On constructible graphs, infinite bridged graphs and weakly cop-win graphs

Abstract

A (finite or infinite) graph G is constructible if there exists a well-ordering ⩽ of its vertices such that, for every vertex x which is not the smallest element, there is a vertex y< x which is adjacent to x and to every neighbor z of x with z< x. We prove that every Helly graph and every connected bridged graph are constructible. From the latter result we deduce new characterizations of bridged graphs, and also that any connected bridged graph is ‘moorable’, a property which implies various fixed-point properties (see Chastand, Classes de graphes compacts faiblements modulaires, These de doctorat, Université Claude Bernard, Lyon 1, 1997.), and thus that any connected bridged graph is a retract of the Cartesian product of its blocks. We also solve a problem of Hahn et al. (personal communication) by proving that any finite subgraph of a bridged (resp. constructible) graph G is contained in a finite induced subgraph K of G which is bridged (resp. constructible). Moreover, the vertex set of K is a geodesically convex subset of V( G) whenever G is locally finite or contains no infinite paths. Finally, we study some relations between constructible graphs and a weakening of the concept of cop-win graphs.