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.

翻译：由Kitaev和Seif在研究Perkins半群时引入的可词表示图类，因其与图定向及词组合学的深刻联系，在组合数学和理论计算机科学领域引起了广泛关注。一个图是可词表示的当且仅当它允许一种半传递定向，即一种无环定向，使得对于任意有向路径 $v_0 \\rightarrow v_1 \\rightarrow \\cdots \\rightarrow v_m$（其中 $m \\ge 2$），要么 $v_0$ 与 $v_m$ 之间不存在弧，要么对于所有 $1 \\le i < j \\le m$，都存在从 $v_i$ 指向 $v_j$ 的弧。分裂图——其顶点集可划分为一个团和一个独立集——是研究可词表示性的一个自然且非平凡的子类。然而，并非所有分裂图都是半传递的，且半传递分裂图的最小禁止诱导子图的刻画问题仍未解决。本文引入了一种新的矩阵性质，称为 $I$-循环性质，该性质与Safe提出的著名 $D$-循环性质密切相关。$I$-循环性质要求矩阵的行以及行之间的两两交在列的某种线性序下形成循环区间。利用这一性质，我们在半传递分裂图的结构与其邻接关系的矩阵表示之间建立了直接联系。我们的主要结果是 $I$-循环性质的完全禁止子矩阵刻画，这进而为半传递分裂图提供了基于最小禁止诱导子图的刻画。