Word-Representable Graphs

• Word-representable graphs are finite simple graphs encoded by alternating symbols in a word, characterized by semi-transitive orientations.
• They encompass key subclasses like circle graphs and comparability graphs, and are studied through uniform representation and pattern-avoidance.
• Recent research highlights computational challenges, NP-completeness of recognition, and novel square-free and p-complete representation methods.

A word-representable graph is a finite simple graph whose adjacencies are encoded via the alternation of symbols in a word. This concept links combinatorics on words, permutation patterns, and graph theory by establishing a correspondence between word structures and the presence or absence of edges in a graph. The field is distinguished by a rigorous structural characterization—semi-transitive orientations—as well as deep connections with notable graph classes including circle graphs, comparability graphs, and chromatic graph families. Recent research has also focused on pattern-avoiding word-representation, the role of uniformity in representations, computational aspects, and extremal questions.

1. Formal Definition and Foundational Equivalence

Let $G = (V, E)$ be a simple, undirected graph. Define a finite word $w$ over the alphabet $V$. For $x \neq y \in V$, $x$ and $y$ are said to alternate in $w$ if, after deleting all other letters, the induced subword has the form $xyxy\cdots$ or $yxyx\cdots$, with no two equal letters adjacent. Formalizing this:

$$\forall x\neq y\in V: \; xy\in E \iff x \text{ and } y \text{ alternate in } w$$

A graph $G$ is word-representable if there exists a word $w$ over $V$ representing it in the sense above (Halldórsson et al., 2015, Kitaev, 2017).

A fundamental equivalence establishes a direct correspondence between word-representability and the existence of a semi-transitive orientation:

• Theorem (Halldórsson–Kitaev–Pyatkin): $G$ is word-representable if and only if $G$ admits an acyclic orientation where, for any directed path $v_0 \to v_1 \to \cdots \to v_k$, the presence of $v_0 \to v_k$ implies the presence of all possible arcs $v_i \to v_j$ with $0 \leq i < j \leq k$ (Halldórsson et al., 2015, Kitaev et al., 2021).

This orientation criterion has become the principal tool for establishing word-representability and constructing representations.

2. Hierarchy, Key Classes, and Structural Examples

2.1 Representation Number and Uniform Representations

Every word-representable graph is $k$-word-representable for some $k$. That is, every vertex can be forced to appear exactly $k$ times in a representing word for minimal such $k$, called the representation number $\mathcal{R}(G)$ (Kitaev, 2017).

• $\mathcal{R}(G) = 1$ iff $G$ is a complete graph.
• $\mathcal{R}(G) = 2$ iff $G$ is a circle graph (intersection graph of chords on a circle) (Kitaev, 2014).
• For connected graphs, $\mathcal{R}(G) \leq 2n$, with extremal values for families such as crown graphs and their apexes (Halldórsson et al., 2015, Akgün et al., 2018).

2.2 Notable Subclasses

• Circle graphs: Exactly those with a 2-uniform representation.
• Comparability graphs: Exactly those representable by a concatenation of permutations ("permutationally representable"); equivalently, graphs admitting a transitive orientation (Halldórsson et al., 2015, Kitaev, 2014).
• 3-colorable graphs: Every 3-colorable graph is word-representable; the canonical semi-transitive orientation is obtained by orienting edges along a total order of the color classes (Halldórsson et al., 2015, Kitaev, 2017).
• Bipartite graphs: Permutationally representable with minimal permutation-representation number equal to the size of the smaller part (Mozhui et al., 2021).

2.3 Closure and Hereditary Properties

Word-representable graphs form a hereditary class: all induced subgraphs of a word-representable graph are word-representable. The class is closed under split recomposition and stable under several natural graph operations, with representation number controlled by the maximal component's number (Dwary et al., 2024).

3. Pattern-Avoiding Word-Representability

Recent research has investigated graph families that admit representing words avoiding specific permutation patterns, especially classical patterns of length 3 such as 132, 123, etc. (Mandelshtam, 2016, Gao et al., 2016).

Let $\tau$ be a permutation pattern. A word $w$ over a totally ordered alphabet is $\tau$-avoiding if no subsequence of the word is order-isomorphic to $\tau$.

3.1 τ-Representable Graphs

A graph $G$ is $\tau$-representable if there is a labeling of its vertices and a representing word that is $\tau$-avoiding.

3.2 Main Results for Length-3 Patterns

• 132-Representability: Every 132-representable graph is a circle graph, but not all circle graphs are 132-representable (the disjoint union $K_4 \sqcup K_4$ is a minimal example). All trees and cycles are 132-representable. The class of 132-representable graphs is strictly between the class of circle graphs and all word-representable graphs (Mandelshtam, 2016, Gao et al., 2016).
• 123-Representability: Every 123-representable graph is a circle graph, each admits a 2-uniform 123-avoiding representation. However, not all circle graphs are 123-representable (the star $K_{1,6}$ is a minimal example and not 123-representable) (Mandelshtam, 2016).
• The classes of 132- and 123-representable graphs are distinct proper subclasses of circle graphs, with various explicit separation examples (Mandelshtam, 2016).

Table: Subclass Relationships for Small Patterns

Pattern	All Graphs	Word-Representable	Circle Graphs	τ-Representable	Minimal Non-τ-Repr.
132	⊇	⊇	⊇	⊂	$K_4\sqcup K_4$
123	⊇	⊇	⊇	⊂	$K_{1,6}$

Both 132- and 123-representable families are not closed under taking all circle graphs.

3.3 2-Uniform τ-Avoidance

For small patterns, 2-uniform τ-avoiding representation theorems hold:

• Trees are 132-representable by 2-uniform words.
• Complete graphs are 123-representable by 2-uniform 123-avoiding words, but not 132-representable by 2-uniform 132-avoiding words for $n>3$ (Mandelshtam, 2016).

4. Structural, Algorithmic, and Enumeration Results

4.1 Semi-Transitive Orientation Algorithms and NP-Completeness

• Existence of a semi-transitive orientation is both necessary and sufficient for word-representability (Halldórsson et al., 2015).
• Recognition is in NP: a semi-transitive orientation or a suitable k-uniform word serves as a polynomial certificate (Kitaev et al., 2021, Halldórsson et al., 2015).
• The recognition problem is NP-complete, both by orientation-based and word-based formulations (Akgün et al., 2018, Kitaev et al., 2021).
• For graphs on up to 9 vertices, the complete distribution of representation numbers is computed; there exist graphs with $\mathcal{R}(G)=4$ for 9 vertices (Akgün et al., 2018).

4.2 Enumeration and Extremal Speed

The number of word-representable graphs on $n$ vertices satisfies:

$$\log_2 a_n = \bigl(\tfrac23 + o(1)\bigr) \binom{n}{2}$$

induced by the fact that the class shares index 3 with 3-colorable graphs in the sense of the Alekseev–Bollobás–Thomason theorem (Collins et al., 2013).

4.3 Minimal Non-Word-Representable Graphs

A complete classification of minimal non-word-representable graphs that are non-comparability graphs is established, isolating precisely the intersection of Gallai's forbidden subgraphs for comparability with the non-semi-transitive cases (Kenkireth et al., 10 Feb 2025). Conversely, adding a universal vertex to a semi-transitive minimal non-comparability graph yields minimal non-word-representable graphs with an apex, resulting in several infinite families.

5. Extensions, Variants, and Open Directions

5.1 Pattern-Avoidance Generalizations

The landscape of τ-representable graphs is largely uncharted beyond 132 and 123. Open questions involve the classification for longer patterns, e.g., 1342, 1324, and mixed pattern avoidance (Mandelshtam, 2016, Gao et al., 2016).

5.2 Square-Free and p-Complete Square-Free Representations

• Every non-empty word-representable graph has a representation that is square-free (no non-trivial squares as factors in the word) (Das et al., 2024).
• For uniform representations, minimal-length words, and complete graphs, exact enumerations and constructions of square-free representants are available (Das et al., 2024).
• The concept of p-complete square-free representations is a new generalization, requiring square-freeness not just globally, but for every induced subword on every subset of the alphabet of size $p$ or more. For small $p$, there is a structural characterization: only complete graphs are 1-complete square-free, only edgeless graphs are 2-complete square-free, and 3-complete square-free uniform word-representable graphs are exactly the $K_3$-free circle graphs (Das et al., 8 May 2025).

5.3 Word-Representability in Large Graph Operations

Word-representability is preserved under split recomposition, with the representation number of the recomposed graph equaling the maximum of the components' numbers (Dwary et al., 2024). Parity graphs, a subclass of perfect graphs, are thus word-representable (Dwary et al., 2024). There are exact criteria for recomposed comparability graphs and their permutation-representation numbers.

5.4 Forbidden Subgraph Characterizations for Structured Graph Families

• Split graphs and threshold graphs are well-understood, with forbidden subgraph characterizations known for split graphs with clique number up to 5 (Chen et al., 2019).
• For $K_m$-$K_n$ (two-clique) graphs, explicit forbidden subgraph characterizations are completed for $m \leq 4$, with extension to higher $m$ open (Chen et al., 21 Aug 2025).

5.5 Computational Methods and Human-Checkable Obstructions

Human-verifiable, algorithmically generated proofs of non-word-representability using semi-transitive orientation branching, cycle forcing, and shortcut detection are established as systematic tools; corrections to the minimal non-representable graphs on small vertex sets are leveraged using these methods (Kitaev et al., 2021).

6. Research Directions and Open Problems

• Classify word-representable graphs by forbidden induced subgraphs (analogous to Gallai’s classification for comparability graphs) (Kenkireth et al., 10 Feb 2025, Collins et al., 2013).
• Determine the rate of growth of the maximal representation number $\mathcal{R}(G)$ as a function of $n$.
• Investigate the behavior of word-representability under further graph operations: edge subdivisions, contractions, line graphs, etc.
• Extend pattern-avoidance theory, including enumeration, forbidden subgraph characterizations, and computational complexity, to longer and richer classes of patterns (Mandelshtam, 2016, Gao et al., 2016).
• Develop efficient algorithms for constructing square-free and p-complete square-free word-representations and understand the precise containment hierarchy among their hereditary graph classes (Das et al., 8 May 2025, Das et al., 2024).
• Characterize word-representability in application-relevant graph families, such as generalized de Bruijn graphs, split graphs, and intersection graphs arising from combinatorics on words (Petyuk, 2022, Huang et al., 2023, Chen et al., 2019).
• Analyze further the combinatorial boundary between word-representable, comparability, and circle graphs, particularly through the lens of τ-pattern-avoiding words.

Word-representable graphs thus comprise a richly structured class that unifies and extends significant families in graph theory, governed by alternation phenomena in words and captured algorithmically via semi-transitive orientations. The thematic interplay between word combinatorics and graph structure continues to yield new subfields, structural insights, and computational challenges.