J. Ai, S. Gerke, G. Gutin, Anders Yeo, Yacong Zhou
{"title":"关于小拟核猜想的结果","authors":"J. Ai, S. Gerke, G. Gutin, Anders Yeo, Yacong Zhou","doi":"10.48550/arXiv.2207.12157","DOIUrl":null,"url":null,"abstract":"A {\\em quasi-kernel} of a digraph $D$ is an independent set $Q\\subseteq V(D)$ such that for every vertex $v\\in V(D)\\backslash Q$, there exists a directed path with one or two arcs from $v$ to a vertex $u\\in Q$. In 1974, Chv\\'{a}tal and Lov\\'{a}sz proved that every digraph has a quasi-kernel. In 1976, Erd\\H{o}s and S\\'zekely conjectured that every sink-free digraph $D=(V(D),A(D))$ has a quasi-kernel of size at most $|V(D)|/2$. In this paper, we give a new method to show that the conjecture holds for a generalization of anti-claw-free digraphs. For any sink-free one-way split digraph $D$ of order $n$, when $n\\geq 3$, we show a stronger result that $D$ has a quasi-kernel of size at most $\\frac{n+3}{2} - \\sqrt{n}$, and the bound is sharp.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2022-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Results on the Small Quasi-Kernel Conjecture\",\"authors\":\"J. Ai, S. Gerke, G. Gutin, Anders Yeo, Yacong Zhou\",\"doi\":\"10.48550/arXiv.2207.12157\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A {\\\\em quasi-kernel} of a digraph $D$ is an independent set $Q\\\\subseteq V(D)$ such that for every vertex $v\\\\in V(D)\\\\backslash Q$, there exists a directed path with one or two arcs from $v$ to a vertex $u\\\\in Q$. In 1974, Chv\\\\'{a}tal and Lov\\\\'{a}sz proved that every digraph has a quasi-kernel. In 1976, Erd\\\\H{o}s and S\\\\'zekely conjectured that every sink-free digraph $D=(V(D),A(D))$ has a quasi-kernel of size at most $|V(D)|/2$. In this paper, we give a new method to show that the conjecture holds for a generalization of anti-claw-free digraphs. For any sink-free one-way split digraph $D$ of order $n$, when $n\\\\geq 3$, we show a stronger result that $D$ has a quasi-kernel of size at most $\\\\frac{n+3}{2} - \\\\sqrt{n}$, and the bound is sharp.\",\"PeriodicalId\":21749,\"journal\":{\"name\":\"SIAM J. Discret. Math.\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-07-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"SIAM J. Discret. Math.\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.48550/arXiv.2207.12157\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM J. Discret. Math.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.48550/arXiv.2207.12157","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
A {\em quasi-kernel} of a digraph $D$ is an independent set $Q\subseteq V(D)$ such that for every vertex $v\in V(D)\backslash Q$, there exists a directed path with one or two arcs from $v$ to a vertex $u\in Q$. In 1974, Chv\'{a}tal and Lov\'{a}sz proved that every digraph has a quasi-kernel. In 1976, Erd\H{o}s and S\'zekely conjectured that every sink-free digraph $D=(V(D),A(D))$ has a quasi-kernel of size at most $|V(D)|/2$. In this paper, we give a new method to show that the conjecture holds for a generalization of anti-claw-free digraphs. For any sink-free one-way split digraph $D$ of order $n$, when $n\geq 3$, we show a stronger result that $D$ has a quasi-kernel of size at most $\frac{n+3}{2} - \sqrt{n}$, and the bound is sharp.