On bridged graphs and cop-win graphs

Abstract

A subgraph H of a graph G is isometric if the distance between any pair of vertices in H is the same as that in G. A graph is bridged if it contains no isometric cycles of length greater than three (roughly, each cycle of length greater than three has a shortcut). It is proved that every nontrivial bridged graph has a pair of adjacent vertices u, v, with v adjacent to everything to which u is adjacent. (This result was conjectured by P. Hell, and independently raised as a question, though in a different form, by R. E. Jamison). From this, it follows that every bridged graph G has a vertex u such that G − u is an isometric subgraph, and hence also bridged. The latter is a result of Farber. It is also proved that a connected graph is bridged if and only if every isometric subgraph is a cop-win graph, as defined by Nowakowski and Winkler.