COMPLEXITY OF CERTAIN FUNCTIONAL VARIANTS OF TOTAL DOMINATION IN CHORDAL BIPARTITE GRAPHS

In this paper, we consider minimum total domination problem along with two of its variations namely, minimum signed total domination problem and minimum minus total domination problem for chordal bipartite graphs. In the minimum total domination problem, the objective is to find a smallest size subset TD subset of V of a given graph G -(V, E) such that | TD boolean AND N-G(v)| >= 1 for every v. V. In the minimum signed (minus) total domination problem for a graph G -(V, E), it is required to find a function f : V -> {-1, 1} ({-1, 0, 1}) such that f(N-G(v)) = Sigma(u is an element of NG(v)) f(u) >= 1 for each v is an element of V, and the cost f(V) = Sigma(v is an element of V f(v)) is minimized. We first show that for a given chordal bipartite graph G -(V, E) with a weak elimination ordering, a minimum total dominating set can be computed in O(n + m) time, where n = | V | and m = |E|. This improves the complexity of the minimum total domination problem for chordal bipartite graphs from O(n(2)) time to O(n + m) time. We then adopt a unified approach to solve the minimum signed (minus) total domination problem for chordal bipartite graphs in O(n+m) time. The method is also able to solve the minimum k-tuple total domination problem for chordal bipartite graphs in O(n + m) time. For a fixed integer k >= 1 and a graph G = (V, E), the minimum k-tuple total domination problem is to find a smallest subset TDk subset of V such that |TDk boolean AND N-G(v)| >= k for every v is an element of V