Julien Bensmail, François Dross, H. Hocquard, É. Sopena
A strong edge-colouring of an undirected graph $G$ is an edge-colouring where every two edges at distance at most~$2$ receive distinct colours. The strong chromatic index of $G$ is the least number of colours in a strong edge-colouring of $G$. A conjecture of ErdH{o}s and Nev{s}etv{r}il, stated back in the $80$'s, asserts that every graph with maximum degree $Delta$ should have strong chromatic index at most roughly $1.25 Delta^2$. Several works in the last decades have confirmed this conjecture for various graph classes. In particular, lots of attention have been dedicated to planar graphs, for which the strong chromatic index decreases to roughly $4Delta$, and even to smaller values under additional structural requirements. In this work, we initiate the study of the strong chromatic index of $1$-planar graphs, which are those graphs that can be drawn on the plane in such a way that every edge is crossed at most once. We provide constructions of $1$-planar graphs with maximum degree~$Delta$ and strong chromatic index roughly $6Delta$. As an upper bound, we prove that the strong chromatic index of a $1$-planar graph with maximum degree $Delta$ is at most roughly $24Delta$ (thus linear in $Delta$). The proof of this result is based on the existence of light edges in $1$-planar graphs with minimum degree at least~$3$.
{"title":"From light edges to strong edge-colouring of 1-planar graphs","authors":"Julien Bensmail, François Dross, H. Hocquard, É. Sopena","doi":"10.23638/DMTCS-22-1-2","DOIUrl":"https://doi.org/10.23638/DMTCS-22-1-2","url":null,"abstract":"A strong edge-colouring of an undirected graph $G$ is an edge-colouring where every two edges at distance at most~$2$ receive distinct colours. The strong chromatic index of $G$ is the least number of colours in a strong edge-colouring of $G$. A conjecture of ErdH{o}s and Nev{s}etv{r}il, stated back in the $80$'s, asserts that every graph with maximum degree $Delta$ should have strong chromatic index at most roughly $1.25 Delta^2$. Several works in the last decades have confirmed this conjecture for various graph classes. In particular, lots of attention have been dedicated to planar graphs, for which the strong chromatic index decreases to roughly $4Delta$, and even to smaller values under additional structural requirements. \u0000 \u0000In this work, we initiate the study of the strong chromatic index of $1$-planar graphs, which are those graphs that can be drawn on the plane in such a way that every edge is crossed at most once. We provide constructions of $1$-planar graphs with maximum degree~$Delta$ and strong chromatic index roughly $6Delta$. As an upper bound, we prove that the strong chromatic index of a $1$-planar graph with maximum degree $Delta$ is at most roughly $24Delta$ (thus linear in $Delta$). The proof of this result is based on the existence of light edges in $1$-planar graphs with minimum degree at least~$3$.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"87 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122046972","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}
Edmonds, Lov'asz, and Pulleyblank showed that if a matching covered graph has a nontrivial tight cut, then it also has a nontrivial ELP-cut. Carvalho et al. gave a stronger conjecture: if a matching covered graph has a nontrivial tight cut $C$, then it also has a nontrivial ELP-cut that does not cross $C$. Chen, et al gave a proof of the conjecture. This note is inspired by the paper of Carvalho et al. We give a simplified proof of the conjecture, and prove the following result which is slightly stronger than the conjecture: if a nontrivial tight cut $C$ of a matching covered graph $G$ is not an ELP-cut, then there is a sequence $G_1=G, G_2,ldots,G_r, rgeq2$ of matching covered graphs, such that for $i=1, 2,ldots, r-1$, $G_i$ has an ELP-cut $C_i$, and $G_{i+1}$ is a $C_i$-contraction of $G_i$, and $C$ is a $2$-separation cut of $G_r$. Comment: 7pages
{"title":"A note on tight cuts in matching-covered graphs","authors":"Xiao Zhao, Sheng Chen","doi":"10.46298/dmtcs.6013","DOIUrl":"https://doi.org/10.46298/dmtcs.6013","url":null,"abstract":"Edmonds, Lov'asz, and Pulleyblank showed that if a matching covered graph\u0000has a nontrivial tight cut, then it also has a nontrivial ELP-cut. Carvalho et\u0000al. gave a stronger conjecture: if a matching covered graph has a nontrivial\u0000tight cut $C$, then it also has a nontrivial ELP-cut that does not cross $C$.\u0000Chen, et al gave a proof of the conjecture. This note is inspired by the paper\u0000of Carvalho et al. We give a simplified proof of the conjecture, and prove the\u0000following result which is slightly stronger than the conjecture: if a\u0000nontrivial tight cut $C$ of a matching covered graph $G$ is not an ELP-cut,\u0000then there is a sequence $G_1=G, G_2,ldots,G_r, rgeq2$ of matching covered\u0000graphs, such that for $i=1, 2,ldots, r-1$, $G_i$ has an ELP-cut $C_i$, and\u0000$G_{i+1}$ is a $C_i$-contraction of $G_i$, and $C$ is a $2$-separation cut of\u0000$G_r$.\u0000\u0000 Comment: 7pages","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129352483","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}
Catalan words are particular growth-restricted words counted by the eponymous integer sequence. In this article we consider Catalan words avoiding a pair of patterns of length 3, pursuing the recent initiating work of the first and last authors and of S. Kirgizov where (among other things) the enumeration of Catalan words avoiding a patterns of length 3 is completed. More precisely, we explore systematically the structural properties of the sets of words under consideration and give enumerating results by means of recursive decomposition, constructive bijections or bivariate generating functions with respect to the length and descent number. Some of the obtained enumerating sequences are known, and thus the corresponding results establish new combinatorial interpretations for them.
{"title":"Catalan words avoiding pairs of length three patterns","authors":"Jean-Luc Baril, Carine Khalil, V. Vajnovszki","doi":"10.46298/dmtcs.6002","DOIUrl":"https://doi.org/10.46298/dmtcs.6002","url":null,"abstract":"Catalan words are particular growth-restricted words counted by the eponymous integer sequence. In this article we consider Catalan words avoiding a pair of patterns of length 3, pursuing the recent initiating work of the first and last authors and of S. Kirgizov where (among other things) the enumeration of Catalan words avoiding a patterns of length 3 is completed. More precisely, we explore systematically the structural properties of the sets of words under consideration and give enumerating results by means of recursive decomposition, constructive bijections or bivariate generating functions with respect to the length and descent number. Some of the obtained enumerating sequences are known, and thus the corresponding results establish new combinatorial interpretations for them.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"84 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115145006","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}
Two permutation classes, the X-class and subpermutations of the increasing oscillation are shown to exhibit an exponential Wilf-collapse. This means that the number of distinct enumerations of principal subclasses of each of these classes grows much more slowly than the class itself whereas a priori, based only on symmetries of the class, there is no reason to expect this. The underlying cause of the collapse in both cases is the ability to apply some form of local symmetry which, combined with a greedy algorithm for detecting patterns in these classes, yields a Wilf-collapse. Comment: Final version as accepted by DMTCS. Formatting changes only
{"title":"Two examples of Wilf-collapse","authors":"M. Albert, V'it Jel'inek, Michal Opler","doi":"10.46298/dmtcs.5986","DOIUrl":"https://doi.org/10.46298/dmtcs.5986","url":null,"abstract":"Two permutation classes, the X-class and subpermutations of the increasing\u0000oscillation are shown to exhibit an exponential Wilf-collapse. This means that\u0000the number of distinct enumerations of principal subclasses of each of these\u0000classes grows much more slowly than the class itself whereas a priori, based\u0000only on symmetries of the class, there is no reason to expect this. The\u0000underlying cause of the collapse in both cases is the ability to apply some\u0000form of local symmetry which, combined with a greedy algorithm for detecting\u0000patterns in these classes, yields a Wilf-collapse.\u0000\u0000 Comment: Final version as accepted by DMTCS. Formatting changes only","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125917055","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}
In this paper, we study the staircase encoding of permutations, which maps a permutation to a staircase grid with cells filled with permutations. We study many cases where restricted to a permutation class, the staircase encoding becomes a bijection to its image. We describe the image of those restrictions using independent sets of graphs weighted with permutations. We derive the generating function for the independent sets and then for their weighted counterparts. The bijections we establish provide the enumeration of permutation classes. We use our results to uncover some unbalanced Wilf-equivalences of permutation classes and outline how to do random sampling in the permutation classes. In particular, we cover the classes $mathrm{Av}(2314,3124)$, $mathrm{Av}(2413,3142)$, $mathrm{Av}(2413,3124)$, $mathrm{Av}(2413,2134)$ and $mathrm{Av}(2314,2143)$, as well as many subclasses.
{"title":"Enumeration of Permutation Classes and Weighted Labelled Independent Sets","authors":"Christian Bean, Émile Nadeau, Henning Úlfarsson","doi":"10.46298/dmtcs.5995","DOIUrl":"https://doi.org/10.46298/dmtcs.5995","url":null,"abstract":"In this paper, we study the staircase encoding of permutations, which maps a permutation to a staircase grid with cells filled with permutations. We study many cases where restricted to a permutation class, the staircase encoding becomes a bijection to its image. We describe the image of those restrictions using independent sets of graphs weighted with permutations. We derive the generating function for the independent sets and then for their weighted counterparts. The bijections we establish provide the enumeration of permutation classes. We use our results to uncover some unbalanced Wilf-equivalences of permutation classes and outline how to do random sampling in the permutation classes. In particular, we cover the classes $mathrm{Av}(2314,3124)$, $mathrm{Av}(2413,3142)$, $mathrm{Av}(2413,3124)$, $mathrm{Av}(2413,2134)$ and $mathrm{Av}(2314,2143)$, as well as many subclasses.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115548505","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}
The enumeration of inversion sequences avoiding a single pattern was initiated by Corteel--Martinez--Savage--Weselcouch and Mansour--Shattuck independently. Their work has sparked various investigations of generalized patterns in inversion sequences, including patterns of relation triples by Martinez and Savage, consecutive patterns by Auli and Elizalde, and vincular patterns by Lin and Yan. In this paper, we carried out the systematic study of inversion sequences avoiding two patterns of length $3$. Our enumerative results establish further connections to the OEIS sequences and some classical combinatorial objects, such as restricted permutations, weighted ordered trees and set partitions. Since patterns of relation triples are some special multiple patterns of length $3$, our results complement the work by Martinez and Savage. In particular, one of their conjectures regarding the enumeration of $(021,120)$-avoiding inversion sequences is solved.
{"title":"Inversion sequences avoiding pairs of patterns","authors":"Chunyan Yan, Zhicong Lin","doi":"10.23638/DMTCS-22-1-23","DOIUrl":"https://doi.org/10.23638/DMTCS-22-1-23","url":null,"abstract":"The enumeration of inversion sequences avoiding a single pattern was initiated by Corteel--Martinez--Savage--Weselcouch and Mansour--Shattuck independently. Their work has sparked various investigations of generalized patterns in inversion sequences, including patterns of relation triples by Martinez and Savage, consecutive patterns by Auli and Elizalde, and vincular patterns by Lin and Yan. In this paper, we carried out the systematic study of inversion sequences avoiding two patterns of length $3$. Our enumerative results establish further connections to the OEIS sequences and some classical combinatorial objects, such as restricted permutations, weighted ordered trees and set partitions. Since patterns of relation triples are some special multiple patterns of length $3$, our results complement the work by Martinez and Savage. In particular, one of their conjectures regarding the enumeration of $(021,120)$-avoiding inversion sequences is solved.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133312001","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}
The family of generalized Petersen graphs $G(n,k)$, introduced by Coxeter et al. [4] and named by Mark Watkins (1969), is a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. The Kronecker cover $KC(G)$ of a simple undirected graph $G$ is a a special type of bipartite covering graph of $G$, isomorphic to the direct (tensor) product of $G$ and $K_2$. We characterize all the members of generalized Petersen graphs that are Kronecker covers, and describe the structure of their respective quotients. We observe that some of such quotients are again generalized Petersen graphs, and describe all such pairs.The results of this paper have been presented at EUROCOMB 2019 and an extended abstract has been published elsewhere.
{"title":"Generalized Petersen graphs and Kronecker covers","authors":"Matjaž Krnc, T. Pisanski","doi":"10.23638/DMTCS-21-4-15","DOIUrl":"https://doi.org/10.23638/DMTCS-21-4-15","url":null,"abstract":"The family of generalized Petersen graphs $G(n,k)$, introduced by Coxeter et al. [4] and named by Mark Watkins (1969), is a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. The Kronecker cover $KC(G)$ of a simple undirected graph $G$ is a a special type of bipartite covering graph of $G$, isomorphic to the direct (tensor) product of $G$ and $K_2$. We characterize all the members of generalized Petersen graphs that are Kronecker covers, and describe the structure of their respective quotients. We observe that some of such quotients are again generalized Petersen graphs, and describe all such pairs.The results of this paper have been presented at EUROCOMB 2019 and an extended abstract has been published elsewhere.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128836038","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}
Let $D$ be a strong balanced digraph on $2a$ vertices. Adamus et al. have proved that $D$ is hamiltonian if $d(u)+d(v)ge 3a$ whenever $uvnotin A(D)$ and $vunotin A(D)$. The lower bound $3a$ is tight. In this paper, we shall show that the extremal digraph on this condition is two classes of digraphs that can be clearly characterized. Moreover, we also show that if $d(u)+d(v)geq 3a-1$ whenever $uvnotin A(D)$ and $vunotin A(D)$, then $D$ is traceable. The lower bound $3a-1$ is tight.
{"title":"Extremal digraphs on Meyniel-type condition for hamiltonian cycles in balanced bipartite digraphs","authors":"Ruixia Wang, Linxin Wu, Wei Meng","doi":"10.46298/dmtcs.5851","DOIUrl":"https://doi.org/10.46298/dmtcs.5851","url":null,"abstract":"Let $D$ be a strong balanced digraph on $2a$ vertices. Adamus et al. have\u0000proved that $D$ is hamiltonian if $d(u)+d(v)ge 3a$ whenever $uvnotin A(D)$\u0000and $vunotin A(D)$. The lower bound $3a$ is tight. In this paper, we shall\u0000show that the extremal digraph on this condition is two classes of digraphs\u0000that can be clearly characterized. Moreover, we also show that if\u0000$d(u)+d(v)geq 3a-1$ whenever $uvnotin A(D)$ and $vunotin A(D)$, then $D$ is\u0000traceable. The lower bound $3a-1$ is tight.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133067123","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}
G. Jäger, K. Markström, Denys Shcherbak, Lars–Daniel Öhman
In this paper we first study $k times n$ Youden rectangles of small orders. We have enumerated all Youden rectangles for a range of small parameter values, excluding the almost square cases where $k = n-1$, in a large scale computer search. In particular, we verify the previous counts for $(n,k) = (7,3), (7,4)$, and extend this to the cases $(11,5), (11,6), (13,4)$ and $(21,5)$. For small parameter values where no Youden rectangles exist, we also enumerate rectangles where the number of symbols common to two columns is always one of two possible values, differing by 1, which we call emph{near Youden rectangles}. For all the designs we generate, we calculate the order of the autotopism group and investigate to which degree a certain transformation can yield other row-column designs, namely double arrays, triple arrays and sesqui arrays. Finally, we also investigate certain Latin rectangles with three possible pairwise intersection sizes for the columns and demonstrate that these can give rise to triple and sesqui arrays which cannot be obtained from Youden rectangles, using the transformation mentioned above.
本文首先研究了$k times n$小阶约登矩形。我们已经为一系列小参数值列举了所有的约登矩形，排除了在大规模计算机搜索中$k = n-1$几乎是方形的情况。特别地，我们将验证$(n,k) = (7,3),(7,4)$的先前计数，并将其扩展到$(11,5), (11,6), (13,4)$和$(21,5)$。对于不存在约登矩形的小参数值，我们也枚举两个矩形，其中两列共有的符号数总是两个可能值中的一个，差为1，我们称之为emph{近约登矩形}。对于我们生成的所有设计，我们计算了自拓群的阶数，并研究了某个变换在多大程度上优于其他行列设计，即双列阵列、三列阵列和倍列阵列。最后，我们还研究了某些具有三种可能的列对交叉大小的拉丁矩形，并证明这些矩形可以产生三重数组和倍数组，这些数组不能使用上面提到的转换从Youdenrectangles中获得。
{"title":"Small Youden Rectangles, Near Youden Rectangles, and Their Connections to Other Row-Column Designs","authors":"G. Jäger, K. Markström, Denys Shcherbak, Lars–Daniel Öhman","doi":"10.46298/dmtcs.6754","DOIUrl":"https://doi.org/10.46298/dmtcs.6754","url":null,"abstract":"In this paper we first study $k times n$ Youden rectangles of small orders.\u0000We have enumerated all Youden rectangles for a range of small parameter values,\u0000excluding the almost square cases where $k = n-1$, in a large scale computer\u0000search. In particular, we verify the previous counts for $(n,k) = (7,3),\u0000(7,4)$, and extend this to the cases $(11,5), (11,6), (13,4)$ and $(21,5)$. For\u0000small parameter values where no Youden rectangles exist, we also enumerate\u0000rectangles where the number of symbols common to two columns is always one of\u0000two possible values, differing by 1, which we call emph{near Youden\u0000rectangles}. For all the designs we generate, we calculate the order of the\u0000autotopism group and investigate to which degree a certain transformation can\u0000yield other row-column designs, namely double arrays, triple arrays and sesqui\u0000arrays. Finally, we also investigate certain Latin rectangles with three\u0000possible pairwise intersection sizes for the columns and demonstrate that these\u0000can give rise to triple and sesqui arrays which cannot be obtained from Youden\u0000rectangles, using the transformation mentioned above.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115529794","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}
Ear decompositions of graphs are a standard concept related to several major problems in graph theory like the Traveling Salesman Problem. For example, the Hamiltonian Cycle Problem, which is notoriously N P-complete, is equivalent to deciding whether a given graph admits an ear decomposition in which all ears except one are trivial (i.e. of length 1). On the other hand, a famous result of Lovasz states that deciding whether a graph admits an ear decomposition with all ears of odd length can be done in polynomial time. In this paper, we study the complexity of deciding whether a graph admits an ear decomposition with prescribed ear lengths. We prove that deciding whether a graph admits an ear decomposition with all ears of length at most is polynomial-time solvable for all fixed positive integer. On the other hand, deciding whether a graph admits an ear decomposition without ears of length in F is N P-complete for any finite set F of positive integers. We also prove that, for any k ≥ 2, deciding whether a graph admits an ear decomposition with all ears of length 0 mod k is N P-complete. We also consider the directed analogue to ear decomposition, which we call handle decomposition, and prove analogous results : deciding whether a digraph admits a handle decomposition with all handles of length at most is polynomial-time solvable for all positive integer ; deciding whether a digraph admits a handle decomposition without handles of length in F is N P-complete for any finite set F of positive integers (and minimizing the number of handles of length in F is not approximable up to n(1 −)); for any k ≥ 2, deciding whether a digraph admits a handle decomposition with all handles of length 0 mod k is N P-complete. Also, in contrast with the result of Lovasz, we prove that deciding whether a digraph admits a handle decomposition with all handles of odd length is N P-complete. Finally, we conjecture that, for every set A of integers, deciding whether a digraph has a handle decomposition with all handles of length in A is N P-complete, unless there exists h ∈ N such that A = {1, · · · , h}.
图的耳分解是图论中几个主要问题的标准概念，比如旅行商问题。例如，众所周知的np完备的哈密顿循环问题，等价于判定一个给定的图是否允许除一个耳朵以外的所有耳朵都是平凡的(即长度为1)的耳分解。另一方面，Lovasz的一个著名结果表明，判定一个图是否允许所有耳朵长度为奇数的耳分解可以在多项式时间内完成。在本文中，我们研究了在给定耳长下图是否允许耳分解的复杂性。证明了对于所有固定正整数，判定一个图是否允许最大长度为所有耳朵的耳朵分解是多项式时间可解的。另一方面，对于任意正整数的有限集合F，判定图是否允许在F中没有耳的耳分解是np完全的。我们还证明了，对于任意k≥2，判定一个图是否允许所有耳长为0 mod k的耳分解是np完全的。我们还考虑了对耳分解的有向模拟，我们称之为柄分解，并证明了类似的结果:判定一个有向图是否允许柄分解，且所有柄的长度最多为正整数，是多项式时间可解的;判定一个有向图是否允许无长度为F的句柄分解对任何正整数有限集F是np完全的(且最小化长度为F的句柄的数量不逼近到N(1−));对于任意k≥2，判定一个有向图是否允许所有句柄长度为0 mod k的句柄分解为np完全。同时，与Lovasz的结果相反，我们证明了判定一个有向图是否允许所有柄长为奇数的柄分解是np完全的。最后，我们推测，对于每一个整数集合A，判定一个有向图是否有一个句柄分解，且所有句柄的长度都在A中，是np完全的，除非存在h∈N使得A ={1，···，h}。
{"title":"Constrained ear decompositions in graphs and digraphs","authors":"F. Havet, N. Nisse","doi":"10.23638/DMTCS-21-4-3","DOIUrl":"https://doi.org/10.23638/DMTCS-21-4-3","url":null,"abstract":"Ear decompositions of graphs are a standard concept related to several major problems in graph theory like the Traveling Salesman Problem. For example, the Hamiltonian Cycle Problem, which is notoriously N P-complete, is equivalent to deciding whether a given graph admits an ear decomposition in which all ears except one are trivial (i.e. of length 1). On the other hand, a famous result of Lovasz states that deciding whether a graph admits an ear decomposition with all ears of odd length can be done in polynomial time. In this paper, we study the complexity of deciding whether a graph admits an ear decomposition with prescribed ear lengths. We prove that deciding whether a graph admits an ear decomposition with all ears of length at most is polynomial-time solvable for all fixed positive integer. On the other hand, deciding whether a graph admits an ear decomposition without ears of length in F is N P-complete for any finite set F of positive integers. We also prove that, for any k ≥ 2, deciding whether a graph admits an ear decomposition with all ears of length 0 mod k is N P-complete. We also consider the directed analogue to ear decomposition, which we call handle decomposition, and prove analogous results : deciding whether a digraph admits a handle decomposition with all handles of length at most is polynomial-time solvable for all positive integer ; deciding whether a digraph admits a handle decomposition without handles of length in F is N P-complete for any finite set F of positive integers (and minimizing the number of handles of length in F is not approximable up to n(1 −)); for any k ≥ 2, deciding whether a digraph admits a handle decomposition with all handles of length 0 mod k is N P-complete. Also, in contrast with the result of Lovasz, we prove that deciding whether a digraph admits a handle decomposition with all handles of odd length is N P-complete. Finally, we conjecture that, for every set A of integers, deciding whether a digraph has a handle decomposition with all handles of length in A is N P-complete, unless there exists h ∈ N such that A = {1, · · · , h}.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"111 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125571910","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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