Representing Graphs with Binary Languages: A Formal Language Approach to Generalizing Word-Representable Graphs

Core Concepts

This paper introduces a novel framework for representing graphs using binary languages, generalizing the concept of word-representable graphs and offering a unified approach to studying various graph classes and their properties.

Abstract

Feng, Z., Fernau, H., Fleischmann, P., Mann, K., & Sacher, S. C. (2024). Generalized Word-Representable Graphs: A Formal Language Approach. Leibniz International Proceedings in Informatics, WRG, WRG:1–WRG:49. https://doi.org/10.4230/LIPIcs.CVIT.2016.WRG

This paper aims to generalize the concept of word-representable graphs by linking it to formal language theory, proposing a framework where any binary language can define a graph class.

Key Insights Distilled From

Generalized Word-Representable Graphs

by Zhidan Feng,... at arxiv.org 11-06-2024

https://arxiv.org/pdf/2411.03274.pdf

Deeper Inquiries

How could this framework be extended to represent directed graphs or hypergraphs?

This framework can be extended to represent directed graphs and hypergraphs by modifying the core definitions of L-representation and leveraging additional symbols or structures within the language:
Directed Graphs:

Modified Morphism: Instead of using the morphism h_u,v which leads to a symmetric edge relation, we can define two morphisms: h_u->v and h_v->u. h_u->v would map 'u' to '0', 'v' to '1', and all other symbols to λ, while h_v->u would map 'v' to '0', 'u' to '1', and the rest to λ.
Directed Edge Definition: A directed edge (u,v) (from u to v) would exist in the graph G(L,w) if and only if h_u->v(w) ∈ L.

Hypergraphs:

Extended Alphabet: Introduce a new symbol, for example, '|', to denote the separation between edges in the word representation. Each section of the word delimited by '|' would represent a hyperedge.
Hyperedge Definition: A hyperedge {u_1, u_2, ..., u_n} would exist in the hypergraph H(L,w) if and only if there exists a section within w, delimited by '|', containing all the symbols u_1, u_2, ..., u_n at least once. The language L could impose further restrictions on the order or arrangement of these symbols within the section.

Example for Directed Graphs:
Let's consider the language L = {01}. The word "abbc" would represent the following directed graph:

Vertices: {a, b, c}
Edges: {(a,b), (b,c)}
This is because h_a->b("abbc") = "01" ∈ L and h_b->c("abbc") = "01" ∈ L, but no other h_x->y(w) belongs to L.
Challenges and Considerations:

Complexity: Extending the framework to hypergraphs might significantly increase the complexity of analyzing the resulting graph class.
Language Design: Carefully designing the language L is crucial to ensure that it captures the desired properties of directed graphs or hypergraphs.
Canonical Representation: Defining a canonical L-representation for directed graphs and hypergraphs might be more challenging than for undirected graphs.

Could there be limitations in representing graphs with specific properties, such as planarity or bounded degree, using this framework?

Yes, there are likely limitations in representing certain graph classes with specific properties using this framework. Here's why:

Hereditary Property: Theorem 4.7 states that all language-representable graph classes are hereditary, meaning that they are closed under induced subgraphs. This inherent property imposes limitations:

Planar Graphs: Planar graphs are not hereditary. For example, a grid graph is planar, but it contains a K4 as a minor, which is not planar. Therefore, no language L can represent all planar graphs.
Bounded Degree Graphs: Similar to planar graphs, graphs with bounded degree are not hereditary. For instance, a cycle graph has maximum degree 2, but it can contain an arbitrarily large complete graph as a minor, violating the bounded degree property.

Language Constraints: The structure of languages and the way they define edges through morphisms might not be expressive enough to capture certain graph properties:

Non-Local Properties: Properties like planarity or bounded degree depend on the overall structure of the graph, not just local relationships between vertices. The word representation and morphisms primarily capture local information.
Counting and Arrangement: While languages can enforce constraints on the number of occurrences of symbols, capturing complex relationships between vertex degrees or global arrangements required for planarity might be difficult or impossible.
Possible Approaches and Open Questions:

Restricted Graph Classes: It might be possible to represent subclasses of planar graphs or bounded degree graphs with additional constraints. For example, representing outerplanar graphs or graphs with maximum degree 3 might be feasible with carefully designed languages.
Alternative Frameworks: Exploring alternative language-based frameworks or extending the existing one with additional mechanisms might be necessary to represent graph classes with such properties effectively.
Decidability: The question of whether a given language L can represent a graph class with a specific property (like planarity) might be undecidable in the general case.

What are the implications of this framework for developing more efficient algorithms for graph problems, such as graph isomorphism or finding shortest paths?

This framework offers both potential advantages and challenges for developing efficient graph algorithms:
Potential Advantages:

Compact Representation: For specific graph classes representable by simple languages, the word representation could be significantly more compact than traditional adjacency lists or matrices, especially for dense graphs. This could lead to reduced storage space and potentially faster algorithms.
Word-Based Algorithms: The framework allows leveraging the rich toolbox of algorithms and techniques from formal language theory and stringology. Problems like finding paths or cycles might be solvable using pattern matching or automata-based approaches on the word representation.
Structural Insights: The connection between languages and graph classes provides structural insights that could be exploited for algorithm design. For instance, understanding the properties of a language L might directly translate to efficient algorithms for graphs in GL.
Challenges:

General Graph Problems: For general graph problems like graph isomorphism or finding shortest paths, it's unclear whether this framework would provide significant advantages. These problems are already challenging for general graphs, and the word representation might not offer substantial benefits.
Language Complexity: The efficiency of algorithms would heavily depend on the complexity of the language L. For complex languages, even basic operations on the word representation could be computationally expensive.
Translating Algorithms: Translating existing graph algorithms to work directly on the word representation might not always be straightforward or efficient. New algorithms specifically designed for this framework might be necessary.
Specific Implications:

Graph Isomorphism: For graph classes with unique or canonical word representations, graph isomorphism might be solvable in polynomial time by simply comparing the corresponding words. However, finding such representations for general graph classes is a long-standing open problem.
Shortest Paths: For graph classes representable by languages with certain structural properties, it might be possible to develop efficient shortest path algorithms based on analyzing the word representation. For example, if the language enforces a specific ordering of vertices, finding shortest paths might reduce to simpler string operations.
Overall, the framework's implications for algorithm design are promising but require further investigation. The key lies in identifying graph classes with suitable language representations and developing algorithms that exploit the connection between languages and graph properties.