Terrain-like graphs and the median Genocchi numbers

Abstract

A graph with vertex set {1,…,n} is terrain-like if, for any edge pair {a,c}, {b,d} with a<b<c<d, the edge {a,d} also exists. Terrain-like graphs frequently appear in geometry in the context of visibility graphs. We show that terrain-like graphs are counted by the median Genocchi numbers. To this end, we show a natural bijection between terrain-like graphs and Dumont derangements of the second kind, with the graph’s edges representing individual transpositions.