On the signed strong Roman domination number of graphs

Let G = (V, E) be a finite and simple graph of order n and maximum degree Delta. A signed strong Roman dominating function on a graph G is a function f : V(G) {-1, 1, 2, ..., inverted left perpendicular Delta/2inverted right perpendicular + 1} satisfying the conditions that (i) for every vertex v of G, f[v] = Sigma(u is an element of N[v]) f(u) >= 1, where N[v] is the closed neighborhood of v and (ii) every vertex v for which f(v) = -1 is adjacent to at least one vertex w for which f(w) ( )>= 1 + inverted left perpendicular1/2 vertical bar N(w)boolean AND V(-1 vertical bar)inverted right perpendicular, where V-1 = {v is an element of V : f(v) = -1}. The minimum of the values w(f) = Sigma(v is an element of V) f(v), taken over all signed strong Roman dominating functions f of G, is called the signed strong Roman domination number of G and is denoted by gamma(ssR)(G). In this paper, we continue the study signed strong Roman domination number of a graph and give several bounds for this parameter. Then, among other results, we determine the signed strong Roman domination number of special classes of graphs.