Connected graph searching in chordal graphs

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 G (e.g., a network containing a hostile intruder), the search number s ( 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 ) ( 2 s ( 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 ) ( 2 s ( T ) − 1 ) searchers where T is an optimal tree-decomposition of G .