On fixed-parameter tractability of the mixed domination problem for graphs with bounded tree-width

A mixed dominating set for a graph G = (V,E) is a set S ⊆ V ∪ E such that every element x ∈ (V ∪E)\S is either adjacent or incident to an element of S. The mixed domination number of a graphG, denoted by γm(G), is the minimum cardinality of mixed dominating sets ofG and any mixed dominating set with cardinality of γm(G) is called a minimum mixed dominating set. The mixed domination problem is to find a minimum mixed dominating set for graphG and is known to be an NPcomplete problem. In this paper, we present a novel approach to solve the mixed domination problem. Our new technique of assigning power to edges and vertices, and charging vertices for edges combined with with dynamic programming leads to a fixed-parameter algorithm of time O(3tw 2 × tw2 × |V |), which proves our next result, which is the mixed domination problem is fixed-parameter tractable with respect to tree-width.