Veto Interval Graphs and Variations

TL;DR

This paper introduces veto interval (VI) graphs, a new class of graphs based on interval intersections with veto marks, and explores their properties, related graph families, and chromatic number bounds.

Contribution

It defines veto interval graphs and related variants, establishing their properties, relationships with other graph classes, and bounds on chromatic numbers.

Findings

Identified families of graphs that are VI graphs.

Proved bounds on the maximum chromatic number of VI graphs.

Explored relationships between approval graphs and tolerance graphs.

Abstract

We introduce a variation of interval graphs, called veto interval (VI) graphs. A VI graph is represented by a set of closed intervals, each containing a point called a veto mark. The edge $ab$ is in the graph if the intervals corresponding to the vertices $a$ and $b$ intersect, and neither contains the veto mark of the other. We find families of graphs which are VI graphs, and prove results towards characterizing the maximum chromatic number of a VI graph. We define and prove similar results about several related graph families, including unit VI graphs, midpoint unit VI (MUVI) graphs, and single and double approval graphs. We also highlight a relationship between approval graphs and a family of tolerance graphs.