Safe sets, network majority on weighted trees

Let G = (V, E) be a graph and let w : V → ℝ>0 be a positive weight function on the vertices of G. For every subset X of V, let w(X) ≔ ∑v∈Gw(v). A non-empty subset ∑ 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. If the subgraph G(S) induced by a weighted safe set S is connected, then the set S is called a weighted connected safe set. In this article, we show that the problem of computing the minimum weight of a safe set is NP-hard for trees, even if the underlying tree is restricted to be a star, but it is polynomially solvable for paths. We also give an O(n log n) time 2-approximation algorithm for finding a weighted connected safe set with minimum weight in a weighted tree. Then, as a generalization of the concept of a minimum safe set, we define the concept of a parameterized infinite family of proper central subgraphs on weighted trees, whose polar ends are the vertex set of the tree and the centroid points. We show that each of these central subgraphs includes a centroid point. © 2017 Wiley Periodicals, Inc.