Let H = be a hypergraph, where G = (V, E) is a complete undirected graph and S is a set of not necessarily disjoint clusters Si ⊆ V. The Clustered Spanning Tree problem is to find a spanning tree of G which satisifes that each cluster induces a subtree, when it exists. We provide an efficient and unique algorithm which finds a feasible solution tree for H when it exists, or states that no feasible solution exists. The paper also uses special structures of the intersection graph of H to construct a feasible solution more efficiently. For cases when the hypergraph does not have a feasible solution tree, we consider adding vertices to exactly one cluster in order to gain feasibility. We characterize when such addition can gain feasibility, find the appropriate cluster and a possible set of vertices to be added.
{"title":"Clustered Spanning Tree - Conditions for Feasibility","authors":"Nili Guttmann-Beck, Zeev Sorek, Michal Stern","doi":"10.23638/DMTCS-21-1-15","DOIUrl":"https://doi.org/10.23638/DMTCS-21-1-15","url":null,"abstract":"Let H = be a hypergraph, where G = (V, E) is a complete undirected graph and S is a set of not necessarily disjoint clusters Si ⊆ V. The Clustered Spanning Tree problem is to find a spanning tree of G which satisifes that each cluster induces a subtree, when it exists. We provide an efficient and unique algorithm which finds a feasible solution tree for H when it exists, or states that no feasible solution exists. The paper also uses special structures of the intersection graph of H to construct a feasible solution more efficiently. For cases when the hypergraph does not have a feasible solution tree, we consider adding vertices to exactly one cluster in order to gain feasibility. We characterize when such addition can gain feasibility, find the appropriate cluster and a possible set of vertices to be added.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116029402","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 rearrangement operation makes a small graph-theoretical change to a phylogenetic network to transform it into another one. For unrooted phylogenetic trees and networks, popular rearrangement operations are tree bisection and reconnection (TBR) and prune and regraft (PR) (called subtree prune and regraft (SPR) on trees). Each of these operations induces a metric on the sets of phylogenetic trees and networks. The TBR-distance between two unrooted phylogenetic trees $T$ and $T'$ can be characterised by a maximum agreement forest, that is, a forest with a minimum number of components that covers both $T$ and $T'$ in a certain way. This characterisation has facilitated the development of fixed-parameter tractable algorithms and approximation algorithms. Here, we introduce maximum agreement graphs as a generalisations of maximum agreement forests for phylogenetic networks. While the agreement distance -- the metric induced by maximum agreement graphs -- does not characterise the TBR-distance of two networks, we show that it still provides constant-factor bounds on the TBR-distance. We find similar results for PR in terms of maximum endpoint agreement graphs.
{"title":"The agreement distance of unrooted phylogenetic networks","authors":"J. Klawitter","doi":"10.23638/DMTCS-22-1-22","DOIUrl":"https://doi.org/10.23638/DMTCS-22-1-22","url":null,"abstract":"A rearrangement operation makes a small graph-theoretical change to a phylogenetic network to transform it into another one. For unrooted phylogenetic trees and networks, popular rearrangement operations are tree bisection and reconnection (TBR) and prune and regraft (PR) (called subtree prune and regraft (SPR) on trees). Each of these operations induces a metric on the sets of phylogenetic trees and networks. The TBR-distance between two unrooted phylogenetic trees $T$ and $T'$ can be characterised by a maximum agreement forest, that is, a forest with a minimum number of components that covers both $T$ and $T'$ in a certain way. This characterisation has facilitated the development of fixed-parameter tractable algorithms and approximation algorithms. Here, we introduce maximum agreement graphs as a generalisations of maximum agreement forests for phylogenetic networks. While the agreement distance -- the metric induced by maximum agreement graphs -- does not characterise the TBR-distance of two networks, we show that it still provides constant-factor bounds on the TBR-distance. We find similar results for PR in terms of maximum endpoint agreement graphs.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"99 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127600341","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 flower at a point x in a Steiner triple system (X; B) is the set of all triples containing x. Denote by J3F(r) the set of all integers k such that there exists a collection of three STS(2r+1) mutually intersecting in the same set of k + r triples, r of them being the triples of a common flower. In this article we determine the set J3F(r) for any positive integer r = 0, 1 (mod 3) (only some cases are left undecided for r = 6, 7, 9, 24), and establish that J3F(r) = I3F(r) for r = 0, 1 (mod 3) where I3F(r) = {0, 1,..., 2r(r-1)/3-8, 2r(r-1)/3-6, 2r(r-1)/3}.
{"title":"The 3-way flower intersection problem for Steiner triple systems","authors":"H. Amjadi, N. Soltankhah","doi":"10.23638/DMTCS-22-1-5","DOIUrl":"https://doi.org/10.23638/DMTCS-22-1-5","url":null,"abstract":"The flower at a point x in a Steiner triple system (X; B) is the set of all triples containing x. Denote by J3F(r) the set of all integers k such that there exists a collection of three STS(2r+1) mutually intersecting in the same set of k + r triples, r of them being the triples of a common flower. \u0000In this article we determine the set J3F(r) for any positive integer r = 0, 1 (mod 3) (only some cases are left undecided for r = 6, 7, 9, 24), and establish that J3F(r) = I3F(r) for r = 0, 1 (mod 3) where I3F(r) = {0, 1,..., 2r(r-1)/3-8, 2r(r-1)/3-6, 2r(r-1)/3}.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114778580","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 enumerate classes of uniquely sorted permutations that avoid a pattern of length three and a pattern of length four by establishing bijections between these classes and various lattice paths. This allows us to prove nine conjectures of Defant.
{"title":"Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations","authors":"H. Mularczyk","doi":"10.46298/dmtcs.6494","DOIUrl":"https://doi.org/10.46298/dmtcs.6494","url":null,"abstract":"We enumerate classes of uniquely sorted permutations that avoid a pattern of length three and a pattern of length four by establishing bijections between these classes and various lattice paths. This allows us to prove nine conjectures of Defant.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121440902","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}
Fici, Restivo, Silva, and Zamboni define a $k$-antipower to be a word composed of $k$ pairwise distinct, concatenated words of equal length. Berger and Defant conjecture that for any sufficiently well-behaved aperiodic morphic word $w$, there exists a constant $c$ such that for any $k$ and any index $i$, a $k$-antipower with block length at most $ck$ starts at the $i$th position of $w$. They prove their conjecture in the case of binary words, and we extend their result to alphabets of arbitrary finite size and characterize those words for which the result does not hold. We also prove their conjecture in the specific case of the Fibonacci word.
{"title":"Antipowers in Uniform Morphic Words and the Fibonacci Word","authors":"Swapnil Garg","doi":"10.46298/dmtcs.7134","DOIUrl":"https://doi.org/10.46298/dmtcs.7134","url":null,"abstract":"Fici, Restivo, Silva, and Zamboni define a $k$-antipower to be a word\u0000composed of $k$ pairwise distinct, concatenated words of equal length. Berger\u0000and Defant conjecture that for any sufficiently well-behaved aperiodic morphic\u0000word $w$, there exists a constant $c$ such that for any $k$ and any index $i$,\u0000a $k$-antipower with block length at most $ck$ starts at the $i$th position of\u0000$w$. They prove their conjecture in the case of binary words, and we extend\u0000their result to alphabets of arbitrary finite size and characterize those words\u0000for which the result does not hold. We also prove their conjecture in the\u0000specific case of the Fibonacci word.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124097636","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}
Y. Manoussakis (J. Graph Theory 16, 1992, 51-59) proposed the following conjecture. noindenttextbf{Conjecture}. {it Let $D$ be a 2-strongly connected digraph of order $n$ such that for all distinct pairs of non-adjacent vertices $x$, $y$ and $w$, $z$, we have $d(x)+d(y)+d(w)+d(z)geq 4n-3$. Then $D$ is Hamiltonian.} In this paper, we confirm this conjecture. Moreover, we prove that if a digraph $D$ satisfies the conditions of this conjecture and has a pair of non-adjacent vertices ${x,y}$ such that $d(x)+d(y)leq 2n-4$, then $D$ contains cycles of all lengths $3, 4, ldots , n$.
Y. Manoussakis (J. Graph Theory 16, 1992,51 -59)提出了以下猜想。noindenttextbf{猜想}。{it设$D$是一个二阶强连通有向图，其阶为$n$，使得对于所有不同的不相邻顶点对$x$, $y$和$w$, $z$，我们有$d(x)+d(y)+d(w)+d(z)geq 4n-3$。那么$D$就是汉密尔顿函数。}在本文中，我们证实了这一猜想。此外，我们证明了如果一个有向图$D$满足这个猜想的条件并且有一对不相邻的顶点${x,y}$使得$d(x)+d(y)leq 2n-4$，那么$D$包含所有长度$3, 4, ldots , n$的环。
{"title":"A new sufficient condition for a Digraph to be Hamiltonian-A proof of Manoussakis Conjecture","authors":"S. Darbinyan","doi":"10.23638/DMTCS-22-4-12","DOIUrl":"https://doi.org/10.23638/DMTCS-22-4-12","url":null,"abstract":"Y. Manoussakis (J. Graph Theory 16, 1992, 51-59) proposed the following conjecture. \u0000noindenttextbf{Conjecture}. {it Let $D$ be a 2-strongly connected digraph of order $n$ such that for all distinct pairs of non-adjacent vertices $x$, $y$ and $w$, $z$, we have $d(x)+d(y)+d(w)+d(z)geq 4n-3$. Then $D$ is Hamiltonian.} \u0000In this paper, we confirm this conjecture. Moreover, we prove that if a digraph $D$ satisfies the conditions of this conjecture and has a pair of non-adjacent vertices ${x,y}$ such that $d(x)+d(y)leq 2n-4$, then $D$ contains cycles of all lengths $3, 4, ldots , n$.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"96 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116894226","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}
Julien Bensmail, T. Blanc, Nathann Cohen, F. Havet, L. Rocha
We investigate graph colouring models for the purpose of optimizing TDMA link scheduling in Wireless Networks. Inspired by the BPRN-colouring model recently introduced by Rocha and Sasaki, we introduce a new colouring model, namely the BMRN-colouring model, which can be used to model link scheduling problems where particular types of collisions must be avoided during the node transmissions. In this paper, we initiate the study of the BMRN-colouring model by providing several bounds on the minimum number of colours needed to BMRN-colour digraphs, as well as several complexity results establishing the hardness of finding optimal colourings. We also give a special focus on these considerations for planar digraph topologies, for which we provide refined results.
{"title":"Backbone colouring and algorithms for TDMA scheduling","authors":"Julien Bensmail, T. Blanc, Nathann Cohen, F. Havet, L. Rocha","doi":"10.23638/DMTCS-21-3-24","DOIUrl":"https://doi.org/10.23638/DMTCS-21-3-24","url":null,"abstract":"We investigate graph colouring models for the purpose of optimizing TDMA link scheduling in Wireless Networks. Inspired by the BPRN-colouring model recently introduced by Rocha and Sasaki, we introduce a new colouring model, namely the BMRN-colouring model, which can be used to model link scheduling problems where particular types of collisions must be avoided during the node transmissions. \u0000 \u0000In this paper, we initiate the study of the BMRN-colouring model by providing several bounds on the minimum number of colours needed to BMRN-colour digraphs, as well as several complexity results establishing the hardness of finding optimal colourings. We also give a special focus on these considerations for planar digraph topologies, for which we provide refined results.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116167469","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}
Nicolas Grelier, S. Ilchi, Tillmann Miltzow, Shakhar Smorodinsky
A family S of convex sets in the plane defines a hypergraph H = (S, E) as follows. Every subfamily S' of S defines a hyperedge of H if and only if there exists a halfspace h that fully contains S' , and no other set of S is fully contained in h. In this case, we say that h realizes S'. We say a set S is shattered, if all its subsets are realized. The VC-dimension of a hypergraph H is the size of the largest shattered set. We show that the VC-dimension for pairwise disjoint convex sets in the plane is bounded by 3, and this is tight. In contrast, we show the VC-dimension of convex sets in the plane (not necessarily disjoint) is unbounded. We provide a quadratic lower bound in the number of pairs of intersecting sets in a shattered family of convex sets in the plane. We also show that the VC-dimension is unbounded for pairwise disjoint convex sets in R^d , for d > 2. We focus on, possibly intersecting, segments in the plane and determine that the VC-dimension is always at most 5. And this is tight, as we construct a set of five segments that can be shattered. We give two exemplary applications. One for a geometric set cover problem and one for a range-query data structure problem, to motivate our findings.
{"title":"On the VC-dimension of half-spaces with respect to convex sets","authors":"Nicolas Grelier, S. Ilchi, Tillmann Miltzow, Shakhar Smorodinsky","doi":"10.46298/dmtcs.6631","DOIUrl":"https://doi.org/10.46298/dmtcs.6631","url":null,"abstract":"A family S of convex sets in the plane defines a hypergraph H = (S, E) as\u0000follows. Every subfamily S' of S defines a hyperedge of H if and only if there\u0000exists a halfspace h that fully contains S' , and no other set of S is fully\u0000contained in h. In this case, we say that h realizes S'. We say a set S is\u0000shattered, if all its subsets are realized. The VC-dimension of a hypergraph H\u0000is the size of the largest shattered set. We show that the VC-dimension for\u0000pairwise disjoint convex sets in the plane is bounded by 3, and this is tight.\u0000In contrast, we show the VC-dimension of convex sets in the plane (not\u0000necessarily disjoint) is unbounded. We provide a quadratic lower bound in the\u0000number of pairs of intersecting sets in a shattered family of convex sets in\u0000the plane. We also show that the VC-dimension is unbounded for pairwise\u0000disjoint convex sets in R^d , for d > 2. We focus on, possibly intersecting,\u0000segments in the plane and determine that the VC-dimension is always at most 5.\u0000And this is tight, as we construct a set of five segments that can be\u0000shattered. We give two exemplary applications. One for a geometric set cover\u0000problem and one for a range-query data structure problem, to motivate our\u0000findings.\u0000","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128364333","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 Hausdorff distance is a relatively new measure of similarity of graphs. The notion of the Hausdorff distance considers a special kind of a common subgraph of the compared graphs and depends on the structural properties outside of the common subgraph. There was no known efficient algorithm for the problem of determining the Hausdorff distance between two trees, and in this paper we present a polynomial-time algorithm for it. The algorithm is recursive and it utilizes the divide and conquer technique. As a subtask it also uses the procedure that is based on the well known graph algorithm of finding the maximum bipartite matching.
{"title":"Determining the Hausdorff Distance Between Trees in Polynomial Time","authors":"Aleksander Kelenc","doi":"10.46298/dmtcs.6952","DOIUrl":"https://doi.org/10.46298/dmtcs.6952","url":null,"abstract":"The Hausdorff distance is a relatively new measure of similarity of graphs.\u0000The notion of the Hausdorff distance considers a special kind of a common\u0000subgraph of the compared graphs and depends on the structural properties\u0000outside of the common subgraph. There was no known efficient algorithm for the\u0000problem of determining the Hausdorff distance between two trees, and in this\u0000paper we present a polynomial-time algorithm for it. The algorithm is recursive\u0000and it utilizes the divide and conquer technique. As a subtask it also uses the\u0000procedure that is based on the well known graph algorithm of finding the\u0000maximum bipartite matching.\u0000","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"77 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125907582","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}
Neumann-Lara and Skrekovski conjectured that every planar digraph is 2-colourable. We show that this conjecture is equivalent to the more general statement that all oriented K_5-minor-free graphs are 2-colourable.
{"title":"A Note on Graphs of Dichromatic Number 2","authors":"R. Steiner","doi":"10.23638/DMTCS-22-4-11","DOIUrl":"https://doi.org/10.23638/DMTCS-22-4-11","url":null,"abstract":"Neumann-Lara and Skrekovski conjectured that every planar digraph is 2-colourable. We show that this conjecture is equivalent to the more general statement that all oriented K_5-minor-free graphs are 2-colourable.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"2013 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125668672","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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