A Helly theorem for geodesic convexity in strongly dismantlable graphs

AbstractA (finite or infinite) 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 with xij and with all neighbors of xij in the subgraph of G induced by {xy: 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