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∈Gw(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.
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