Weak signed Roman domination in graphs

Abstract

A weak signed Roman dominating function (WSRDF) of a graph $G$ with vertex set $V(G)$ is defined as a function $f:V(G)\rightarrow\{-1,1,2\}$ having the property that $\sum_{x\in N[v]}f(x)\ge 1$ for each $v\in V(G)$, where $N[v]$ is the closed neighborhood of $v$. The weight of a WSRDF is the sum of its function values over all vertices. The weak signed Roman domination number of $G$, denoted by $\gamma_{wsR}(G)$, is the minimum weight of a WSRDF in $G$. We initiate the study of the weak signed Roman domination number, and we present different sharp bounds on $\gamma_{wsR}(G)$. In addition, we determine the weak signed Roman domination number of some classes of graphs.

Keywords

• domination
• signed roman domination
• weak signed roman domination