direction, if is a subgraph of an interval graph with clique number , then has a path decomposition of width whose nodes are given by the maximal cliques of the interval graph. For instance, the interval graph shown with its interval representation in the figure has a path decomposition with five nodes, corresponding to its five maximal cliques , , , , and ; the maximum clique size is three and the width of this path decomposition is two. This equivalence between pathwidth and interval thickness is closely analogous to the equivalence between treewidth and the minimum clique number (minus one) of a chordal graph of which the given graph is a subgraph. Interval graphs are a special case of chordal graphs, and chordal graphs can be represented as intersection graphs of subtrees of a common tree generalizing the way that interval graphs are intersection graphs of subpaths of a path. Vertex separation number Suppose that the vertices of a graph are linearly ordered. Then the vertex separation number of is the smallest number such that, for each vertex , at most vertices are earlier than in the ordering but that have or a later vertex as a neighbor. The vertex separation number of is the minimum vertex separation number of any linear ordering of . The vertex separation number was defined by , and is equal to the pathwidth of . This follows from the earlier equivalence with interval graph clique numbers: if is a subgraph of an interval graph , represented (as in the figure) in such a way that all interval endpoints are distinct, then the ordering of the left endpoints of the intervals of has vertex separation number one less than the clique number of . And in the other direction, from a linear ordering of one may derive an interval representation in which the left endpoint of the interval for a vertex is its position in the ordering and the right endpoint is the position of the neighbor of that comes last in the ordering. Node searching number The node searching game on a graph is a form of pursuit–evasion in which a set of searchers collaborate to track down a fugitive hiding in a graph. The searchers are placed on vertices of the graph while the fugitive may be in any edge of the graph, and the fugitive's location and moves are hidden from