S\subseteq V(G)$ of a graph $G$ is a dominating set if each vertex has a neighbor in $S$ or belongs to $S$. Let $\gamma(G)$ be the cardinality of a minimum dominating set in $G$. The bondage number $b(G)$ of a graph $G$ is the…

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 $G$,…

A vertex subset $S$ in a graph $G$ is a dominating set if every vertex not contained in $S$ has a neighbor in $S$. A dominating set $S$ is a connected dominating set if the subgraph $G[S]$ induced by $S$ is connected. A connected dominating…

Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. We attach to $N$ two graphs ${\Gamma}_G(N)$ and ${\Gamma}^{\ast}_G(N)$ related to the conjugacy classes of $G$ contained in $N$ and to the set of primes dividing the sizes…