Word-Representability of Split Graphs with Independent Set of Size 4
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866918088938618880 |
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| author | Roy, Suchanda Hariharasubramanian, Ramesh |
| author_facet | Roy, Suchanda Hariharasubramanian, Ramesh |
| contents | 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 $xyxy\ldots$ (of even or odd length) or $yxyx\ldots$ (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] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_08483 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Word-Representability of Split Graphs with Independent Set of Size 4 Roy, Suchanda Hariharasubramanian, Ramesh Combinatorics Discrete Mathematics 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 $xyxy\ldots$ (of even or odd length) or $yxyx\ldots$ (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] |
| title | Word-Representability of Split Graphs with Independent Set of Size 4 |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2507.08483 |