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Structural and Spectral Properties of Strictly Interval Graphs

Lilian Markenzon, Claudia Justel

TL;DR

The paper develops a new, linear-time recognition framework for strictly interval graphs via the critical clique graph, enriching the theory of chordal graph subclasses. It introduces SI-core graphs as a two-separator, strictly interval subclass and analyzes their Laplacian spectra, showing many members are Laplacian-integral under specific parameter conditions. Using equitable quotient theory, it characterizes fixed spectral components across G(s,p) and provides a constructive approach to realizing spectral features, while also addressing issues of spectral non-uniqueness and spectrum-driven design. This work advances structural and spectral understanding of a tight graph class intersection with potential implications for graph algorithms and spectral graph theory.

Abstract

In this paper we deal with a subclass of chordal graphs, which are simultaneously strictly chordal and interval, the strictly interval graphs. We present a new characterization of the class that leads to a simple linear recognition algorithm. Next we introduce a new subclass of strictly interval graphs, the $SI$-core graphs, that are non-split and non-cograph graphs and show that several elements of the new class are Laplacian integral.

Structural and Spectral Properties of Strictly Interval Graphs

TL;DR

The paper develops a new, linear-time recognition framework for strictly interval graphs via the critical clique graph, enriching the theory of chordal graph subclasses. It introduces SI-core graphs as a two-separator, strictly interval subclass and analyzes their Laplacian spectra, showing many members are Laplacian-integral under specific parameter conditions. Using equitable quotient theory, it characterizes fixed spectral components across G(s,p) and provides a constructive approach to realizing spectral features, while also addressing issues of spectral non-uniqueness and spectrum-driven design. This work advances structural and spectral understanding of a tight graph class intersection with potential implications for graph algorithms and spectral graph theory.

Abstract

In this paper we deal with a subclass of chordal graphs, which are simultaneously strictly chordal and interval, the strictly interval graphs. We present a new characterization of the class that leads to a simple linear recognition algorithm. Next we introduce a new subclass of strictly interval graphs, the -core graphs, that are non-split and non-cograph graphs and show that several elements of the new class are Laplacian integral.

Paper Structure

This paper contains 10 sections, 19 theorems, 13 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Graph Basic Concepts
  3. Subclasses of chordal graphs
  4. Strictly Interval Graphs: new characterization and recognition algorithm
  5. Spectral Basic Concepts
  6. A new subclass: $SI$-core graphs
  7. Laplacian spectrum of $SI$-core graphs
  8. Combinatorial and Spectral Results
  9. Answering some questions
  10. Acknowledgment

Key Result

Theorem 1

HT06 A maximal clique is a leaf in some clique-tree if and only if it is a boundary clique.

Figures (6)

  • Figure 1: 2-net and bipartite claw graphs
  • Figure 2: Gem and dart graphs.
  • Figure 3: A $SI$-core graph in $\mathcal{G}(2,3)$
  • Figure 4: The $min$ and $max$$SI$-core graphs in $\mathcal{G}(2,3)$
  • Figure 5: A $SI$-core graph in $\mathcal{G}(3,10)$
  • ...and 1 more figures

Theorems & Definitions (27)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 4.1
  • Lemma 5
  • Lemma 6
  • Theorem 7
  • Corollary 7.1
  • Theorem 8
  • ...and 17 more