The Roman domination number of comaximal ideal graph

Abstract

Let $G=(V(G), E(G))$ be a graph. A Roman dominating function $\varphi$ is a coloring of the verticesof $G$ with the colors $\{0, 1, 2\}$ such that every vertex colored $0$ is adjacent to atleast one vertex colored $2$. The weight of $\varphi$ is defined to be $\sum _{x\in V(G)}\varphi (x)$. The weight of a Roman dominating functionon $G$ whose weight is minimum, is called the Roman domination numberof $G$. In this paper, we compute the Roman domination number of comaximal ideal graph for all Artinian commutative rings.