Signed mixed Roman domination numbers in graphs

Abstract

Let $G = (V;E)$ be a simple graph with vertex set $V$ and edge set $E$. A signed mixed Roman dominating function (SMRDF) of $G$ is a function $f: V\cup E\rightarrow \{-1,1,2\}$ satisfying the conditions that (i) $\sum _{y\in N_m[x]}f(y)\ge 1$ for each $x\in V\cup E$, where $N_m[x]$ is the set, called mixed closed neighborhood of $x$, consists of $x$ and the elements of $V\cup E$ adjacent or incident to $x$ (ii) every element $x\in V\cup E$ for which $f(x) = -1$ is adjacent or incident to at least one element $y\in V\cup E$ for which $f(y) = 2$. The weight of a SMRDF $f$ is $\omega (f)=\sum _{x\in V\cup E}f(x)$. The signed mixed Roman domination number $\gamma _{sR}^*(G)$ of $G$ is the minimum weight of a SMRDF of $G$. In this paper we initiate the study of the signed mixed Roman domination number and we present bounds for this parameter. In particular, we determine this parameter for some classes of graphs.