Convex Roman Dominating Functions on Graphs under some Binary Operations

Abstract

Let $G$ be a connected graph. A function $f:V(G)\to \{0,1,2\}$ is a \textit{convex Roman dominating function} (or CvRDF) if every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$ and $V_1\cup V_2$ is convex. The weight of a convex Roman dominating function $f$, denoted by ${\omega }_G^{CvR}(f)$, is given by ${\omega }_G^{CvR}(f)=\sum _{v\in V(G)}f(v)$. The minimum weight of a CvRDF on $G$, denoted by ${\gamma }_{CvR}(G)$, is called the \textit{convex Roman domination number} of $G$. In this paper, we specifically study the concept of convex Roman domination in the corona and edge corona of graphs, complementary prism, lexicographic
product, and Cartesian product of graphs.