The Signed Roman Domination Number of Ladder graphs, circular Ladder graphs and their complements

Abstract

Let $G=(V,E)$ be a finite connected simple graph with vertex set $V$ and edge set $E$. A signed Roman dominating function (SRDF) on a graph $G$ is a function $f:V\to \{-1,1,2\}$ that satisfies two conditions: (i) ${\sum }_{y\in N[x]}f(y)\ge 1$ for each $x\in V$, where the set $N[x]$ is the closed neighborhood of $x$ consisting of $x$ and vertices of $V$ that are adjacent to $x$, and (ii) each vertex $x\in V$ where $f(x)=-1$ is adjacent to at least one vertex $y\in V$ where $f(y)=2$. The weight of a SRDF is the sum of its function values over all vertices. The signed Roman domination number of $G$, denoted by ${\gamma }_{SR}(G)$, is the minimum weight of a SRDF on $G$. In this paper, we investigate the signed Roman domination number of the Ladder graph $L{G}_n$, the circular Ladder graph $C{L}_n$ and their complements.

13 pages