{"title":"步行支配和无 HHD 图形","authors":"Silvia B. Tondato","doi":"10.1007/s00373-024-02771-y","DOIUrl":null,"url":null,"abstract":"<p>HHD-free is the class of graphs which contain no house, hole, or domino as induced subgraph. It is known that HHD-free graphs can be characterized via LexBFS-ordering and via <span>\\(m^3\\)</span>-convexity. In this paper we present new characterizations of HHD-free via domination of paths and walks. To achieve this, in particular we concentrate our attention on <span>\\(m_3\\)</span> path, i.e, an induced path of length at least 3 between two non-adjacent vertices in a graph <i>G</i>. We show that the domination between induced paths, paths and walks versus <span>\\(m_3\\)</span> paths, gives rise to characterization of HHD-free. We also characterize the class of graphs in which every <span>\\(m_3\\)</span> path dominates every path, induced path, walk, and <span>\\(m_3\\)</span> path, respectively.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Walk Domination and HHD-Free Graphs\",\"authors\":\"Silvia B. Tondato\",\"doi\":\"10.1007/s00373-024-02771-y\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>HHD-free is the class of graphs which contain no house, hole, or domino as induced subgraph. It is known that HHD-free graphs can be characterized via LexBFS-ordering and via <span>\\\\(m^3\\\\)</span>-convexity. In this paper we present new characterizations of HHD-free via domination of paths and walks. To achieve this, in particular we concentrate our attention on <span>\\\\(m_3\\\\)</span> path, i.e, an induced path of length at least 3 between two non-adjacent vertices in a graph <i>G</i>. We show that the domination between induced paths, paths and walks versus <span>\\\\(m_3\\\\)</span> paths, gives rise to characterization of HHD-free. We also characterize the class of graphs in which every <span>\\\\(m_3\\\\)</span> path dominates every path, induced path, walk, and <span>\\\\(m_3\\\\)</span> path, respectively.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-04-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00373-024-02771-y\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00373-024-02771-y","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
HHD-free is the class of graphs which contain no house, hole, or domino as induced subgraph. It is known that HHD-free graphs can be characterized via LexBFS-ordering and via \(m^3\)-convexity. In this paper we present new characterizations of HHD-free via domination of paths and walks. To achieve this, in particular we concentrate our attention on \(m_3\) path, i.e, an induced path of length at least 3 between two non-adjacent vertices in a graph G. We show that the domination between induced paths, paths and walks versus \(m_3\) paths, gives rise to characterization of HHD-free. We also characterize the class of graphs in which every \(m_3\) path dominates every path, induced path, walk, and \(m_3\) path, respectively.
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