Sasun Hambartsumyan, V. Mkrtchyan, Vahe L. Musoyan, H. Sargsyan
{"title":"The hardness of the independence and matching clutter of a graph","authors":"Sasun Hambartsumyan, V. Mkrtchyan, Vahe L. Musoyan, H. Sargsyan","doi":"10.7494/OPMATH.2016.36.3.375","DOIUrl":null,"url":null,"abstract":"A {\\it clutter} (or {\\it antichain} or {\\it Sperner family}) $L$ is a pair $(V,E)$, where $V$ is a finite set and $E$ is a family of subsets of $V$ none of which is a subset of another. Usually, the elements of $V$ are called {\\it vertices} of $L$, and the elements of $E$ are called {\\it edges} of $L$. A subset $s_e$ of an edge $e$ of a clutter is called {\\it recognizing} for $e$, if $s_e$ is not a subset of another edge. The {\\it hardness} of an edge $e$ of a clutter is the ratio of the size of $e\\textrm{'s}$ smallest recognizing subset to the size of $e$. The hardness of a clutter is the maximum hardness of its edges. We study the hardness of clutters arising from independent sets and matchings of graphs.","PeriodicalId":111854,"journal":{"name":"arXiv: Discrete Mathematics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2009-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: Discrete Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.7494/OPMATH.2016.36.3.375","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
A {\it clutter} (or {\it antichain} or {\it Sperner family}) $L$ is a pair $(V,E)$, where $V$ is a finite set and $E$ is a family of subsets of $V$ none of which is a subset of another. Usually, the elements of $V$ are called {\it vertices} of $L$, and the elements of $E$ are called {\it edges} of $L$. A subset $s_e$ of an edge $e$ of a clutter is called {\it recognizing} for $e$, if $s_e$ is not a subset of another edge. The {\it hardness} of an edge $e$ of a clutter is the ratio of the size of $e\textrm{'s}$ smallest recognizing subset to the size of $e$. The hardness of a clutter is the maximum hardness of its edges. We study the hardness of clutters arising from independent sets and matchings of graphs.
A {\it杂波}(或{\it反链}或{\it Sperner族})$L$是一对$(V,E)$,其中$V$是一个有限集,$E$是$V$的子集族,其中$V$都不是另一个的子集。通常,$V$的元素称为$L$的{\it顶点},$E$的元素称为$L$的{\it边}。如果$s_e$不是另一条边的子集,则杂波的边$s_e$的子集$s_e$对于$e$称为{\it recognition}。杂波边缘$e$的{\it硬度}是$e\textrm{'s}$最小识别子集的大小与$e$的大小之比。杂波的硬度是其边缘的最大硬度。我们研究了图的独立集和匹配所产生的杂波的硬度。