Terrain-like Graphs and the Median Genocchi Numbers

Abstract: A graph with vertex set ${1,\ldots,n}$ 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 prove a bijection between terrain-like graphs and Dumont derangements of the second kind.