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Generalized Word-Representable Graphs

Zhidan Feng, Henning Fernau, Pamela Fleischmann, Kevin Mann, Silas Cato Sacher

TL;DR

It is proved that quite famous languages like the palindromes, the copy language or the Lyndon words can match the efficiency of standard graph representations, and a systematic study of graph classes that can be represented by languages in which each letter occurs at most twice is presented.

Abstract

The literature on word-representable graphs is quite rich, and a number of variations of the original definition have been proposed over the years. We are initiating a systematic study of such variations based on formal languages. In our framework, we can associate a graph class to each language over the binary alphabet \{0,1\}. All graph classes that are language-representable in this sense are hereditary and enjoy further common properties. Besides word-representable graphs and, more generally, 1^k- or k-11-representable graphs, we can identify many more graph classes in our framework, like (co)bipartite graphs, (co)comparability graphs, to name a few. It was already known that any graph is 111- or 2-11-representable. When such representations are considered for storing graphs, 111- or 2-11-representability bears the disadvantage of being significantly inferior to standard adjacency matrices or lists. We prove that quite famous languages like the palindromes, the copy language or the Lyndon words can match the efficiency of standard graph representations. The perspective of language theory allows us to prove general results that hold for all graph classes that can be defined in this way. This includes certain closure properties (e.g., all language-definable graph classes are hereditary) as well as certain limitations (e.g., all language-representable graph classes contain graphs of arbitrarily large treewidth and of arbitrarily large degeneracy, except a trivial case). As each language describes a graph class, we can also ask decidability questions concerning graph classes, given a concrete presentation of a formal language. We also present a systematic study of graph classes that can be represented by languages in which each letter occurs at most twice. Here, we find graph classes like interval, permutation, circle, bipartite chain, convex, and threshold graphs.

Generalized Word-Representable Graphs

TL;DR

It is proved that quite famous languages like the palindromes, the copy language or the Lyndon words can match the efficiency of standard graph representations, and a systematic study of graph classes that can be represented by languages in which each letter occurs at most twice is presented.

Abstract

The literature on word-representable graphs is quite rich, and a number of variations of the original definition have been proposed over the years. We are initiating a systematic study of such variations based on formal languages. In our framework, we can associate a graph class to each language over the binary alphabet \{0,1\}. All graph classes that are language-representable in this sense are hereditary and enjoy further common properties. Besides word-representable graphs and, more generally, 1^k- or k-11-representable graphs, we can identify many more graph classes in our framework, like (co)bipartite graphs, (co)comparability graphs, to name a few. It was already known that any graph is 111- or 2-11-representable. When such representations are considered for storing graphs, 111- or 2-11-representability bears the disadvantage of being significantly inferior to standard adjacency matrices or lists. We prove that quite famous languages like the palindromes, the copy language or the Lyndon words can match the efficiency of standard graph representations. The perspective of language theory allows us to prove general results that hold for all graph classes that can be defined in this way. This includes certain closure properties (e.g., all language-definable graph classes are hereditary) as well as certain limitations (e.g., all language-representable graph classes contain graphs of arbitrarily large treewidth and of arbitrarily large degeneracy, except a trivial case). As each language describes a graph class, we can also ask decidability questions concerning graph classes, given a concrete presentation of a formal language. We also present a systematic study of graph classes that can be represented by languages in which each letter occurs at most twice. Here, we find graph classes like interval, permutation, circle, bipartite chain, convex, and threshold graphs.

Paper Structure

This paper contains 47 sections, 66 theorems, 13 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries: Fixing General Notions and Notation
  3. Definitions and Simple Observations About the Framework
  4. Operations on Languages and Graphs
  5. Structure and Limitations of Language-Representable Graph Classes
  6. A Decomposition Theorem for Language-Representable Graphs
  7. Limitations of Language-Representable Graphs
  8. Special Representable Graph Classes from Finite Languages
  9. Interval models with respect to $(1,2)$- and $2$-uniform languages
  10. $(1,2)$-uniform languages
  11. 2-uniform languages
  12. Languages with at most two occurrences of $0$ or $1$
  13. Some simple infinite languages and their graph classes
  14. Consequences of this section for general questions
  15. Palindromes and Other Well-known Languages
  16. ...and 32 more sections

Key Result

Corollary 7

Let $L\subseteq \{0,1\}^*$ be a language with $L=L^R$. Then for each word $w$, $G(L,w)= G(L,w^R)$.

Figures (4)

  • Figure 1: Graph classes represented by certain length-uniform and nearly length-uniform languages
  • Figure 2: Interval pairs matching words from $(1,2)$- and $2$-uniform languages (with points $p$ considered intervals of the form $[p,p]$)
  • Figure 3: An interval model with respect to the language $\langle 010 \rangle$ and the graph described by the model
  • Figure 4: Language $L$ represents $C_4$ with the word $w$.

Theorems & Definitions (95)

  • Remark 1
  • Definition 2
  • Remark 3
  • Definition 4
  • Remark 5
  • Example 6
  • Corollary 7
  • Example 8
  • Lemma 9
  • Lemma 10
  • ...and 85 more