Word-Representable Co-Bipartite Graphs

• Word-representable co-bipartite graphs are graphs whose vertices split into two cliques and admit an alternating word representation linking vertex order to edge presence.
• They leverage semi-transitive orientations and the circularly compatible ones property in adjacency matrices to enable linear-time recognition and explicit 3-uniform word constructions.
• Forbidden induced subgraph characterizations and vertex-ordering criteria provide actionable insights into structural constraints and algorithmic representations in these graphs.

A word-representable co-bipartite graph is a finite simple graph whose vertex set can be partitioned into two disjoint cliques and that admits a word representation of the alternation type: there exists a word over the vertex set such that two distinct vertices alternate in the word if and only if they are adjacent. This class forms the intersection of word-representable graphs and co-bipartite graphs, and is intimately linked to combinatorial, algebraic, and matrix-theoretic perspectives, especially via the concept of semi-transitive orientations and forbidden substructure characterizations.

1. Definitions and Preliminaries

Let $G = (V, E)$ be a simple undirected graph. For any word $w$ over the alphabet $V$, two distinct letters $x, y \in V$ are said to alternate in $w$ if, after deleting all letters except $x$ and $y$, the resulting subword is either $xyxy\dots$ or $yxyx\dots$; specifically, no double occurrence ($xx$ or $yy$) appears. A graph $G$ is word-representable if there exists such a word $w$ such that $\{ x, y \} \in E$ if and only if $x$ and $y$ alternate in $w$.

A k-uniform word is a word in which each letter occurs exactly $k$ times. The representation number of a word-representable graph is the minimum $k$ such that a $k$-uniform representing word exists.

A graph is co-bipartite if its complement is bipartite; that is, $V$ can be partitioned into two disjoint cliques $X$ and $Y$. The central structural tool in word-representability is the concept of a semi-transitive orientation: an acyclic orientation in which any shortcut is prohibited—specifically, there is no induced path $$v_0 \rightarrow v_1 \rightarrow \dots \rightarrow v_m$$ ($m \geq 2$) together with an arc $v_0 \rightarrow v_m$, unless the subgraph induced by $v_0, v_1, \dots, v_m$ is a complete transitive tournament (Halldórsson et al., 2015).

2. Structural and Forbidden Subgraph Characterizations

Semi-transitive Orientations and Matrix Criteria

A fundamental result is that $G$ is word-representable if and only if it admits a semi-transitive orientation (Halldórsson et al., 2015, Srinivasan et al., 13 Dec 2025). For co-bipartite graphs $G = (X, Y; E)$ with adjacency matrix $M(G)$, this is equivalent to $M(G)$ having the circularly compatible ones property: there exist linear orders on rows ($X$) and columns ($Y$) such that in each order, the 1s in every row (resp., column) form a circular interval, and, critically, the left and right endpoints of these intervals in the respective orders form circularly monotone sequences (Srinivasan et al., 13 Dec 2025).

Forbidden induced subgraphs arise from the minimal obstructions to this property and to the existence of a semi-transitive orientation. Specifically, a co-bipartite graph is word-representable if and only if it contains no induced subgraph isomorphic to one arising from a forbidden configuration (called CCO^∞ in the matrix language) (Srinivasan et al., 13 Dec 2025).

For the subclass of circle graphs, an important structural finding is that in the co-bipartite setting, circle graphs coincide precisely with permutation graphs. The minimal forbidden induced subgraphs for co-bipartite permutation/circle graphs are $\overline{C_{2k}}$ ($k \geq 3$) and three small explicit graphs $G_1, G_2, G_3$ (Srinivasan et al., 13 Dec 2025).

Finite Forbidden Subgraphs for Small Cliques

For graphs partitioned into $K_m$ and $K_n$, complete characterizations by forbidden subgraphs are known for $m \leq 4$ (Chen et al., 21 Aug 2025):

• For $m = 1$ or $2$, all $K_m$-$K_n$ graphs are word-representable.
• For $m = 3$, word-representability is characterized by the absence of a single minimal forbidden induced subgraph $A_3$.
• For $m = 4$, there are seven minimal forbidden induced subgraphs $B_1, \dots, B_7$ whose structures are explicitly described.

For larger cliques ($m \geq 5$), the forbidden subgraph characterization becomes intractable due to combinatorial explosion, and only partial results are known (Chen et al., 21 Aug 2025).

3. Vertex Ordering and Interval Characterization

A recent breakthrough is the vertex-ordering characterization for word-representable co-bipartite graphs (Das et al., 3 Sep 2025). For $G = (X, Y; E)$, define the type of vertex $a \in X$ with respect to its neighborhood $N(a) \subseteq Y$:

• Type A: $N(a)$ is a contiguous interval $[x,y]$ in $Y$ (with some ordering).
• Type C: $N(a) = [1,x] \cup [y,n]$, a union of beginning and end intervals.

$G$ is word-representable if and only if there exists a total order $<$ on $X$ such that:

1. All type A vertices precede all type C vertices.
2. Among type C vertices, the endpoints of their intervals exhibit coordinatewise monotonicity in the order.
3. The same coordinatewise monotonicity holds among type A vertices.
4. For any type A $a$ preceding type C $c$, certain cross-interval endpoint inequalities are satisfied.

This ordering criterion both provides a structural explanation of word-representability for co-bipartite graphs and underpins an explicit representation algorithm (Das et al., 3 Sep 2025).

4. Algorithmic Aspects and Recognition Algorithms

Recognition of word-representable co-bipartite graphs is algorithmically efficient: testing the circularly compatible ones property for the adjacency matrix $M(G)$ can be performed in linear time via Safe’s algorithm, which either yields the required biorders or returns a forbidden configuration as a certificate for non-word-representability (Srinivasan et al., 13 Dec 2025). The recognition procedure operates in $O(m+n)$ time, where $m = |E|$ and $n = |V|$, aligning with the optimal complexity for graph representation in adjacency list or matrix form.

The 3-uniform word-representation construction for any word-representable co-bipartite graph (given an appropriate vertex ordering) is also executable in linear time (Das et al., 3 Sep 2025).

5. Representation Number and Uniform Representation

A central numerical invariant is the representation number. By their interval structure and algorithmic construction, all word-representable co-bipartite graphs not corresponding to permutation (circle) graphs have representation number exactly 3; that is, every such graph admits a word representation in which each vertex appears three times, but not fewer (Das et al., 3 Sep 2025). The only co-bipartite graphs with representation number 2 are precisely the permutation graphs, coinciding with the circle graphs (Das et al., 3 Sep 2025, Srinivasan et al., 13 Dec 2025).

These results resolve earlier upper and lower bounds which only guaranteed existence of $k$-uniform words for potentially much larger $k$, especially since general word-representable graphs have representation number at most $2n$ and can be as large as $\lceil n/2 \rceil$ in some cases (Halldórsson et al., 2015).

Class of Co-bipartite Graph	Representation Number	Construction Paradigm
Permutation (Circle) graphs	2	via permutation representation
All other word-representable	3	explicit 3-uniform construction

6. Special Cases, Complements, and Structural Perspective

Not all co-bipartite graphs are word-representable. Examples exist where the complement operation does not preserve word-representability. For instance, the complement of bipartite chain graphs is always word-representable; more generally, the characterizations for complement preservation include complements of paths, even cycles, and generalized crown graphs (Das et al., 17 Jan 2025). The structural underpinnings unify both orientation (semi-transitive) and matrix (interval or circular-ones) approaches through the lens of forbidden submatrices and endpoint monotonicity.

For small clique sizes, explicit descriptions of representable structures and their word-representations are fully worked out, and semi-transitive orientations are described constructively (Das et al., 17 Jan 2025).

7. Research Developments and Outlook

The past several years have seen the resolution of key open problems regarding vertex ordering, recognition, uniform representability, and forbidden subgraph characterization for word-representable co-bipartite graphs (Das et al., 3 Sep 2025, Chen et al., 21 Aug 2025, Srinivasan et al., 13 Dec 2025, Das et al., 17 Jan 2025). The full extension of explicit forbidden subgraph characterizations to all clique sizes remains open due to complexity growth (Chen et al., 21 Aug 2025). The tight correspondence between matrix-theoretic properties and orientation-based conditions in the co-bipartite case sets a pattern for further study in other graph classes, as does the linear-time recognizability via combinatorial properties of adjacency matrices.

These advances solidify the crucial role of semi-transitive orientations and related interval concepts in understanding the landscape of word-representability, particularly for co-bipartite graphs.