Total 2-Rainbow Domination in Graphs: Complexity and Algorithms.

For a simple, undirected graph G (V , E) without isolated vertices, a function h : V → { ϕ , { 1 } , { 2 } , { 1 , 2 } } which satisfies the following two conditions is called a total 2-rainbow dominating function (T2RDF) of G. (i) For all u ∈ V , if h (u) = ϕ then ∪ v ∈ N (u) h (v) = { 1 , 2 } and (ii) Every u ∈ V with h (u) ≠ ϕ is adjacent to a vertex v with h (v) ≠ ϕ. The weight of a T2RDF h of G is the value h (V) = ∑ u ∈ V | h (u) |. The total 2-rainbow domination number is the minimum weight of a T2RDF on G and is denoted by γ t r 2 (G). The minimum total 2-rainbow domination problem (MT2RDP) is to find a T2RDF of minimum weight in the input graph. In this article, we show that the problem of deciding if G has a T2RDF of weight at most l for star convex bipartite graphs, comb convex bipartite graphs, split graphs and planar graphs is NP-complete. On the positive side, we show that MT2RDP is linear time solvable for threshold graphs, chain graphs and bounded tree-width graphs. On the approximation point of view, we show that MT2RDP cannot be approximated within (1 −) ln | V | for any > 0 unless P=NP and also propose 2 (ln (Δ − 0. 5) + 1. 5) -approximation algorithm for it. Further, we show that MT2RDP is APX-complete for graphs with maximum degree 4. Next, it is shown that domination problem and the total 2-rainbow domination problems are not equivalent in computational complexity aspects. Finally, an integer linear programming formulation for MT2RDP is presented. [ABSTRACT FROM AUTHOR]