Word-Representability of Split Graphs with Independent Set of Size 4

A pair of letters $x$ and $y$ are said to alternate in a word $w$ if, after removing all letters except for the copies of $x$ and $y$ from $w$, the resulting word is of the form $xyxy\dots$ (of even or odd length) or $yxyx\dots$ (of even or odd length). A graph $G=(V(G),E(G))$ is word-representable if there exists a word $w$ over the alphabet $V(G)$, such that any two distinct vertices $x,y\in V(G)$ are adjacent in $G$ (i.e., $xy\in E(G)$) if and only if the letters $x$ and $y$ alternate in $w$. A split graph is a graph in which the vertices can be partitioned into a clique and an independent set. Word-representability of split graphs has been studied in a series of papers [2, 5, 7, 9] in the literature. In this work, we give a minimal forbidden induced subgraph characterization of word-representable split graphs with an independent set of size 4, which is an open problem posed by Kitaev and Pyatkin in [9]