Improper Interval Graphs and the Corresponding Minimal Forbidden Interval Subgraphs

TL;DR

This paper explores the structure of p-improper interval graphs by classifying their minimal forbidden subgraphs into four categories, extending previous work on 1-improper graphs.

Contribution

It generalizes the classification of minimal forbidden subgraphs from 1-improper to all p-improper interval graphs, revealing diverse subgraph types.

Findings

Identification of four broad categories of minimal forbidden subgraphs.

Extension of previous results from 1-improper to general p-improper interval graphs.

Enhanced understanding of the structural properties of improper interval graphs.

Abstract

An interval graph is considered improper if and only if it has a representation such that an interval contains another interval. Previously these have been investigated in terms of balance and minimal forbidden interval subgraphs for the class of 1-improper interval graphs. This paper investigates the minimal forbidden interval sub-graphs further, generalizing results to all p-improper interval graphs. It is apparent that there are many different types of possible minimal forbidden subgraphs that fall into four broad categories.