On Congruent Domination Number of Disjoint and One Point Union of Graphs

Abstract

A dominating set $D \subseteq V(G)$ is said to be a congruent dominating set of $G$ if $$\sum_{v \in V(G)} d(v) \equiv 0 \left( \bmod\;\sum_{v \in D} d(v)\right).$$The minimum cardinality of a minimal congruent dominating set of $G$ is called the congruent domination number of $G$ which is denoted by $\gamma_{cd}(G)$. We establish the bounds on congruent domination number in terms of order of disjoint union of graphs as well as one point union of graphs.