Isometric cycles, cutsets, and crowning of bridged graphs

A graph G is bridged if every cycle C of length at least 4 has vertices x; y such that dG(x; y )< dC (x ; y ). A cycle C is isometric if dG(x; y )1⁄4 dC (x; y ) for all x; y 2 V (C). We show that every graph contractible to a graph with girth g has an isometric cycle of length at least g. We use this to show that every minimal cutset S in a bridged graph G induces a connected subgraph. We introduce a ‘‘crowning’’ construction to enlarge