6 research outputs found

    The Firebreak Problem

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    Suppose we have a network that is represented by a graph GG. Potentially a fire (or other type of contagion) might erupt at some vertex of GG. We are able to respond to this outbreak by establishing a firebreak at kk other vertices of GG, so that the fire cannot pass through these fortified vertices. The question that now arises is which kk vertices will result in the greatest number of vertices being saved from the fire, assuming that the fire will spread to every vertex that is not fully behind the kk vertices of the firebreak. This is the essence of the {\sc Firebreak} decision problem, which is the focus of this paper. We establish that the problem is intractable on the class of split graphs as well as on the class of bipartite graphs, but can be solved in linear time when restricted to graphs having constant-bounded treewidth, or in polynomial time when restricted to intersection graphs. We also consider some closely related problems

    The Structure of Minimum Vertex Cuts

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    In this paper we continue a long line of work on representing the cut structure of graphs. We classify the types of minimum vertex cuts, and the possible relationships between multiple minimum vertex cuts. As a consequence of these investigations, we exhibit a simple O(? n)-space data structure that can quickly answer pairwise (?+1)-connectivity queries in a ?-connected graph. We also show how to compute the "closest" ?-cut to every vertex in near linear O?(m+poly(?)n) time

    (t,k)-Shredders in k-Connected Graphs

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