Stable Structure on Safe Set Problems in Vertex-Weighted Graphs II –Recognition and Complexity–

Abstract

Let G be a graph, and let w be a non-negative real-valued weight function on V(G). For every subset X of V(G), let \(w(X)=\sum _{v \in X} w(v).\) A non-empty subset \(S \subset V(G)\) is a weighted safe set of (G, w) if for every component C of the subgraph induced by S and every component D of \(G-S\), we have \(w(C) \ge w(D)\) whenever there is an edge between C and D. If the subgraph of G induced by a weighted safe set S is connected, then the set S is called a connected weighted safe set of (G, w). The weighted safe number \(\mathrm {s}(G,w)\) and connected weighted safe number \(\mathrm {cs}(G,w)\) of (G, w) are the minimum weights w(S) among all weighted safe sets and all connected weighted safe sets of (G, w), respectively. It is easy to see that for every pair (G, w), \(\mathrm {s}(G,w) \le \mathrm {cs}(G,w)\) by their definitions. In [Journal of Combinatorial Optimization, 37:685–701, 2019], the authors asked which pair (G, w) satisfies the equality \(\mathrm {s}(G,w)=\mathrm {cs}(G,w)\) and it was shown that every weighted cycle satisfies the equality. In the companion paper [European Journal of Combinatorics, in press] of this paper, we give a complete list of connected bipartite graphs G such that \(\mathrm {s}(G,w)=\mathrm {cs}(G,w)\) for every weight function w on V(G). In this paper, as is announced in the companion paper, we show that, for any graph G in this list and for any weight function w on V(G), there exists an FPTAS for calculating a minimum connected safe set of (G, w). In order to prove this result, we also prove that for any tree T and for any weight function \(w^{\prime }\) on V(T), there exists an FPTAS for calculating a minimum connected safe set of \((T,w^{\prime })\). This gives a complete answer to a question posed by Bapat et al. [Networks, 71:82–92, 2018] and disproves a conjecture by Ehard and Rautenbach [Discrete Applied Mathematics, 281:216–223, 2020]. We also show that determining whether a graph is in the above list or not can be done in linear time.