Network Majority on Tree Topological Network

Abstract Let G = ( V , E ) be a graph, and w : V → Q > 0 be a positive weight function on the vertices of G. For every subset X of V, let w ( X ) = ∑ v ∈ G w ( v ) . A non-empty subset S ⊂ V ( G ) is a weighted safe set if, for every component C of the subgraph induced by S and every component D of G \ S , we have w ( C ) ≥ w ( D ) whenever there is an edge between C and D. In this paper we show that the problem of computing the minimum weight of a safe set is NP -hard for trees, even if the underlining tree is restricted to be a star, but it is polynomially solvable for paths. Then we define the concept of a parameterized infinite family of “proper central subgraphs” on trees, whose polar ends are the minimum-weight connected safe sets and the centroids. We show that each of these central subgraphs includes a centroid. We also give a linear-time algorithm to find all of these subgraphs on unweighted trees.