Terrain-like graphs and the median Genocchi numbers

A graph with vertex set {1,center dot center dot center dot, 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. (c) 2023 Elsevier Ltd. All rights reserved.