New complexity results on Roman {2}-domination.

The study of a variant of Roman domination was initiated by Chellali et al. [Discrete Appl. Math. 204 (2016) 22-28]. Given a graph G with vertex set V, a Roman {2}-dominating function f : V & RARR; {0, 1, 2} has the property that for every vertex v & ISIN; V with f(v) = 0, either there exists a vertex u adjacent to v with f(u) = 2, or at least two vertices x, y adjacent to v with f(x) = f(y) = 1. The weight of a Roman {2}-dominating function is the value f(V) = n-ary sumation (v & ISIN;V) f(v). The minimum weight of a Roman {2}-dominating function is called the Roman {2}-domination number and is denoted by & gamma;({R2})(G). In this work we find several NP-complete instances of the Roman {2}-domination problem: chordal graphs, bipartite planar graphs, chordal bipartite graphs, bipartite with maximum degree 3 graphs, among others. A result by Chellali et al. [Discrete Appl. Math. 204 (2016) 22-28] shows that & gamma;({R2})(G) and the 2-rainbow domination number of G coincide when G is a tree, and thus, the linear time algorithm for k-rainbow domination due to Bresar et al. [Taiwan J. Math. 12 (2008) 213-225] can be followed to compute & gamma;({R2})(G). In this work we develop an efficient algorithm that is independent of k-rainbow domination and computes the Roman {2}-domination number on a subclass of trees called caterpillars.