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Let be a graph and let be a positive weight function on the vertices of G. For every subset X of V, let . A non‐empty subset is a weighted safe set if, for every component C of the subgraph induced by S and every component D of , we have whenever there is an edge between C and D. If the subgraph induced by a weighted safe set S is connected,...
A safe set of a graph $$G=(V,E)$$ 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 $$G[V {\setminus } S]$$ G[V\S] , we have $$|A| \ge |B|$$ |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...
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