Forbidden Induced Subgraph Characterization of Word-Representable Split Graphs

The class of word-representable graphs, introduced in connection with the study of the Perkins semigroup by Kitaev and Seif, has attracted significant attention in combinatorics and theoretical computer science due to its deep connections with graph orientations and combinatorics on words. A graph is word-representable if and only if it admits a semi-transitive orientation, which is an acyclic orientation such that for any directed path $v_0 \rightarrow v_1 \rightarrow \cdots \rightarrow v_m$ with $m \ge 2$, either there is no arc between $v_0$ and $v_m$, or, for all $1 \le i < j \le m$, there exists an arc from $v_i$ to $v_j$. Split graphs, whose vertex set can be partitioned into a clique and an independent set, constitute a natural yet nontrivial subclass for studying word-representability. However, not all split graphs are semi-transitive, and the characterization of minimal forbidden induced subgraphs for semi-transitive split graphs remains an open problem. In this paper, we introduce a new matrix property called the $I$-circular property, which is closely related to the well-known $D$-circular property introduced by Safe. The $I$-circular property requires that both the rows of a matrix and the pairwise intersections of rows form circular intervals under some linear ordering of the columns. Using this property, we establish a direct connection between the structure of semi-transitive split graphs and the matrix representation of their adjacency relationships. Our main result is a complete forbidden submatrix characterization of the $I$-circular property, which in turn provides a characterization for semi-transitive split graphs in terms of minimal forbidden induced subgraphs.