[Paper Review] Interval Graphs with Containment Restrictions

Jeffrey Beyerl, Robert E. Jamison|arXiv (Cornell University)|Sep 29, 2011

Advanced Graph Theory Research20 references3 citations

TL;DR

This paper introduces $p$-improper interval graphs, defined by interval representations where no interval contains more than $p$ others, and characterizes the minimal forbidden subgraphs for the $\mathscr{I}_p$ class. It establishes that $p$-improper interval graphs exhibit a rich diversity of minimal forbidden subgraphs, with a complete characterization provided for the balanced case using a novel construction based on vertex types and component weights.

ABSTRACT

An interval graph is proper iff it has a representation in which no interval contains another. Fred Roberts characterized the proper interval graphs as those containing no induced star $K_{1,3}$. Proskurowski and Telle have studied $q$-proper graphs, which are interval graphs having a representation in which no interval is properly contained in more than $q$ other intervals. Like Roberts they found that their classes of graphs where characterized, each by a single minimal forbidden subgraph. This paper initiates the study of $p$-improper interval graphs where no interval contains more than $p$ other intervals. This paper will focus on a special case of $p$-improper interval graphs for which the minimal forbidden subgraphs are readily described. Even in this case, it is apparent that a very wide variety of minimal forbidden subgraphs are possible.

Motivation & Objective

• To define and study $p$-improper interval graphs, where no interval contains more than $p$ other intervals in a representation.
• To identify the minimal forbidden subgraphs (MFISGs) that characterize the class $\mathscr{I}_p$ of $p$-improper interval graphs.
• To analyze the structural properties of $p$-improper interval graphs, particularly focusing on the balanced case where component weights and vertex types are used to determine impropriety.
• To provide a constructive characterization of $p$-critical and balanced interval graphs using the $\textsf{BAL}_k(\mathcal{H})$ construction for $k=0,1,2$.
• To establish that the weight of a vertex provides a lower bound on the impropriety of the graph, and that equality holds precisely in the balanced case.

Proposed method

• Define the impropriety $\mbox{imp}(G)$ as the minimum, over all interval representations, of the maximum number of intervals contained within any single interval.
• Introduce the weight $\mbox{wt}(z)$ of a vertex $z$ as the sum of the sizes of the $n-2$ smallest non-exterior local components in $G\setminus\{z\}$, where $n$ is the number of local components.
• Use the weight function to derive a lower bound on $\mbox{imp}(G)$, with equality if and only if the graph is balanced.
• Define $\textsf{BAL}_k(\mathcal{H})$ as a construction where a central vertex $z$ is joined to a sequence of interval graphs $H_i$, with $k$ pendant $P_3$ paths attached for $k=1,2$, and cliques among the largest $H_i$ to ensure criticality.
• Prove that $p$-critical and balanced graphs are exactly those isomorphic to $\textsf{BAL}_k(\mathcal{H})$ for $k=0,1,2$, with appropriate clique conditions on the largest components.
• Demonstrate that removing any interval from a $\textsf{BAL}_k(\mathcal{H})$ representation reduces the impropriety, proving criticality.

Experimental results

Research questions

• RQ1 What are the minimal forbidden subgraphs that characterize the class of $p$-improper interval graphs?
• RQ2 How does the weight of a vertex relate to the impropriety of the graph, and when is this bound tight?
• RQ3 What structural conditions define $p$-critical and balanced interval graphs?
• RQ4 Can a constructive characterization of $p$-critical and balanced interval graphs be given using a parameterized family of graph constructions?
• RQ5 How do the number and type of local components (especially exterior and interior) affect the impropriety of a vertex and the overall graph?

Key findings

• The class $\mathscr{I}_p$ of $p$-improper interval graphs has a minimal forbidden subgraph characterization, with $K_{1,p+3}$ being a minimal forbidden subgraph for each $p$.
• For $p=1$, the paper provides a complete list of 10 minimal forbidden subgraphs, demonstrating the complexity and diversity of such subgraphs even at low $p$.
• The weight $\mbox{wt}(z)$ of a vertex $z$ provides a lower bound on $\mbox{imp}(G)$, and equality $\mbox{wt}(G) = \mbox{imp}(G)$ holds if and only if the graph is balanced.
• A graph is $p$-critical and balanced if and only if it is isomorphic to $\textsf{BAL}_k(\mathcal{H})$ for $k=0,1,2$, with the largest components among the $H_i$ being cliques.
• In $\textsf{BAL}_k(\mathcal{H})$ constructions, removing any interval from the representation reduces the impropriety, confirming the criticality of these graphs.
• The construction $\textsf{BAL}_k(\mathcal{H})$ ensures that the impropriety equals the weight of the central vertex, and this value is minimal across all representations, confirming the minimality of the representation.