Word-Representable Split Graphs

• The paper presents a complete forbidden induced subgraph characterization for semi-transitive split graphs via matrix encodings and orientation theory.
• Word-representable split graphs are defined by encoding the adjacency relation through letter alternation, linking combinatorial word properties with graph orientations.
• The study establishes efficient polynomial recognition algorithms and bounds the representation number to at most 3, offering practical insights for structural graph analysis.

Word-representable split graphs are split graphs whose adjacency relation can be encoded by alternation in a word over the vertex set: for distinct vertices $x$ and $y$, the edge $xy$ is present if and only if the letters $x$ and $y$ alternate in the representing word. Equivalently, they are the split graphs that admit a semi-transitive orientation, so the subject sits at the intersection of combinatorics on words, graph orientations, comparability theory, and structural graph theory (Halldórsson et al., 2015). Research on the class has evolved from orientation-based partial characterizations for restricted split graphs to matrix and forbidden-subgraph descriptions, and ultimately to a complete forbidden induced subgraph characterization for semi-transitive split graphs (Kitaev et al., 2017, Srinivasan et al., 13 Dec 2025).

1. Definitions and foundational equivalences

Let $G=(V,E)$ be a simple graph. A word $w$ over alphabet $V$ represents $G$ when, for all distinct $x,y\in V$, the edge $y$0 is present exactly when the restriction $y$1 is of the form $y$2 or $y$3; equivalently, $y$4 contains no factor $y$5 or $y$6. A graph is word-representable if it has such a word. If each letter appears exactly $y$7 times, then $y$8 is $y$9-uniform, $xy$0 is $xy$1-word-representable, and the minimum such $xy$2 is the representation number $xy$3 (Halldórsson et al., 2015).

A split graph is a graph whose vertex set can be partitioned as $xy$4, where $xy$5 induces a clique and $xy$6 induces an independent set. In matrix-based work on split graphs, the bipartite adjacencies between $xy$7 and $xy$8 are encoded by the $xy$9 $x$0-matrix $x$1 with $x$2 (Srinivasan et al., 13 Dec 2025).

The central structural fact is that a graph is word-representable if and only if it admits a semi-transitive orientation: an acyclic orientation in which every directed path

$x$3

either has no arc between $x$4 and $x$5, or else induces full transitivity on the path vertices. In the equivalent shortcut formulation, semi-transitivity means acyclicity together with the absence of an induced shortcut. This equivalence converts a problem about alternation in words into a problem about acyclic orientations and is the main organizing principle for the theory (Halldórsson et al., 2015).

A nearby subclass is given by comparability graphs, i.e. graphs admitting a transitive orientation. These are exactly the permutationally representable graphs, meaning graphs representable by a concatenation of permutations. For split graphs, comparability structure is especially important when one studies permutation-representation number and the boundary between representation numbers $x$6 and $x$7 (Dwary et al., 2024).

2. Orientation structure specific to split graphs

For split graphs, semi-transitivity imposes a particularly rigid local form. If $x$8 is semi-transitively oriented, then the clique $x$9 is necessarily oriented transitively; equivalently, its vertices can be arranged on a directed Hamiltonian path

$y$0

Every vertex of the independent set then falls into one of three types relative to $y$1 (Kitaev et al., 2017).

Type A vertices are adjacent to a consecutive interval $y$2 and are oriented uniformly as sources toward that interval. Type B vertices are adjacent to a consecutive interval $y$3 and are oriented uniformly as sinks from that interval. Type C vertices are adjacent to a prefix and a suffix of $y$4: there exist $y$5 such that

$y$6

with edges $y$7 for $y$8 and $y$9 for $G=(V,E)$0. Type C is therefore the split-graph analogue of a two-interval pattern at the ends of the clique order (Kitaev et al., 2017).

This orientation picture has an equivalent neighborhood formulation. A split graph is semi-transitive if and only if the clique vertices can be labeled $G=(V,E)$1 so that every $G=(V,E)$2 has one of the two neighborhood forms

$G=(V,E)$3

together with two compatibility rules. If $G=(V,E)$4 and $G=(V,E)$5, then

$G=(V,E)$6

If

$G=(V,E)$7

then

$G=(V,E)$8

These conditions encode exactly when interval-type and two-end neighborhoods can coexist without producing shortcuts (Roy et al., 11 Jul 2025).

A common misconception is that the split partition itself is already sufficient to force word-representability. The orientation theory shows that this is false: not all split graphs are semi-transitive, and the obstruction lies in how neighborhoods in the independent set interleave along the transitive orientation of the clique, not in the existence of the partition alone (Kitaev et al., 2017).

3. From partial forbidden families to complete forbidden induced subgraphs

The earliest structural results for word-representable split graphs were size-restricted. For split graphs in which every vertex of the independent set has degree at most $G=(V,E)$9, word-representability is equivalent to avoiding the graph $w$0 and the infinite family $w$1; for split graphs with clique $w$2, it is equivalent to avoiding exactly $w$3 as induced subgraphs (Kitaev et al., 2017).

This program was extended in several directions. Threshold graphs, a subclass of split graphs obtained by repeatedly adding an isolated or a dominating vertex, were shown to be word-representable. The same paper characterized split graphs with clique $w$4: a split graph $w$5 is word-representable if and only if it avoids the nine induced subgraphs $w$6 (Chen et al., 2019).

A complementary size restriction fixes the independent set rather than the clique. For $w$7, word-representability is equivalent to avoiding the finite family

$w$8

Here $w$9 is the new obstruction specific to the $V$0 case: its independent set is $V$1, its clique has size $V$2, the vertices $V$3 are adjacent to three disjoint consecutive pairs of clique vertices, and $V$4 is adjacent to the alternating triple $V$5. The graph is minimal non-word-representable, because deleting any vertex restores the interval compatibility demanded by the split-graph neighborhood theorem (Roy et al., 11 Jul 2025).

The decisive later step is a complete forbidden induced subgraph characterization for semi-transitive split graphs. The paper introduces a finite family $V$6 derived from a finite family of forbidden submatrices, and proves that a split graph $V$7 is semi-transitive, hence word-representable, if and only if it contains none of the graphs in $V$8 as an induced subgraph. Each graph in $V$9 is minimal non-semi-transitive (Srinivasan et al., 13 Dec 2025). This closes the earlier pattern of partial results by turning the hereditary nature of word-representability into a complete obstruction theory inside the split class.

4. Matrix encodings, circular orders, and the $G$0-circular property

A major conceptual shift was to reformulate split-graph word-representability as a property of the adjacency matrix between the independent set and the clique. An early matrix characterization states that, after a suitable permutation of the clique columns, each row must have one of the forms

$G$1

and whenever a row has the form $G$2 with $G$3, no other row may have $G$4s simultaneously in positions $G$5 and $G$6. This already captures the interval and two-end neighborhood patterns appearing in the orientation description (Iamthong, 2021).

The later matrix theory rephrases the same phenomenon more cleanly. A $G$7-matrix $G$8 has the $G$9-circular property if there exists a linear order of the columns such that every row is a circular interval and, moreover, the intersection of every pair of rows is also a circular interval. If $x,y\in V$0 denotes the matrix obtained by adding all relevant pairwise intersections as extra rows, then

$x,y\in V$1

This turns split-graph word-representability into a circular-interval problem (Srinivasan et al., 13 Dec 2025).

The corresponding graph-theoretic statement is exact: if $x,y\in V$2 is a split graph with clique $x,y\in V$3, independent set $x,y\in V$4, and adjacency matrix $x,y\in V$5 between $x,y\in V$6 and $x,y\in V$7, then

$x,y\in V$8

The same paper proves a finite forbidden submatrix characterization for $x,y\in V$9-circularity. The forbidden family is

$y$00

and the associated split graphs $y$01, $y$02, generate the finite obstruction family $y$03 mentioned above (Srinivasan et al., 13 Dec 2025).

Algorithmically, this matters because circular-ones recognition is linear-time, and the reduction $y$04 yields a polynomial recognition method for word-representable split graphs. A second polynomial method is immediate from the finite family $y$05: one may test $y$06-freeness instead of searching for a semi-transitive orientation directly (Srinivasan et al., 13 Dec 2025).

5. Representation number inside the split class

The representation number of a general word-representable graph can be large, and determining it is NP-complete in general. For split graphs, however, the situation is markedly sharper: every word-representable split graph has representation number at most $y$07 (Dwary et al., 2 Feb 2025).

The proof is constructive. Starting from the neighborhood characterization of split graphs, the authors partition the independent set into

$y$08

and build three words $y$09 from the clique order $y$10. Vertices in $y$11 and $y$12 are inserted at specific neighborhood endpoints in $y$13 and $y$14, while $y$15 contains a reversed control block that enforces the required alternation and non-alternation patterns. The final word

$y$16

is 3-uniform and represents the split graph (Dwary et al., 2 Feb 2025).

The same paper gives an exact description of when the upper bound is tight. A word-representable split graph $y$17 has

$y$18

if and only if it contains at least one of the following induced subgraphs:

• Even-$y$19-sun for even $y$20,
• $y$21,
• $y$22 for odd $y$23,
• $y$24 for odd $y$25 (Dwary et al., 2 Feb 2025).

For split comparability graphs the criterion is even cleaner. Since split comparability graphs have permutation-representation number at most $y$26, the paper proves that

$y$27

within this subclass (Dwary et al., 2 Feb 2025). By contrast, threshold graphs are permutation graphs and hence circle graphs, so their representation number is at most $y$28 (Dwary et al., 2 Feb 2025).

6. Structural subclasses, split decomposition, and recomposition

Several important subclasses of split graphs are automatically word-representable. Threshold graphs are one such class (Chen et al., 2019). Another sufficient condition is degree-theoretic: if $y$29 is a split graph such that every clique vertex has degree at most $y$30, meaning each vertex of the clique is adjacent to at most one vertex of the independent set outside the clique, then $y$31 is word-representable (Chen et al., 2019).

A distinct but relevant structural perspective comes from split decomposition rather than classical split partitions. The class of word-representable graphs is closed under split recomposition, and if $y$32 has split decomposition components $y$33, then

$y$34

and

$y$35

Although this theory is not restricted to classical split graphs, it is directly relevant to them whenever a split graph admits a decomposition into cliques, stars, bipartite pieces, or other word-representable components (Dwary et al., 2024).

This viewpoint is especially effective for parity graphs. Since every component in the minimal split decomposition of a parity graph is either a clique or a bipartite graph, parity graphs are word-representable. The synthesis explicitly notes that many split graphs are parity graphs, including bipartite split graphs and distance-hereditary split graphs, so the decomposition theorem yields a structural route to word-representability for these subclasses as well (Dwary et al., 2024).

A common source of confusion is the word “split” itself. In the classical sense, a split graph has a clique–independent-set partition. In split decomposition theory, “split” refers to a decomposition operation on connected graphs. The two notions are different, but the decomposition results still feed back into the study of classical split graphs whenever such graphs can be assembled from word-representable building blocks (Dwary et al., 2024).

7. Computation, extensions, and surrounding research directions

Computational work on general word-representable graphs provides useful context for the split-graph class. Semi-transitive orientations were used to enumerate connected non-word-representable graphs up to $y$36 vertices and to compute distributions of representation numbers up to $y$37 vertices; the same work introduced $y$38-semi-transitive orientations and showed computationally that $y$39-semi-transitivity is strictly weaker than semi-transitivity (Akgün et al., 2018). For split graphs, these methods suggest a practical route to exhaustive experimentation: generate split graphs, search for semi-transitive orientations, and then test word constructions.

Another surrounding direction is $y$40-11-representability, which relaxes alternation by allowing up to $y$41 occurrences of the consecutive pattern $y$42 in the two-letter restriction. Word-representable graphs are precisely $y$43-11-representable graphs. The hierarchy is strict at the first step,

$y$44

and every graph is $y$45-11-representable by a concatenation of permutations (Cheon et al., 2018). This places word-representable split graphs inside a broader relaxation framework. A natural extension is whether split graphs that fail word-representability still admit low-level $y$46-11 representations.

Historically, several open problems were posed first in restricted split-graph settings and then resolved. The $y$47 case was posed and later characterized by the forbidden family $y$48 (Roy et al., 11 Jul 2025). The broader problem of a forbidden induced subgraph characterization for semi-transitive split graphs was then resolved through the $y$49-circular property and the finite obstruction family $y$50 (Srinivasan et al., 13 Dec 2025). What remains open in the matrix direction are refinements such as direct linear-time recognition of $y$51-circularity without explicitly constructing $y$52, and the relation between the $y$53-circular and $y$54-circular properties (Srinivasan et al., 13 Dec 2025).

Taken together, the modern picture is unusually sharp for a nontrivial hereditary class. Word-representable split graphs now admit equivalent descriptions by semi-transitive orientations, by neighborhood intervals on an ordered clique, by column-permuted $y$55-matrices, by finite forbidden induced subgraphs, and by bounded representation number. The class therefore serves as one of the clearest laboratories for the general program of understanding word-representability through structural graph theory (Srinivasan et al., 13 Dec 2025).