Signed Double Roman Domination of Graphs

In this paper we continue the study of signed double Roman dominating functions in graphs. A signed double Roman dominating function (SDRDF) on a graph G = (V, E) is a function f : V(G) -> {-1, 1, 2, 3} having the property that for each v is an element of V(G), f[v] >= 1, and if f(v) = -1, then vertex v has at least two neighbors assigned 2 under f or one neighbor w with f(w) = 3, and if f(v) = 1, then vertex v must have at leat one neighbor w with f(w) >= 2. The weight of a SDRDF is the sum of its function values over all vertices. The signed double Roman domination number gamma(sdR)(G) is the minimum weight of a SDRDF on G. We present several lower bounds on the signed double Roman domination number of a graph in terms of various graph invariants. In particular, we show that if G is a graph of order n and size m with no isolated vertex, then gamma(sdR)(G) >= 19n-24m/9 and gamma(sdR)(G) >= 4 root n/3 - n. Moreover, we characterize the graphs attaining equality in these two bounds.