Approximating Connected Safe Sets in Weighted Trees

For a graph G and a non-negative integral weight function w on the vertex set of G, a set S of vertices of G is w-safe if w(C) ≥ w(D) for every component C of the subgraph of G induced by S and every component D of the subgraph of G induced by the complement of S such that some vertex in C is adjacent to some vertex of D. The minimum weight w(S) of a w-safe set S is the safe number s(G,w) of the weighted graph (G,w), and the minimum weight of a w-safe set that induces a connected subgraph of G is its connected safe number cs(G,w). Bapat et al. showed that computing cs(G,w) is NP-hard even when G is a star. For a given weighted tree (T,w), they described an efficient 2-approximation algorithm for cs(T,w) as well as an efficient 4-approximation algorithm for s(T,w). Addressing a problem they posed, we present a PTAS for the connected safe number of a weighted tree. Our PTAS partly relies on an exact pseudopolynomial time algorithm, which also allows to derive an asymptotic FPTAS for restricted instances. Finally, we extend a bound due to Fujita et al. from trees to block graphs.