Connected Graph Searching in Chordal Graphs

Description

Graph searching was introduced by Parson [T. Parson, Pursuit-evasion in a graph, in: Theory and Applications of Graphs, in: Lecture Notes in Mathematics, Springer-Verlag, 1976, pp. 426--441]: given a "contaminated†graph G (e.g., a network containing a hostile intruder), the search number View the MathML source of the graph G is the minimum number of searchers needed to "clear†the graph (or to capture the intruder). A search strategy is connected if, at every step of the strategy, the set of cleared edges induces a connected subgraph. The connected search number View the MathML source of a graph G is the minimum k such that there exists a connected search strategy for the graph G using at most k searchers. This paper is concerned with the ratio between the connected search number and the search number. We prove that, for any chordal graph G of treewidth View the MathML source, View the MathML source. More precisely, we propose a polynomial-time algorithm that, given any chordal graph G, computes a connected search strategy for G using at most View the MathML source searchers. Our main tool is the notion of connected tree-decomposition. We show that, for any connected graph G of chordality k, there exists a connected search strategy using at most View the MathML source searchers where T is an optimal tree-decomposition of G.

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