Terrain-like Graphs and the Median Genocchi Numbers

TL;DR

This paper introduces terrain-like graphs, a class of graphs relevant in geometry, and establishes a connection with median Genocchi numbers through a bijection with Dumont derangements.

Contribution

It characterizes terrain-like graphs and proves they are counted by median Genocchi numbers, linking graph theory with combinatorial number sequences.

Findings

Terrain-like graphs are counted by median Genocchi numbers.

A bijection between terrain-like graphs and Dumont derangements is established.

The work connects geometric visibility graphs with combinatorial enumeration.

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.