Connected Graph Searching in Outerplanar Graphs

Search games are a powerfull tool for studying various connectivity parameters of graphs. In the classical search game, we consider an undirected graph G = (V, E) whose edges are initially contaminated. A set of searchers try to clean the graph. At the beginning the graph contains no searchers. At each step of the game, a searcher can be placed on an arbitrary vertex of the graph or, if the searcher is already on a vertex v it can slide through an edge e incident to v. In the former case the edge e is cleaned by the searcher. If, for some clean edge e, there is a path from e to a contaminated edge such that no searcher separates the two edges on the path, then e becomes recontaminated. The search number s(G) of G is the minimum number of searchers required to clean all the edges of G. The search number differs by at most one from another well-known graph parameter, namely the pathwidth. The treewidth, the branchwidth and several parameters of the same flavour can be defined by versions of the search game. In this paper we consider a variant of the search game introduced by Barrière et al. [1], called connected search. It requires that, at each step of the search game,