A vertex ordering characterization of simple-triangle graphs

TL;DR

This paper characterizes simple-triangle graphs through a specific vertex ordering involving alternating and transitive orientations, providing a new structural understanding of these intersection graphs.

Contribution

It introduces a novel vertex ordering characterization of simple-triangle graphs based on alternating and transitive orientations.

Findings

A graph is a simple-triangle graph if and only if it admits a vertex ordering with specific orientation properties.

Provides a structural characterization that can aid in recognition algorithms.

Enhances understanding of intersection graph classes through orientation-based properties.

Abstract

Consider two horizontal lines in the plane. A pair of a point on the top line and an interval on the bottom line defines a triangle between two lines. The intersection graph of such triangles is called a simple-triangle graph. This paper shows a vertex ordering characterization of simple-triangle graphs as follows: a graph is a simple-triangle graph if and only if there is a linear ordering of the vertices that contains both an alternating orientation of the graph and a transitive orientation of the complement of the graph.