Roman {3}-domination in graphs: Complexity and algorithms

A Roman {3}-dominating function on a graph G is a function f:V(G)→{0,1,2,3} having the property that for any vertex u∈V(G), if f(u)=0, then ∑_v∈NG(u)f(v)≥3, and if f(u)=1, then ∑_v∈NG(u)f(v)≥2. The weight of a Roman {3}-dominating function f is the sum f(V(G))=∑_v∈V(G)f(v) and the minimum weight of a Roman {3}-dominating function on G is called the Roman {3}-domination number of G and is denoted by γ_{R3}(G). Given a graph G, ROMAN{3}-DOMINATION asks to find the minimum weight of a Roman {3}-dominating function on G. In this paper, we study the algorithmic aspects of ROMAN{3}-DOMINATION on various graph classes. We show that the decision version of ROMAN{3}-DOMINATION remains NP-complete for chordal bipartite graphs, planar graphs, star-convex bipartite graphs, and chordal graphs. We show that ROMAN{3}-DOMINATION cannot be approximated within a ratio of ([Formula presented.]−ɛ)ln|V(G)| for any ɛ>0 unless P=NP for bipartite graphs as well as chordal graphs, whereas ROMAN{3}-DOMINATION can be approximated within a factor of O(lnΔ) for graphs having maximum degree Δ. We also show that ROMAN{3}-DOMINATION is APX-complete for graphs with maximum degree 4. On the positive side, we show that ROMAN{3}-DOMINATION can be solved in linear time for chain graphs and cographs.