Structural properties of signed graphs admitting Roman dominating function

Abstract

A Roman dominating function(RDF) on a signed graph S = (G, σ) is a function f : V (S) → {0, 1, 2} such that (i) f(N[v]) = f(v) + ∑u∈N(v)σ(uv)f(u) ≥ 1 for every vertex v ∈ V (S) and (ii) for any vertex v with f(v) = 0 there exists a vertex u ∈ N⁺(v) having f(u) = 2. In this article we explore structural properties of signed graphs admitting an RDF. Further, signed graphs with 3-regular graph as their underlying graph are examined and characterisation of one of its subclasses, net-regular signed graphs admitting an RDF is obtained.