We assign a relational structure to any finite algebra in a canonical way, using solution sets of equations, and we prove that this relational structure is polymorphism-homogeneous if and only if the algebra itself is polymorphism-homogeneous. We show that polymorphism-homogeneity is also equivalent to the property that algebraic sets (i.e., solution sets of systems of equations) are exactly those sets of tuples that are closed under the centralizer clone of the algebra. Furthermore, we prove that the aforementioned properties hold if and only if the algebra is injective in the category of its finite subpowers. We also consider two additional conditions: a stronger variant for polymorphism-homogeneity and for injectivity, and we describe explicitly the finite semilattices, lattices, Abelian groups and monounary algebras satisfying any one of these three conditions.
{"title":"Polymorphism-homogeneity and universal algebraic geometry","authors":"Endre T'oth, Tamás Waldhauser","doi":"10.46298/dmtcs.6904","DOIUrl":"https://doi.org/10.46298/dmtcs.6904","url":null,"abstract":"We assign a relational structure to any finite algebra in a canonical way, using solution sets of equations, and we prove that this relational structure is polymorphism-homogeneous if and only if the algebra itself is polymorphism-homogeneous. We show that polymorphism-homogeneity is also equivalent to the property that algebraic sets (i.e., solution sets of systems of equations) are exactly those sets of tuples that are closed under the centralizer clone of the algebra. Furthermore, we prove that the aforementioned properties hold if and only if the algebra is injective in the category of its finite subpowers. We also consider two additional conditions: a stronger variant for polymorphism-homogeneity and for injectivity, and we describe explicitly the finite semilattices, lattices, Abelian groups and monounary algebras satisfying any one of these three conditions.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"5 2 Suppl 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126103938","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 forbidden number $mathrm{forb}(m,F)$, which denotes the maximum number of unique columns in an $m$-rowed $(0,1)$-matrix with no submatrix that is a row and column permutation of $F$, has been widely studied in extremal set theory. Recently, this function was extended to $r$-matrices, whose entries lie in ${0,1,dots,r-1}$. The combinatorics of the generalized forbidden number is less well-studied. In this paper, we provide exact bounds for many $(0,1)$-matrices $F$, including all $2$-rowed matrices when $r > 3$. We also prove a stability result for the $2times 2$ identity matrix. Along the way, we introduce some interesting qualitative differences between the cases $r=2$, $r = 3$, and $r > 3$.
{"title":"Exponential multivalued forbidden configurations","authors":"Travis Dillon, A. Sali","doi":"10.46298/dmtcs.6613","DOIUrl":"https://doi.org/10.46298/dmtcs.6613","url":null,"abstract":"The forbidden number $mathrm{forb}(m,F)$, which denotes the maximum number of unique columns in an $m$-rowed $(0,1)$-matrix with no submatrix that is a row and column permutation of $F$, has been widely studied in extremal set theory. Recently, this function was extended to $r$-matrices, whose entries lie in ${0,1,dots,r-1}$. The combinatorics of the generalized forbidden number is less well-studied. In this paper, we provide exact bounds for many $(0,1)$-matrices $F$, including all $2$-rowed matrices when $r > 3$. We also prove a stability result for the $2times 2$ identity matrix. Along the way, we introduce some interesting qualitative differences between the cases $r=2$, $r = 3$, and $r > 3$.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125295356","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}
Given a partially order set (poset) $P$, and a pair of families of ideals $mathcal{I}$ and filters $mathcal{F}$ in $P$ such that each pair $(I,F)in mathcal{I}timesmathcal{F}$ has a non-empty intersection, the dualization problem over $P$ is to check whether there is an ideal $X$ in $P$ which intersects every member of $mathcal{F}$ and does not contain any member of $mathcal{I}$. Equivalently, the problem is to check for a distributive lattice $L=L(P)$, given by the poset $P$ of its set of joint-irreducibles, and two given antichains $mathcal{A},mathcal{B}subseteq L$ such that no $ainmathcal{A}$ is dominated by any $binmathcal{B}$, whether $mathcal{A}$ and $mathcal{B}$ cover (by domination) the entire lattice. We show that the problem can be solved in quasi-polynomial time in the sizes of $P$, $mathcal{A}$ and $mathcal{B}$, thus answering an open question in Babin and Kuznetsov (2017). As an application, we show that minimal infrequent closed sets of attributes in a rational database, with respect to a given implication base of maximum premise size of one, can be enumerated in incremental quasi-polynomial time.
给定一个偏序集(偏序集)$P$和一对理想族$mathcal{I}$，并在$P$中过滤$mathcal{F}$，使得$(I,F) mathcal{I}$中每一对$(I,F)乘以$ mathcal{F}$有一个非空相交，则$P$上的对偶问题是检查$P$中是否存在一个理想$X$，该理想$X$与$mathcal{F}$中的每一个成员相交，并且不包含$mathcal{I}$中的任何成员。同样地，问题是检查一个由其联合不可约集合的偏序集$P$给出的分配格$L=L(P)$，以及两个给定的反链$mathcal{a}，mathcal{B}subseteq L$使得mathcal{a}$中没有$a被mathcal{B}$中的任何$ B 支配，以及$mathcal{a}$和$mathcal{B}$是否(通过支配)覆盖了整个格。我们证明了这个问题可以在拟多项式时间内以$P$、$mathcal{A}$和$mathcal{B}$的大小来解决，从而回答了Babin和kuznetsov(2017)中的一个开放问题。作为一个应用，我们证明了在一个给定的最大前提大小为1的隐含基下，在增量拟多项式时间内可以枚举出有理数据库中最小的非频繁闭集属性。
{"title":"On Dualization over Distributive Lattices","authors":"Khaled M. Elbassioni","doi":"10.46298/dmtcs.6742","DOIUrl":"https://doi.org/10.46298/dmtcs.6742","url":null,"abstract":"Given a partially order set (poset) $P$, and a pair of families of ideals\u0000$mathcal{I}$ and filters $mathcal{F}$ in $P$ such that each pair $(I,F)in\u0000mathcal{I}timesmathcal{F}$ has a non-empty intersection, the dualization\u0000problem over $P$ is to check whether there is an ideal $X$ in $P$ which\u0000intersects every member of $mathcal{F}$ and does not contain any member of\u0000$mathcal{I}$. Equivalently, the problem is to check for a distributive lattice\u0000$L=L(P)$, given by the poset $P$ of its set of joint-irreducibles, and two\u0000given antichains $mathcal{A},mathcal{B}subseteq L$ such that no\u0000$ainmathcal{A}$ is dominated by any $binmathcal{B}$, whether $mathcal{A}$\u0000and $mathcal{B}$ cover (by domination) the entire lattice. We show that the\u0000problem can be solved in quasi-polynomial time in the sizes of $P$,\u0000$mathcal{A}$ and $mathcal{B}$, thus answering an open question in Babin and\u0000Kuznetsov (2017). As an application, we show that minimal infrequent closed\u0000sets of attributes in a rational database, with respect to a given implication\u0000base of maximum premise size of one, can be enumerated in incremental\u0000quasi-polynomial time.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115225224","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}
Loic Dubois, G. Joret, G. Perarnau, Marcin Pilipczuk, Franccois Pitois
Given a graph $G$ and an integer $p$, a coloring $f : V(G) to mathbb{N}$ is $p$-centered if for every connected subgraph $H$ of $G$, either $f$ uses more than $p$ colors on $H$ or there is a color that appears exactly once in $H$. The notion of $p$-centered colorings plays a central role in the theory of sparse graphs. In this note we show two lower bounds on the number of colors required in a $p$-centered coloring. First, we consider monotone classes of graphs whose shallow minors have average degree bounded polynomially in the radius, or equivalently (by a result of Dvořak and Norin), admitting strongly sublinear separators. We construct such a class such that $p$-centered colorings require a number of colors exponential in $p$. This is in contrast with a recent result of Pilipczuk and Siebertz, who established a polynomial upper bound in the special case of graphs excluding a fixed minor. Second, we consider graphs of maximum degree $Delta$. Debski, Felsner, Micek, and Schroder recently proved that these graphs have $p$-centered colorings with $O(Delta^{2-1/p} p)$ colors. We show that there are graphs of maximum degree $Delta$ that require $Omega(Delta^{2-1/p} p ln^{-1/p}Delta)$ colors in any $p$-centered coloring, thus matching their upper bound up to a logarithmic factor.
{"title":"Two lower bounds for $p$-centered colorings","authors":"Loic Dubois, G. Joret, G. Perarnau, Marcin Pilipczuk, Franccois Pitois","doi":"10.23638/DMTCS-22-4-9","DOIUrl":"https://doi.org/10.23638/DMTCS-22-4-9","url":null,"abstract":"Given a graph $G$ and an integer $p$, a coloring $f : V(G) to mathbb{N}$ is $p$-centered if for every connected subgraph $H$ of $G$, either $f$ uses more than $p$ colors on $H$ or there is a color that appears exactly once in $H$. The notion of $p$-centered colorings plays a central role in the theory of sparse graphs. In this note we show two lower bounds on the number of colors required in a $p$-centered coloring. \u0000First, we consider monotone classes of graphs whose shallow minors have average degree bounded polynomially in the radius, or equivalently (by a result of Dvořak and Norin), admitting strongly sublinear separators. We construct such a class such that $p$-centered colorings require a number of colors exponential in $p$. This is in contrast with a recent result of Pilipczuk and Siebertz, who established a polynomial upper bound in the special case of graphs excluding a fixed minor. \u0000Second, we consider graphs of maximum degree $Delta$. Debski, Felsner, Micek, and Schroder recently proved that these graphs have $p$-centered colorings with $O(Delta^{2-1/p} p)$ colors. We show that there are graphs of maximum degree $Delta$ that require $Omega(Delta^{2-1/p} p ln^{-1/p}Delta)$ colors in any $p$-centered coloring, thus matching their upper bound up to a logarithmic factor.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"66 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114125267","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}
Finding a solution to a Constraint Satisfaction Problem (CSP) is known to be an NP-hard task. This has motivated the multitude of works that have been devoted to developing techniques that simplify CSP instances before or during their resolution. The present work proposes rigidly enforced schemes for simplifying binary CSPs that allow the narrowing of value domains, either via value merging or via value suppression. The proposed schemes can be viewed as parametrized generalizations of two widely studied CSP simplification techniques, namely, value merging and neighbourhood substitutability. Besides, we show that both schemes may be strengthened in order to allow variable elimination, which may result in more significant simplifications. This work contributes also to the theory of tractable CSPs by identifying a new tractable class of binary CSP.
{"title":"New schemes for simplifying binary constraint satisfaction problems","authors":"Wady Naanaa","doi":"10.23638/DMTCS-22-1-10","DOIUrl":"https://doi.org/10.23638/DMTCS-22-1-10","url":null,"abstract":"Finding a solution to a Constraint Satisfaction Problem (CSP) is known to be an NP-hard task. This has motivated \u0000the multitude of works that have been devoted to developing techniques that simplify CSP instances before or during \u0000their resolution. \u0000The present work proposes rigidly enforced schemes for simplifying binary CSPs that allow the narrowing of value \u0000domains, either via value merging or via value suppression. The proposed schemes can be viewed as parametrized \u0000generalizations of two widely studied CSP simplification techniques, namely, value merging and neighbourhood \u0000substitutability. Besides, we show that both schemes may be strengthened in order to allow variable elimination, \u0000which may result in more significant simplifications. This work contributes also to the theory of tractable CSPs by \u0000identifying a new tractable class of binary CSP.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126543337","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 $H=(V,F)$ be a simple hypergraph without loops. $H$ is called linear if $|fcap g|le 1$ for any $f,gin F$ with $fnot=g$. The $2$-section of $H$, denoted by $[H]_2$, is a graph with $V([H]_2)=V$ and for any $ u,vin V([H]_2)$, $uvin E([H]_2)$ if and only if there is $ fin F$ such that $u,vin f$. The treewidth of a graph is an important invariant in structural and algorithmic graph theory. In this paper, we consider the treewidth of the $2$-section of a linear hypergraph. We will use the minimum degree, maximum degree, anti-rank and average rank of a linear hypergraph to determine the upper and lower bounds of the treewidth of its $2$-section. Since for any graph $G$, there is a linear hypergraph $H$ such that $[H]_2cong G$, we provide a method to estimate the bound of treewidth of graph by the parameters of the hypergraph.
{"title":"The treewidth of 2-section of hypergraphs","authors":"Ke Liu, Mei Lu","doi":"10.46298/dmtcs.6499","DOIUrl":"https://doi.org/10.46298/dmtcs.6499","url":null,"abstract":"Let $H=(V,F)$ be a simple hypergraph without loops. $H$ is called linear if\u0000$|fcap g|le 1$ for any $f,gin F$ with $fnot=g$. The $2$-section of $H$,\u0000denoted by $[H]_2$, is a graph with $V([H]_2)=V$ and for any $ u,vin\u0000V([H]_2)$, $uvin E([H]_2)$ if and only if there is $ fin F$ such that $u,vin\u0000f$. The treewidth of a graph is an important invariant in structural and\u0000algorithmic graph theory. In this paper, we consider the treewidth of the\u0000$2$-section of a linear hypergraph. We will use the minimum degree, maximum\u0000degree, anti-rank and average rank of a linear hypergraph to determine the\u0000upper and lower bounds of the treewidth of its $2$-section. Since for any graph\u0000$G$, there is a linear hypergraph $H$ such that $[H]_2cong G$, we provide a\u0000method to estimate the bound of treewidth of graph by the parameters of the\u0000hypergraph.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124892281","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 prove a collection of results on graphical indices. We determine the extremal graphs attaining the maximal generalized Wiener index (e.g. the hyper-Wiener index) among all graphs with given matching number or independence number. This generalizes some work of Dankelmann, as well as some work of Chung. We also show alternative proofs for two recents results on maximizing the Wiener index and external Wiener index by deriving it from earlier results. We end with proving two conjectures. We prove that the maximum for the difference of the Wiener index and the eccentricity is attained by the path if the order $n$ is at least $9$ and that the maximum weighted Szeged index of graphs of given order is attained by the balanced complete bipartite graphs.
{"title":"Five results on maximizing topological indices in graphs","authors":"Stijn Cambie","doi":"10.46298/dmtcs.6896","DOIUrl":"https://doi.org/10.46298/dmtcs.6896","url":null,"abstract":"In this paper, we prove a collection of results on graphical indices. We\u0000determine the extremal graphs attaining the maximal generalized Wiener index\u0000(e.g. the hyper-Wiener index) among all graphs with given matching number or\u0000independence number. This generalizes some work of Dankelmann, as well as some\u0000work of Chung. We also show alternative proofs for two recents results on\u0000maximizing the Wiener index and external Wiener index by deriving it from\u0000earlier results. We end with proving two conjectures. We prove that the maximum\u0000for the difference of the Wiener index and the eccentricity is attained by the\u0000path if the order $n$ is at least $9$ and that the maximum weighted Szeged\u0000index of graphs of given order is attained by the balanced complete bipartite\u0000graphs.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125105508","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}
Ruy Fabila Monroy, J. Leaños, A. L. Trujillo-Negrete
Let $k$ and $n$ be integers such that $1leq k leq n-1$, and let $G$ be a simple graph of order $n$. The $k$-token graph $F_k(G)$ of $G$ is the graph whose vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$ whenever their symmetric difference is an edge of $G$. In this paper we show that if $G$ is a tree, then the connectivity of $F_k(G)$ is equal to the minimum degree of $F_k(G)$.
设$k$和$n$为整数，使得$1leq k leq n-1$，设$G$为阶为$n$的简单图。$G$的$k$ -token图$F_k(G)$是其顶点是$V(G)$的$k$ -子集的图，其中两个顶点在$F_k(G)$中相邻，只要它们的对称差是$G$的一条边。本文证明了如果$G$是树，那么$F_k(G)$的连通性等于$F_k(G)$的最小度。
{"title":"On the Connectivity of Token Graphs of Trees","authors":"Ruy Fabila Monroy, J. Leaños, A. L. Trujillo-Negrete","doi":"10.46298/dmtcs.7538","DOIUrl":"https://doi.org/10.46298/dmtcs.7538","url":null,"abstract":"Let $k$ and $n$ be integers such that $1leq k leq n-1$, and let $G$ be a\u0000simple graph of order $n$. The $k$-token graph $F_k(G)$ of $G$ is the graph\u0000whose vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent\u0000in $F_k(G)$ whenever their symmetric difference is an edge of $G$. In this\u0000paper we show that if $G$ is a tree, then the connectivity of $F_k(G)$ is equal\u0000to the minimum degree of $F_k(G)$.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115192020","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 consider non-trivial homomorphisms to reflexive oriented graphs in which some pair of adjacent vertices have the same image. Using a notion of convexity for oriented graphs, we study those oriented graphs that do not admit such homomorphisms. We fully classify those oriented graphs with tree-width $2$ that do not admit such homomorphisms and show that it is NP-complete to decide if a graph admits an orientation that does not admit such homomorphisms. We prove analogous results for $2$-edge-coloured graphs. We apply our results on oriented graphs to provide a new tool in the study of chromatic number of orientations of planar graphs -- a long-standing open problem.
{"title":"On the existence and non-existence of improper homomorphisms of oriented and $2$-edge-coloured graphs to reflexive targets","authors":"Christopher Duffy, S. Shan","doi":"10.46298/dmtcs.6773","DOIUrl":"https://doi.org/10.46298/dmtcs.6773","url":null,"abstract":"We consider non-trivial homomorphisms to reflexive oriented graphs in which some pair of adjacent vertices have the same image. Using a notion of convexity for oriented graphs, we study those oriented graphs that do not admit such homomorphisms. We fully classify those oriented graphs with tree-width $2$ that do not admit such homomorphisms and show that it is NP-complete to decide if a graph admits an orientation that does not admit such homomorphisms. We prove analogous results for $2$-edge-coloured graphs. We apply our results on oriented graphs to provide a new tool in the study of chromatic number of orientations of planar graphs -- a long-standing open problem.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"2005 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128808438","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 equitable partition into branchings in a digraph is a partition of the arc set into branchings such that the sizes of any two branchings differ at most by one. For a digraph whose arc set can be partitioned into $k$ branchings, there always exists an equitable partition into $k$ branchings. In this paper, we present two extensions of equitable partitions into branchings in digraphs: those into matching forests in mixed graphs; and into $b$-branchings in digraphs. For matching forests, Kir'{a}ly and Yokoi (2022) considered a tricriteria equitability based on the sizes of the matching forest, and the matching and branching therein. In contrast to this, we introduce a single-criterion equitability based on the number of covered vertices, which is plausible in the light of the delta-matroid structure of matching forests. While the existence of this equitable partition can be derived from a lemma in Kir'{a}ly and Yokoi, we present its direct and simpler proof. For $b$-branchings, we define an equitability notion based on the size of the $b$-branching and the indegrees of all vertices, and prove that an equitable partition always exists. We then derive the integer decomposition property of the associated polytopes.
{"title":"Notes on Equitable Partitions into Matching Forests in Mixed Graphs and into $b$-branchings in Digraphs","authors":"Kenjiro Takazawa","doi":"10.46298/dmtcs.8719","DOIUrl":"https://doi.org/10.46298/dmtcs.8719","url":null,"abstract":"An equitable partition into branchings in a digraph is a partition of the arc\u0000set into branchings such that the sizes of any two branchings differ at most by\u0000one. For a digraph whose arc set can be partitioned into $k$ branchings, there\u0000always exists an equitable partition into $k$ branchings. In this paper, we\u0000present two extensions of equitable partitions into branchings in digraphs:\u0000those into matching forests in mixed graphs; and into $b$-branchings in\u0000digraphs. For matching forests, Kir'{a}ly and Yokoi (2022) considered a\u0000tricriteria equitability based on the sizes of the matching forest, and the\u0000matching and branching therein. In contrast to this, we introduce a\u0000single-criterion equitability based on the number of covered vertices, which is\u0000plausible in the light of the delta-matroid structure of matching forests.\u0000While the existence of this equitable partition can be derived from a lemma in\u0000Kir'{a}ly and Yokoi, we present its direct and simpler proof. For\u0000$b$-branchings, we define an equitability notion based on the size of the\u0000$b$-branching and the indegrees of all vertices, and prove that an equitable\u0000partition always exists. We then derive the integer decomposition property of\u0000the associated polytopes.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"25 10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125764561","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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