On infinite bridged graphs and strongly dismantlable graphs Original Research Article

A graph G is bridged if it contains no isometric cycle of length greater than three. A graph G is strongly dismantlable if its vertices can be linearly ordered x0,…,xα so that, for each ordinal β<α, there exists a strictly increasing finite sequence (ij)0⩽j⩽n of ordinals such that i0=β, in=α and xij+1 is adjacent to xij and to all neighbors of xij in the subgraph of G induced by {xγ: β⩽γ⩽α}. We show that if a connected bridged graph G contains no infinite simplices and, if the vertex set of each ray of G contains an infinite bounded subset, then G is strongly dismantlable. Using this result and some properties of strongly dismantlable graphs, we obtain several invariant simplex properties and Helly-type theorems for bridged graphs.