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

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 → \-1, 1, 2\ that satisfies two conditions: (i) ∑y∈ N[x] f(y)≥1 for each x∈ 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∈ V where $f(x) = -1$ is adjacent to at least one vertex y∈ 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 γSR(G), is the minimum weight of a SRDF on G. In this paper, we investigate the signed Roman domination number of the Ladder graph LGₙ, the circular Ladder graph CLₙ and their complements.