Connected graph searching in chordal graphs

• Authors: Nicolas Nisse
• Venue: Discrete Applied Mathematics
• Year: 2009

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 numbers(G) 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 cs(G) 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 tw(G), cs(G)/s(G)=O(tw(G)). More precisely, we propose a polynomial-time algorithm that, given any chordal graph G, computes a connected search strategy for G using at most (tw(G)+2)(2s(G)-1) 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 (tw(G)@?k/2@?+2)(2s(T)-1) searchers where T is an optimal tree-decomposition of G.