FORCING ROMAN DOMINATION IN GRAPHS |Asian Journal of Mathematics and Computer Research

If every vertex in V S has a neighbour in S, the set S of vertices is called a dominant set. On a graph G = (V,E), a Roman dominating function (RDF) is defined as a function f: V 0 1, 2 satisfying the condition that every vertex u where f(u) = 0 is adjacent to at least one vertex v where f(v) = 2. A collection of ordered pairs Sf = (v, f(v)) can likewise be used to describe a Roman dominating function f of G: v V. If Sf is the unique extension of T to a R(G)-function, a subset T of Sf is termed a forcing subset of Sf. The forcing Roman domination number of Sf, indicated by F(Sf, R), is defined as F(Sf, R) = min|T|, where T is aforcing subset of Sf. G's forced Roman dominion number F(G, R) is calculated as F(G; R) = minf(Sf, R), where f is a R(G) function. As a result, F(G,R) 0 for any graph G. We begin a research of this parameter in this work. We also get the number of pathways, cycles, full graphs, and complete multipartite graphs that force Roman dominance.

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