{"title":"3-Anti-Circulant Digraphs Are <i>α</i>-Diperfect and BE-Diperfect","authors":"Lucas Freitas, Orlando Lee","doi":"10.4236/ojdm.2022.123003","DOIUrl":null,"url":null,"abstract":"Let $D$ be a digraph. A subset $S$ of $V(D)$ is a stable set if every pair of vertices in $S$ is non-adjacent in $D$. A collection of disjoint paths $\\mathcal{P}$ of $D$ is a path partition of $V(D)$, if every vertex in $V(D)$ is exactly on a path of $\\mathcal{P}$. We say that a stable set $S$ and a path partition $\\mathcal{P}$ are orthogonal if each path of $P$ contains exactly one vertex of $S$. A digraph $D$ satisfies the $\\alpha$-property if for every maximum stable set $S$ of $D$, there exists a path partition $\\mathcal{P}$ such that $S$ and $\\mathcal{P}$ are orthogonal. A digraph $D$ is $\\alpha$-diperfect if every induced subdigraph of $D$ satisfies the $\\alpha$-property. In 1982, Claude Berge proposed a characterization for $\\alpha$-diperfect digraphs in terms of forbidden anti-directed odd cycles. In 2018, Sambinelli, Silva and Lee proposed a similar conjecture. A digraph $D$ satisfies the Begin-End-property or BE-property if for every maximum stable set $S$ of $D$, there exists a path partition $\\mathcal{P}$ such that (i) $S$ and $\\mathcal{P}$ are orthogonal and (ii) for each path $P \\in \\mathcal{P}$, either the start or the end of $P$ belongs to $S$. A digraph $D$ is BE-diperfect if every induced subdigraph of $D$ satisfies the BE-property. Sambinelli, Silva and Lee proposed a characterization for BE-diperfect digraphs in terms of forbidden blocking odd cycles. In this paper, we verified both conjectures for $3$-anti-circulant digraphs. We also present some structural results for $\\alpha$-diperfect and BE-diperfect digraphs.","PeriodicalId":61712,"journal":{"name":"离散数学期刊(英文)","volume":" ","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2022-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"离散数学期刊(英文)","FirstCategoryId":"1093","ListUrlMain":"https://doi.org/10.4236/ojdm.2022.123003","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 2
Abstract
Let $D$ be a digraph. A subset $S$ of $V(D)$ is a stable set if every pair of vertices in $S$ is non-adjacent in $D$. A collection of disjoint paths $\mathcal{P}$ of $D$ is a path partition of $V(D)$, if every vertex in $V(D)$ is exactly on a path of $\mathcal{P}$. We say that a stable set $S$ and a path partition $\mathcal{P}$ are orthogonal if each path of $P$ contains exactly one vertex of $S$. A digraph $D$ satisfies the $\alpha$-property if for every maximum stable set $S$ of $D$, there exists a path partition $\mathcal{P}$ such that $S$ and $\mathcal{P}$ are orthogonal. A digraph $D$ is $\alpha$-diperfect if every induced subdigraph of $D$ satisfies the $\alpha$-property. In 1982, Claude Berge proposed a characterization for $\alpha$-diperfect digraphs in terms of forbidden anti-directed odd cycles. In 2018, Sambinelli, Silva and Lee proposed a similar conjecture. A digraph $D$ satisfies the Begin-End-property or BE-property if for every maximum stable set $S$ of $D$, there exists a path partition $\mathcal{P}$ such that (i) $S$ and $\mathcal{P}$ are orthogonal and (ii) for each path $P \in \mathcal{P}$, either the start or the end of $P$ belongs to $S$. A digraph $D$ is BE-diperfect if every induced subdigraph of $D$ satisfies the BE-property. Sambinelli, Silva and Lee proposed a characterization for BE-diperfect digraphs in terms of forbidden blocking odd cycles. In this paper, we verified both conjectures for $3$-anti-circulant digraphs. We also present some structural results for $\alpha$-diperfect and BE-diperfect digraphs.