On [k] -Roman domination in graphs

Abstract

For an integer (Formula presented.) let f be a function that assigns labels from the set (Formula presented.) to the vertices of a simple graph (Formula presented.). The active neighborhood AN(v) of a vertex (Formula presented.) with respect to f is the set of all neighbors of v that are assigned non-zero values under f. A (Formula presented.) -Roman dominating function ((Formula presented.) -RDF) is a function (Formula presented.) such that for every vertex (Formula presented.) with f(v) < k, we have (Formula presented.). The weight of a (Formula presented.) -RDF is the sum of its function values over the whole set of vertices, and the (Formula presented.) -Roman domination number (Formula presented.) is the minimum weight of a (Formula presented.) -RDF on G. In this paper we determine various bounds on the (Formula presented.) -Roman domination number. In particular, we show that for any integer (Formula presented.) every connected graph G of order (Formula presented.), satisfies (Formula presented.) and we characterize the graphs G attaining this bound. Moreover, we show that if T is a nontrivial tree, then (Formula presented.) for every integer (Formula presented.) and we characterize the trees attaining the lower bound. Finally, we prove the NP-completeness of the (Formula presented.) -Roman domination problem in bipartite and chordal graphs.