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

TL;DR

This paper studies the signed Roman domination number, a graph invariant related to vertex functions with specific neighborhood conditions, for ladder graphs, circular ladder graphs, and their complements.

Contribution

It determines the signed Roman domination number for ladder graphs, circular ladder graphs, and their complements, providing new exact values for these classes of graphs.

Findings

Exact signed Roman domination numbers for ladder graphs.

Exact signed Roman domination numbers for circular ladder graphs.

Results for the complements of these graphs.

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 $f: V \rightarrow \{-1, 1, 2\}$ that satisfies two conditions: (i) $\sum_{y\in N[x]} f(y)\geq1$ 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 $LG_n$, the circular Ladder graph $CL_n$ and their complements.