Abstract :
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 image (e.g., a network containing a hostile intruder), the search number image of the graph image 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 image of a graph image is the minimum image such that there exists a connected search strategy for the graph image using at most image searchers. This paper is concerned with the ratio between the connected search number and the search number. We prove that, for any chordal graph image of treewidth image, image. More precisely, we propose a polynomial-time algorithm that, given any chordal graph image, computes a connected search strategy for image using at most image searchers. Our main tool is the notion of connected tree-decomposition. We show that, for any connected graph image of chordality image, there exists a connected search strategy using at most image searchers where image is an optimal tree-decomposition of image.
