A Helly theorem for geodesic convexity in strongly dismantlable graphs

Abstract

A (finite or infinite) graph G is strongly dismantlable if its vertices can be linearly ordered x 0,…, x α so that, for each ordinal β < α, there exists a strictly increasing finite sequence ( i j ) 0 ⩽ j ⩽ n of ordinals such that i 0 = β, i n = α and x i j+1 is adjacent with x i j and with all neighbors of x i j in the subgraph of G induced by { x y : β ⩽ γ ⩽ α}. We show that the Helly number for the geodesic convexity of such a graph equals its clique number. This generalizes a result of Bandelt and Mulder (1990) for dismantlable graphs. We also get an analogous equality dealing with infinite families of convex sets.