the searchers. In each turn, some or all of the searchers may move (arbitrarily, not necessarily along edges) from one vertex to another, and then the fugitive may move along any path in the graph that does not pass through a searcher-occupied vertex. The fugitive is caught when both endpoints of his edge are occupied by searchers. The node searching number of a graph is the minimum number of searchers needed to ensure that the fugitive can be guaranteed to be caught, no matter how he moves. As show, the node searching number of a graph equals its interval thickness. The optimal strategy for the searchers is to move the searchers so that in successive turns they form the separating sets of a linear ordering with minimal vertex separation number. Bounds Every -vertex graph with pathwidth has at most edges, and the maximal pathwidth- graphs (graphs to which no more edges can be added without increasing the pathwidth) have exactly this many edges. A maximal pathwidth- graph must be either a -path or a -caterpillar, two special kinds of -tree. A -tree is a chordal graph with exactly maximal cliques, each containing vertices; in a -tree that is not itself a -clique, each maximal clique either separates the graph into two or more components, or it contains a single leaf vertex, a vertex that belongs to only a single maximal clique. A -path is a -tree with at most two leaves, and a -caterpillar is a -tree that can be partitioned into a -path and a set of -leaves each adjacent to a separator -clique of the -path. In particular the maximal graphs of pathwidth one are exactly the caterpillar trees. Since path-decompositions are a special case of tree-decompositions, the pathwidth of any graph is greater than or equal to its treewidth. The pathwidth is also less than or equal to the cutwidth, the minimum number of edges that cross any cut between lower-numbered and higher-numbered vertices in an optimal linear arrangement of the vertices of a graph; this follows because the vertex separation number, the number of lower-numbered vertices with higher-numbered neighbors, can at most equal the number of cut edges. For similar reasons, the cutwidth is at most the pathwidth times the maximum degree of the vertices in a given graph. Any -vertex forest has pathwidth . For, in a forest, one can always find a constant number