Characterization of Roman domination critical unicyclic graphs
Abstract
A Roman dominating function on a graph G is a function f : V(G) → {0,1,2} satisfying the condition that every vertex u of G for which f(u) = 0 is adjacent to at least one vertex v of G for which f(v) = 2. The weight of a Roman dominating function is the value f(V(G)) = Σu⋯V(G)f(u). The Roman domination number, γR(G), of G is the minimum weight of a Roman dominating function on G. A graph G is said to be Roman domination vertex critical or just γR-vertex critical, if γR(G -v) < γR(G) for any vertex v ⋯ V(G). Similarly, G is Roman domination edge critical or just γR-edge critical, if γR(G + e) < γR(G) for any edge e n⋯ E(G). In this paper, we characterize γR-vertex critical connected unicyclic graphs as well γR-edge critical connected unicyclic graphs.
