Safe Sets in Graphs: Graph Classes and Structural Parameters
Résumé
A safe set of a graph G = (V, E) is a non-empty subset S of V such that for every component A of G[S] and every component B of G[V \ S], we have |A| ≥ |B| whenever there exists an edge of G between A and B. In this paper, we show that a minimum safe set can be found in polynomial time for trees. We then further extend the result and present polynomial-time algorithms for graphs of bounded treewidth, and also for interval graphs. We also study the parameterized complexity of the problem. We show that the problem is fixed-parameter tractable when parameterized by the solution size. Furthermore, we show that this parameter lies between tree-depth and vertex cover number.
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