Metric properties of incomparability graphs with an emphasis on paths

TL;DR

This paper investigates metric properties of incomparability graphs derived from posets, focusing on the existence of infinite paths and specific subgraph structures under certain conditions.

Contribution

It establishes conditions under which incomparability graphs contain infinite induced paths or specific subgraphs like combs or kites.

Findings

Connected incomparability graphs with infinite diameter contain infinite induced paths.

Infinite degree-3 vertex sets imply the presence of combs or kites as induced subgraphs.

Results link graph metric properties to structural features of incomparability graphs.

Abstract

We describe some metric properties of incomparability graphs. We consider the problem of the existence of infinite paths, either induced or isometric, in the incomparability graph of a poset. Among other things, we show that if the incomparability graph of a poset is connected and has infinite diameter, then it contains an infinite induced path. Furthermore, if the diameter of the set of vertices of degree at least $3$ is infinite, then the graph contains as an induced subgraph either a comb or a kite.