A Linear Time Algorithm to Solve the Weighted Perfect Domination Problem in Series-Parallel Graphs

Abstract In this paper, we consider the weighted perfect domination problem in series-parallel graphs. Suppose G = ( V , E ) is a graph in which every vertex x ∈ V has a cost c ( x ) and every edge e ∈ E has a cost c ( e ). The weighted perfect domination problem is to find a subset D χ V such that every vertex not in D is adjacent to exactly one vertex in D and its total cost c ( D ) ≡ ∑{ c ( x ): x ∈ D } + ∑{ c ( x , y ): x ∉ D , y ∈ D and ( x , y ) ∈ E } is minimum. This problem is NP-complete for bipartite graphs and chordal graphs. In this paper, we present a linear time algorithm for the problem in series-parallel graphs.