{"title":"The Degrees of Regular Polytopes of Type [4, 4, 4]","authors":"Maria Elisa Fernandes, Claudio Alexandre Piedade","doi":"10.1137/20m1375012","DOIUrl":"https://doi.org/10.1137/20m1375012","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77082870","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 : 2022-04-22DOI: 10.48550/arXiv.2204.10691
Guillaume Mescoff, C. Paul, D. Thilikos
We consider the mixed search game against an agile and visible fugitive. This is the variant of the classic fugitive search game on graphs where searchers may be placed to (or removed from) the vertices or slide along edges. Moreover, the fugitive resides on the edges of the graph and can move at any time along unguarded paths. The mixed search number against an agile and visible fugitive of a graph $G$, denoted $avms(G)$, is the minimum number of searchers required to capture to fugitive in this graph searching variant. Our main result is that this graph searching variant is monotone in the sense that the number of searchers required for a successful search strategy does not increase if we restrict the search strategies to those that do not permit the fugitive to visit an already clean edge. This means that mixed search strategies against an agile and visible fugitive can be polynomially certified, and therefore that the problem of deciding, given a graph $G$ and an integer $k,$ whether $avms(G)leq k$ is in NP. Our proof is based on the introduction of the notion of tight bramble, that serves as an obstruction for the corresponding search parameter. Our results imply that for a graph $G$, $avms(G)$ is equal to the Cartesian tree product number of $G$ that is the minimum $k$ for which $G$ is a minor of the Cartesian product of a tree and a clique on $k$ vertices.
{"title":"The mixed search game against an agile and visible fugitive is monotone","authors":"Guillaume Mescoff, C. Paul, D. Thilikos","doi":"10.48550/arXiv.2204.10691","DOIUrl":"https://doi.org/10.48550/arXiv.2204.10691","url":null,"abstract":"We consider the mixed search game against an agile and visible fugitive. This is the variant of the classic fugitive search game on graphs where searchers may be placed to (or removed from) the vertices or slide along edges. Moreover, the fugitive resides on the edges of the graph and can move at any time along unguarded paths. The mixed search number against an agile and visible fugitive of a graph $G$, denoted $avms(G)$, is the minimum number of searchers required to capture to fugitive in this graph searching variant. Our main result is that this graph searching variant is monotone in the sense that the number of searchers required for a successful search strategy does not increase if we restrict the search strategies to those that do not permit the fugitive to visit an already clean edge. This means that mixed search strategies against an agile and visible fugitive can be polynomially certified, and therefore that the problem of deciding, given a graph $G$ and an integer $k,$ whether $avms(G)leq k$ is in NP. Our proof is based on the introduction of the notion of tight bramble, that serves as an obstruction for the corresponding search parameter. Our results imply that for a graph $G$, $avms(G)$ is equal to the Cartesian tree product number of $G$ that is the minimum $k$ for which $G$ is a minor of the Cartesian product of a tree and a clique on $k$ vertices.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87136241","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 : 2022-04-22DOI: 10.48550/arXiv.2204.10494
Saúl A. Blanco, Charles Buehrle
In this paper, we consider the lengths of cycles that can be embedded on the edges of the generalized pancake graphs which are the Cayley graph of the generalized symmetric group $S(m,n)$, generated by prefix reversals. The generalized symmetric group $S(m,n)$ is the wreath product of the cyclic group of order $m$ and the symmetric group of order $n!$. Our main focus is the underlying emph{undirected} graphs, denoted by $mathbb{P}_m(n)$. In the cases when the cyclic group has one or two elements, these graphs are isomorphic to the pancake graphs and burnt pancake graphs, respectively. We prove that when the cyclic group has three elements, $mathbb{P}_3(n)$ has cycles of all possible lengths, thus resembling a similar property of pancake graphs and burnt pancake graphs. Moreover, $mathbb{P}_4(n)$ has all the even-length cycles. We utilize these results as base cases and show that if $m>2$ is even, $mathbb{P}_m(n)$ has all cycles of even length starting from its girth to a Hamiltonian cycle. Moreover, when $m>2$ is odd, $mathbb{P}_m(n)$ has cycles of all lengths starting from its girth to a Hamiltonian cycle. We furthermore show that the girth of $mathbb{P}_m(n)$ is $min{m,6}$ if $mgeq3$, thus complementing the known results for $m=1,2.$
{"title":"Lengths of Cycles in Generalized Pancake Graphs","authors":"Saúl A. Blanco, Charles Buehrle","doi":"10.48550/arXiv.2204.10494","DOIUrl":"https://doi.org/10.48550/arXiv.2204.10494","url":null,"abstract":"In this paper, we consider the lengths of cycles that can be embedded on the edges of the generalized pancake graphs which are the Cayley graph of the generalized symmetric group $S(m,n)$, generated by prefix reversals. The generalized symmetric group $S(m,n)$ is the wreath product of the cyclic group of order $m$ and the symmetric group of order $n!$. Our main focus is the underlying emph{undirected} graphs, denoted by $mathbb{P}_m(n)$. In the cases when the cyclic group has one or two elements, these graphs are isomorphic to the pancake graphs and burnt pancake graphs, respectively. We prove that when the cyclic group has three elements, $mathbb{P}_3(n)$ has cycles of all possible lengths, thus resembling a similar property of pancake graphs and burnt pancake graphs. Moreover, $mathbb{P}_4(n)$ has all the even-length cycles. We utilize these results as base cases and show that if $m>2$ is even, $mathbb{P}_m(n)$ has all cycles of even length starting from its girth to a Hamiltonian cycle. Moreover, when $m>2$ is odd, $mathbb{P}_m(n)$ has cycles of all lengths starting from its girth to a Hamiltonian cycle. We furthermore show that the girth of $mathbb{P}_m(n)$ is $min{m,6}$ if $mgeq3$, thus complementing the known results for $m=1,2.$","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87537392","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 vector is dyadic if each of its entries is a dyadic rational number, i.e., an integer multiple of 1 2k for some nonnegative integer k. We prove that every clean clutter with a covering number of at least two has a dyadic fractional packing of value two. This result is best possible for there exist clean clutters with a covering number of three and no dyadic fractional packing of value three. Examples of clean clutters include ideal clutters, binary clutters, and clutters without an intersecting minor. Our proof is constructive and leads naturally to an albeit exponential algorithm. We improve the running time to quasi-polynomial in the rank of the input, and to polynomial in the binary case.
{"title":"Clean Clutters and Dyadic Fractional Packings","authors":"Ahmad Abdi, G. Cornuéjols, B. Guenin, L. Tunçel","doi":"10.1137/21m1397325","DOIUrl":"https://doi.org/10.1137/21m1397325","url":null,"abstract":"A vector is dyadic if each of its entries is a dyadic rational number, i.e., an integer multiple of 1 2k for some nonnegative integer k. We prove that every clean clutter with a covering number of at least two has a dyadic fractional packing of value two. This result is best possible for there exist clean clutters with a covering number of three and no dyadic fractional packing of value three. Examples of clean clutters include ideal clutters, binary clutters, and clutters without an intersecting minor. Our proof is constructive and leads naturally to an albeit exponential algorithm. We improve the running time to quasi-polynomial in the rank of the input, and to polynomial in the binary case.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87848653","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 : 2022-04-17DOI: 10.48550/arXiv.2204.07971
Milovs Stojakovi'c, Jelena Stratijev
Given an increasing graph property F , the strong Avoider-Avoider F game is played on the edge set of a complete graph. Two players, Red and Blue, take turns in claiming previously unclaimed edges with Red going first, and the player whose graph possesses F first loses the game. If the property F is “containing a fixed graph H ”, we refer to the game as the H game. We prove that Blue has a winning strategy in two strong Avoider-Avoider games, P 4 game and CC > 3 game, where CC > 3 is the property of having at least one connected component on more than three vertices. We also study a variant, the strong CAvoider-CAvoider games, with additional require-ment that the graph of each of the players must stay connected throughout the game. We prove that Blue has a winning strategy in the strong CAvoider-CAvoider games S 3 and P 4 , as well as in the Cycle game, where the players aim at avoiding all cycles.
{"title":"On strong avoiding games","authors":"Milovs Stojakovi'c, Jelena Stratijev","doi":"10.48550/arXiv.2204.07971","DOIUrl":"https://doi.org/10.48550/arXiv.2204.07971","url":null,"abstract":"Given an increasing graph property F , the strong Avoider-Avoider F game is played on the edge set of a complete graph. Two players, Red and Blue, take turns in claiming previously unclaimed edges with Red going first, and the player whose graph possesses F first loses the game. If the property F is “containing a fixed graph H ”, we refer to the game as the H game. We prove that Blue has a winning strategy in two strong Avoider-Avoider games, P 4 game and CC > 3 game, where CC > 3 is the property of having at least one connected component on more than three vertices. We also study a variant, the strong CAvoider-CAvoider games, with additional require-ment that the graph of each of the players must stay connected throughout the game. We prove that Blue has a winning strategy in the strong CAvoider-CAvoider games S 3 and P 4 , as well as in the Cycle game, where the players aim at avoiding all cycles.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80775008","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":"Spectral Radius on Linear $r$-Graphs without Expanded $K_{r+1}$","authors":"Guorong Gao, A. Chang, Yuan Hou","doi":"10.1137/21m1404740","DOIUrl":"https://doi.org/10.1137/21m1404740","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76354551","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}
An identifying code $C$ of a graph $G$ is a dominating set of $G$ such that any two distinct vertices of $G$ have distinct closed neighbourhoods within $C$. These codes have been widely studied for over two decades. We give an improvement over all the best known upper bounds, some of which have stood for over 20 years, for identifying codes in trees, proving the upper bound of $(n+ell)/2$, where $n$ is the order and $ell$ is the number of leaves (pendant vertices) of the graph. In addition to being an improvement in size, the new upper bound is also an improvement in generality, as it actually holds for bipartite graphs having no twins (pairs of vertices with the same closed or open neighbourhood) of degree 2 or greater. We also show that the bound is tight for an infinite class of graphs and that there are several structurally different families of trees attaining the bound. We then use our bound to derive a tight upper bound of $2n/3$ for twin-free bipartite graphs of order $n$, and characterize the extremal examples, as $2$-corona graphs of bipartite graphs. This is best possible, as there exist twin-free graphs, and trees with twins, that need $n-1$ vertices in any of their identifying codes. We also generalize the existing upper bound of $5n/7$ for graphs of order $n$ and girth at least 5 when there are no leaves, to the upper bound $frac{5n+2ell}{7}$ when leaves are allowed. This is tight for the $7$-cycle $C_7$ and for all stars.
图$G$的识别码$C$是$G$的支配集，使得$G$的任意两个不同的顶点在$C$内具有不同的封闭邻域。这些密码已经被广泛研究了二十多年。我们对所有已知的上界进行了改进，其中一些已经存在了20多年，用于识别树中的代码，证明了$(n+ well)/2$的上界，其中$n$是图的阶数，$ well $是叶子(垂顶点)的数量。除了在大小上有所改进之外，新的上界在通用性上也有所改进，因为它实际上适用于没有2度或更高度的双胞胎(具有相同封闭或开放邻域的顶点对)的二部图。我们还证明了无限类图的界是紧的，并且有几个结构上不同的树族达到了这个界。然后，我们利用我们的界导出了阶为$n$的无孪生二部图的$2n/3$的紧上界，并将其极值例子表征为二部图的$2$-电晕图。这是最好的选择，因为存在无孪生图和具有双胞胎的树，它们的任何标识码都需要$n-1$个顶点。对于阶数为$n$且周长至少为5的图，当不存在叶时，我们也将已有的$5n/7$的上界推广到$frac{5n+2ell}{7}$的上界。这对于$7$周期$C_7$和所有恒星来说都很紧。
{"title":"Revisiting and Improving Upper Bounds for Identifying Codes","authors":"Florent Foucaud, Tuomo Lehtilä","doi":"10.1137/22M148999X","DOIUrl":"https://doi.org/10.1137/22M148999X","url":null,"abstract":"An identifying code $C$ of a graph $G$ is a dominating set of $G$ such that any two distinct vertices of $G$ have distinct closed neighbourhoods within $C$. These codes have been widely studied for over two decades. We give an improvement over all the best known upper bounds, some of which have stood for over 20 years, for identifying codes in trees, proving the upper bound of $(n+ell)/2$, where $n$ is the order and $ell$ is the number of leaves (pendant vertices) of the graph. In addition to being an improvement in size, the new upper bound is also an improvement in generality, as it actually holds for bipartite graphs having no twins (pairs of vertices with the same closed or open neighbourhood) of degree 2 or greater. We also show that the bound is tight for an infinite class of graphs and that there are several structurally different families of trees attaining the bound. We then use our bound to derive a tight upper bound of $2n/3$ for twin-free bipartite graphs of order $n$, and characterize the extremal examples, as $2$-corona graphs of bipartite graphs. This is best possible, as there exist twin-free graphs, and trees with twins, that need $n-1$ vertices in any of their identifying codes. We also generalize the existing upper bound of $5n/7$ for graphs of order $n$ and girth at least 5 when there are no leaves, to the upper bound $frac{5n+2ell}{7}$ when leaves are allowed. This is tight for the $7$-cycle $C_7$ and for all stars.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73933633","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":"Counting and Cutting Rich Lenses in Arrangements of Circles","authors":"Esther Ezra, O. Raz, M. Sharir, Joshua Zahl","doi":"10.1137/21m1409305","DOIUrl":"https://doi.org/10.1137/21m1409305","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81183042","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 : 2022-04-01DOI: 10.1016/j.disc.2021.112748
Raiji Mukae, K. Ozeki, Terukazu Sano, Ryuji Tazume
{"title":"Covering projective planar graphs with three forests","authors":"Raiji Mukae, K. Ozeki, Terukazu Sano, Ryuji Tazume","doi":"10.1016/j.disc.2021.112748","DOIUrl":"https://doi.org/10.1016/j.disc.2021.112748","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79974166","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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