A subset $S \subseteq V(G)$ is called a dual-server dominating set if there exists a partition $\pi_S=\{R,B\}$ of $S$ such that every vertex in $V(G)\setminus S$ has at least one neighbour in $R$ and at least one neighbour in $B.$ The minimum cardinality of a DS-dominating set of $G$ is called the dual-server domination number, denoted by $\gamma_{ds}(G)$. In this paper, we investigate the dual-server domination number of several graph operations. We establish exact values for the dual-server domination number of the join, Cartesian product, and corona of graphs. We also determine the dual-server domination number of the splitting graphs of paths, cycles, and complete bipartite graphs. These results extend the study of dual-server domination and provide further insight into its behaviour under graph operations.
Let $G$ be a graph of minimum degree at least two. A set $D \subseteq V(G)$ is said to be a
\emph{double total dominating set} of $G$, for every vertex $v\in V (G) $ contains at least two neighbors in $D$. The minimum cardinality among all double total dominating sets of $G$ is the \emph{double total
domination number}...
Srinu Irugula, P. Francis· Boletim da Sociedade Paranae...· 0 citations
A paired dominating set of a graph $G$ is a dominating set $D$ such that $G[D]$ has a perfect matching. The minimum size of such a set is the paired domination number $\gpr(G)$. Desormeaux and Henning conjectured that every cubic bipartite graph $G$ of order $n$ satisfies $\gpr(G)\le n/2$. We prove the conjecture in th...
Let $G$ be a graph with vertex set $V(G)$. A set $I\subseteq V(G)$ is an independent dominating set of $G$ if no two vertices in $I$ are adjacent and every vertex in $V(G)\setminus I$ is adjacent to at least one vertex in $I$. The independent domination number of $G$ is the minimum cardinality among all independent dom...
A. Cabrera-Martínez, J. L. López-Carmona, Ismael Rios-Villamar et al.· 1 citation
A rainbow $k$-dominating function assigns the empty set or exactly one of the colors $\{1\},\{2\},\ldots,\{k\}$ to each vertex of a simple graph in such a way that every vertex receiving $\emptyset$ sees all $k$ colors in its neighborhood. This model has several facility-location interpretations where colored vertices...
The total domination number $\gamma_t(G)$ of a graph $G$ is the minimum cardinality of a set $D\subseteq V(G)$ such that every vertex of $G$ has a neighbor in $D$. The annihilation number $a(G)$ is the largest integer $k$ for which the sum of the $k$ smallest degrees of $G$ is at most $|E(G)|$. A well-known conjecture,...
We continue the study of $k$-limited domination in graphs, a domination variant in which each vertex of a dominating set may dominate at most $k$ vertices outside the set. This concept extends classical domination by incorporating capacity constraints on dominating vertices. We improve the general lower bound for the $...
Gordana Radić· 0 citations
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