Sylvia C. Boyd, J. Cheriyan, Robert Cummings, Logan Grout, Sharat Ibrahimpur, Z. Szigeti, Lu Wang
{"title":"A $frac{4}{3}$-Approximation Algorithm for the Minimum 2-Edge Connected Multisubgraph Problem in the Half-Integral Case","authors":"Sylvia C. Boyd, J. Cheriyan, Robert Cummings, Logan Grout, Sharat Ibrahimpur, Z. Szigeti, Lu Wang","doi":"10.1137/20m1372822","DOIUrl":"https://doi.org/10.1137/20m1372822","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81775701","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-25DOI: 10.48550/arXiv.2207.12157
J. Ai, S. Gerke, G. Gutin, Anders Yeo, Yacong Zhou
A {em quasi-kernel} of a digraph $D$ is an independent set $Qsubseteq V(D)$ such that for every vertex $vin V(D)backslash Q$, there exists a directed path with one or two arcs from $v$ to a vertex $uin Q$. In 1974, Chv'{a}tal and Lov'{a}sz proved that every digraph has a quasi-kernel. In 1976, ErdH{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 $ngeq 3$, 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.
{"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":"https://doi.org/10.48550/arXiv.2207.12157","url":null,"abstract":"A {em quasi-kernel} of a digraph $D$ is an independent set $Qsubseteq V(D)$ such that for every vertex $vin V(D)backslash Q$, there exists a directed path with one or two arcs from $v$ to a vertex $uin Q$. In 1974, Chv'{a}tal and Lov'{a}sz proved that every digraph has a quasi-kernel. In 1976, ErdH{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 $ngeq 3$, 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":0.0,"publicationDate":"2022-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73400113","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-24DOI: 10.48550/arXiv.2207.11668
Jiaxin Wang, Zexia Shi, Yadi Wei, Fang-Wei Fu
Linear codes with a few weights are an important class of codes in coding theory and have attracted a lot of attention. In this paper, we present several constructions of $q$-ary linear codes with two or three weights from vectorial dual-bent functions, where $q$ is a power of an odd prime $p$. The weight distributions of the constructed $q$-ary linear codes are completely determined. We illustrate that some known constructions in the literature can be obtained by our constructions. In some special cases, our constructed linear codes can meet the Griesmer bound. Furthermore, based on the constructed $q$-ary linear codes, we obtain secret sharing schemes with interesting access structures.
{"title":"Constructions of linear codes with two or three weights from vectorial dual-bent functions","authors":"Jiaxin Wang, Zexia Shi, Yadi Wei, Fang-Wei Fu","doi":"10.48550/arXiv.2207.11668","DOIUrl":"https://doi.org/10.48550/arXiv.2207.11668","url":null,"abstract":"Linear codes with a few weights are an important class of codes in coding theory and have attracted a lot of attention. In this paper, we present several constructions of $q$-ary linear codes with two or three weights from vectorial dual-bent functions, where $q$ is a power of an odd prime $p$. The weight distributions of the constructed $q$-ary linear codes are completely determined. We illustrate that some known constructions in the literature can be obtained by our constructions. In some special cases, our constructed linear codes can meet the Griesmer bound. Furthermore, based on the constructed $q$-ary linear codes, we obtain secret sharing schemes with interesting access structures.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83921064","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-22DOI: 10.48550/arXiv.2207.10911
H. Chakraborty, T. Miezaki, M. Oura, Yuuho Tanaka
In this paper, we introduce the notion of Jacobi polynomials with multiple reference vectors of a code, and give the MacWilliams type identity for it. Moreover, we derive a formula to obtain the Jacobi polynomials using the Aronhold polarization operator. Finally, we describe some facts obtained from Type III and Type IV codes that interpret the relation between the Jacobi polynomials and designs.
{"title":"Jacobi polynomials and design theory I","authors":"H. Chakraborty, T. Miezaki, M. Oura, Yuuho Tanaka","doi":"10.48550/arXiv.2207.10911","DOIUrl":"https://doi.org/10.48550/arXiv.2207.10911","url":null,"abstract":"In this paper, we introduce the notion of Jacobi polynomials with multiple reference vectors of a code, and give the MacWilliams type identity for it. Moreover, we derive a formula to obtain the Jacobi polynomials using the Aronhold polarization operator. Finally, we describe some facts obtained from Type III and Type IV codes that interpret the relation between the Jacobi polynomials and designs.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88706873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A General Framework for Hypergraph Coloring","authors":"Ian M. Wanless, D. Wood","doi":"10.1137/21m1421015","DOIUrl":"https://doi.org/10.1137/21m1421015","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91237237","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that every convex code realizable by compact sets in the plane admits a realization consisting of polygons, and analogously every open convex code in the plane can be realized by interiors of polygons. We give factorial-type bounds on the number of vertices needed to form such realizations. Consequently we show that there is an algorithm to decide whether a convex code admits a closed or open realization in the plane.
{"title":"Planar Convex Codes are Decidable","authors":"B. Bukh, R. Jeffs","doi":"10.1137/22m1511187","DOIUrl":"https://doi.org/10.1137/22m1511187","url":null,"abstract":"We show that every convex code realizable by compact sets in the plane admits a realization consisting of polygons, and analogously every open convex code in the plane can be realized by interiors of polygons. We give factorial-type bounds on the number of vertices needed to form such realizations. Consequently we show that there is an algorithm to decide whether a convex code admits a closed or open realization in the plane.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84100676","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A matching of a graph is a set of edges without common end vertex. A graph is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. Recently, Biedl and Wittnebel proved that every 1-planar graph with minimum degree 3 and $ngeq 7$ vertices has a matching of size at least $frac{n+12}{7}$, which is tight for some graphs. They also provided tight lower bounds for the sizes of matchings in 1-planar graphs with minimum degree 4 or 5. In this paper, we show that any 1-planar graph with minimum degree 6 and $n geq 36$ vertices has a matching of size at least $frac{3n+4}{7}$, and this lower bound is tight. Our result confirms a conjecture posed by Biedl and Wittnebel.
{"title":"On the Size of Matchings in 1-Planar Graph with High Minimum Degree","authors":"Yuanqiu Huang, Zhangdong Ouyang, F. Dong","doi":"10.1137/21m1459952","DOIUrl":"https://doi.org/10.1137/21m1459952","url":null,"abstract":"A matching of a graph is a set of edges without common end vertex. A graph is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. Recently, Biedl and Wittnebel proved that every 1-planar graph with minimum degree 3 and $ngeq 7$ vertices has a matching of size at least $frac{n+12}{7}$, which is tight for some graphs. They also provided tight lower bounds for the sizes of matchings in 1-planar graphs with minimum degree 4 or 5. In this paper, we show that any 1-planar graph with minimum degree 6 and $n geq 36$ vertices has a matching of size at least $frac{3n+4}{7}$, and this lower bound is tight. Our result confirms a conjecture posed by Biedl and Wittnebel.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88011925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Posets and Spaces of $k$-Noncrossing RNA Structures","authors":"V. Moulton, Taoyang Wu","doi":"10.1137/21m1413316","DOIUrl":"https://doi.org/10.1137/21m1413316","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80312776","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Recently Chase determined the maximum possible number of cliques of size $t$ in a graph on $n$ vertices with given maximum degree. Soon afterward, Chakraborti and Chen answered the version of this question in which we ask that the graph have $m$ edges and fixed maximum degree (without imposing any constraint on the number of vertices). In this paper we address these problems on hypergraphs. For $s$-graphs with $sge 3$ a number of issues arise that do not appear in the graph case. For instance, for general $s$-graphs we can assign degrees to any $i$-subset of the vertex set with $1le ile s-1$. We establish bounds on the number of $t$-cliques in an $s$-graph $mathcal{H}$ with $i$-degree bounded by $Delta$ in three contexts: $mathcal{H}$ has $n$ vertices; $mathcal{H}$ has $m$ (hyper)edges; and (generalizing the previous case) $mathcal{H}$ has a fixed number $p$ of $u$-cliques for some $u$ with $sle u le t$. When $Delta$ is of a special form we characterize the extremal $s$-graphs and prove that the bounds are tight. These extremal examples are the shadows of either Steiner systems or packings. On the way to proving our uniqueness results, we extend results of F"uredi and Griggs on uniqueness in Kruskal-Katona from the shadow case to the clique case.
最近Chase在一个图形中确定了$n$顶点上具有给定最大度的大小为$t$的团的最大可能数量。不久之后，Chakraborti和Chen回答了这个问题的版本，我们要求图有$m$条边和固定的最大度(没有对顶点数量施加任何限制)。本文在超图上讨论了这些问题。对于使用$sge 3$的$s$ -graphs，会出现一些在图的情况下不会出现的问题。例如，对于一般的$s$ -图，我们可以用$1le ile s-1$为顶点集的任何$i$ -子集分配度数。我们在以下三种情况下建立了$s$ -图$mathcal{H}$中$t$ -团的数量界限，其中$i$ -度由$Delta$限定:$mathcal{H}$有$n$个顶点;$mathcal{H}$有$m$(超)边;并且(推广前面的情况)$mathcal{H}$有固定数量的$p$$u$ -对于一些$u$和$sle u le t$的派系。当$Delta$是一种特殊形式时，我们描述了$s$ -图的极值，并证明了边界是紧的。这些极端的例子是斯坦纳系统或填料的阴影。在证明我们的唯一性结果的过程中，我们将f redi和Griggs关于Kruskal-Katona的唯一性的结果从影子情形推广到团情形。
{"title":"Many Cliques in Bounded-Degree Hypergraphs","authors":"R. Kirsch, Jamie Radcliffe","doi":"10.1137/22m1507565","DOIUrl":"https://doi.org/10.1137/22m1507565","url":null,"abstract":"Recently Chase determined the maximum possible number of cliques of size $t$ in a graph on $n$ vertices with given maximum degree. Soon afterward, Chakraborti and Chen answered the version of this question in which we ask that the graph have $m$ edges and fixed maximum degree (without imposing any constraint on the number of vertices). In this paper we address these problems on hypergraphs. For $s$-graphs with $sge 3$ a number of issues arise that do not appear in the graph case. For instance, for general $s$-graphs we can assign degrees to any $i$-subset of the vertex set with $1le ile s-1$. We establish bounds on the number of $t$-cliques in an $s$-graph $mathcal{H}$ with $i$-degree bounded by $Delta$ in three contexts: $mathcal{H}$ has $n$ vertices; $mathcal{H}$ has $m$ (hyper)edges; and (generalizing the previous case) $mathcal{H}$ has a fixed number $p$ of $u$-cliques for some $u$ with $sle u le t$. When $Delta$ is of a special form we characterize the extremal $s$-graphs and prove that the bounds are tight. These extremal examples are the shadows of either Steiner systems or packings. On the way to proving our uniqueness results, we extend results of F\"uredi and Griggs on uniqueness in Kruskal-Katona from the shadow case to the clique case.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78636224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-30DOI: 10.48550/arXiv.2206.15424
Esther Galby, Liana Khazaliya, Fionn Mc Inerney, Roohani Sharma, P. Tale
For a graph $G$, a subset $S subseteq V(G)$ is called a emph{resolving set} if for any two vertices $u,v in V(G)$, there exists a vertex $w in S$ such that $d(w,u) neq d(w,v)$. The {sc Metric Dimension} problem takes as input a graph $G$ and a positive integer $k$, and asks whether there exists a resolving set of size at most $k$. This problem was introduced in the 1970s and is known to be NP-hard~[GT~61 in Garey and Johnson's book]. In the realm of parameterized complexity, Hartung and Nichterlein~[CCC~2013] proved that the problem is W[2]-hard when parameterized by the natural parameter $k$. They also observed that it is FPT when parameterized by the vertex cover number and asked about its complexity under emph{smaller} parameters, in particular the feedback vertex set number. We answer this question by proving that {sc Metric Dimension} is W[1]-hard when parameterized by the combined parameter feedback vertex set number plus pathwidth. This also improves the result of Bonnet and Purohit~[IPEC 2019] which states that the problem is W[1]-hard parameterized by the pathwidth. On the positive side, we show that {sc Metric Dimension} is FPT when parameterized by either the distance to cluster or the distance to co-cluster, both of which are smaller parameters than the vertex cover number.
对于图$G$，如果对于任意两个顶点$u,v in V(G)$，存在一个顶点$w in S$使得$d(w,u) neq d(w,v)$，则子集$S subseteq V(G)$称为emph{解析集}。{scMetric Dimension}问题以一个图$G$和一个正整数$k$作为输入，并询问是否存在一个大小不超过$k$的解析集。这个问题是在20世纪70年代提出的，已知是NP -hard [Garey和Johnson的书中的GT 61]。在参数化复杂度领域，Hartung和Nichterlein [CCC 2013]证明了当用自然参数$k$参数化时问题是W[2] -hard。他们还观察到，当用顶点覆盖数参数化时，它是FPT，并询问了它在emph{较小}参数下的复杂性，特别是反馈顶点集数。我们通过证明用参数反馈顶点集数加路径宽度的组合参数化{sc度量维度}是W[1] -hard来回答这个问题。这也改进了Bonnet和Purohit [IPEC 2019]的结果，该结果指出问题是W[1] -硬参数化的路径宽度。从积极的方面来看，我们表明，当用到簇的距离或到共簇的距离参数化时，{sc度量维度}是FPT，这两个参数都比顶点覆盖数小。
{"title":"Metric Dimension Parameterized by Feedback Vertex Set and Other Structural Parameters","authors":"Esther Galby, Liana Khazaliya, Fionn Mc Inerney, Roohani Sharma, P. Tale","doi":"10.48550/arXiv.2206.15424","DOIUrl":"https://doi.org/10.48550/arXiv.2206.15424","url":null,"abstract":"For a graph $G$, a subset $S subseteq V(G)$ is called a emph{resolving set} if for any two vertices $u,v in V(G)$, there exists a vertex $w in S$ such that $d(w,u) neq d(w,v)$. The {sc Metric Dimension} problem takes as input a graph $G$ and a positive integer $k$, and asks whether there exists a resolving set of size at most $k$. This problem was introduced in the 1970s and is known to be NP-hard~[GT~61 in Garey and Johnson's book]. In the realm of parameterized complexity, Hartung and Nichterlein~[CCC~2013] proved that the problem is W[2]-hard when parameterized by the natural parameter $k$. They also observed that it is FPT when parameterized by the vertex cover number and asked about its complexity under emph{smaller} parameters, in particular the feedback vertex set number. We answer this question by proving that {sc Metric Dimension} is W[1]-hard when parameterized by the combined parameter feedback vertex set number plus pathwidth. This also improves the result of Bonnet and Purohit~[IPEC 2019] which states that the problem is W[1]-hard parameterized by the pathwidth. On the positive side, we show that {sc Metric Dimension} is FPT when parameterized by either the distance to cluster or the distance to co-cluster, both of which are smaller parameters than the vertex cover number.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84091485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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