On the weighted safe set problem on paths and cycles

Abstract

Let G be a graph, and let w be a positive real-valued weight function on V (G). For every subset X of V (G), let w(X) = v∈X w(v). A non-empty subset S ⊂ 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) ≥ 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 s(G, w) and connected weighted safe number 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 any pair (G, w), s(G, w) ≤ cs(G, w) by their definitions. In this paper, we discuss the possible equality when G is a path or a cycle. We also give an answer to a problem due to Tittmann et al. [Eur. J. Combin. Vol. 32 (2011)] concerning subgraph component polynomials for cycles and complete graphs.

Parameterized Complexity of Safe Set

“…Ehard and Rautenbach [14] presented a PTAS for the connected safe number of a weighted tree. Fujita et al [18] showed among other results that the problems can be solved in linear time for weighted cycles.…”

“…That is, contrary to its name, having a small safe set could be unsafe for a network. Both the combinatorial and algorithmic aspects of the safe set problem have already been extensively studied [1,17,14,18].…”

On the Connected Safe Number of Some Classes of Graphs

“…For a vertex-weighted graph, Bapat et al [6] presented the weighted safe set problem by considering the graph as a community network. For further study about the weighted safe set, we refer to [7][8][9]. Furthermore, the study on safe set and weighted safe set was conducted by several authors.…”

Stable structure on safe set problems in vertex-weighted graphs

“…They also constructed an efficient algorithm computing the safe number for a weighted path. Furthermore, Fujita et al [7] constructed a linear time algorithm computing the safe number for a weighted cycle. Ehard and Rautenbach [5] gave a polynomial-time approximation scheme (PTAS) for the connected safe number of a weighted tree.…”

“…Note that for every pair (G, w), s(G, w) ≤ cs(G, w) by their definitions. In [7], it was asked which pair (G, w) satisfies the equality and shown that every weighted cycle satisfies the equality. In this paper, we give a complete list of connected bipartite graphs G such that s(G, w) = cs(G, w) for every weight function w on V (G).…”