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 ∈ N G (u) f (v) ≥ 3 , and if f (u) = 1 , then ∑ v ∈ N G (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 { R 3 } (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 (1 3 − 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 (ln Δ) 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. [ABSTRACT FROM AUTHOR]