Hyperbolic Bridged Graphs

Given a connected graph G, we take, as usual, the distance xy between any two verticesx , y of G to be the length of some geodesic between x and y. The graph G is said to be δ - hyperbolic, for some δ ≥ 0, if for all vertices x,y , u, v in G the inequality xy + uv ≤ max{ xu + yv,xv + yu } + δholds, and G isbridged if it contains no finite isometric cycles of length four or more. In this paper, we will show that a finite connected bridged graph is 1-hyperbolic if and only if it does not contain any of a list of six graphs as an isometric subgraph.