Bipartite Tournaments Research Articles

Every cycle-connected multipartite tournament has a universal arc

A digraph D is cycle-connected if for every pair of vertices u , v ∈ V ( D ) there exists a directed cycle in D containing both u and v . In 1999, Ádám [A. Ádám, On some cyclic connectivity properties of directed graphs, Acta Cybernet. 14 (1) (1999) 1–12] posed the following problem. Let D be a cycle-connected digraph. Does there exist a universal arc in D , i.e., an arc e ∈ A ( D ) such that for every vertex w ∈ V ( D ) there is a directed cycle in D containing both e and w ? A c -partite or multipartite tournament is an orientation of a complete c -partite graph. Recently, Hubenko [A. Hubenko, On a cyclic connectivity property of directed graphs, Discrete Math. 308 (2008) 1018–1024] proved that each cycle-connected bipartite tournament has a universal arc. As an extension of this result, we show in this note that each cycle-connected multipartite tournament has a universal arc.

Minimum cost homomorphisms to semicomplete multipartite digraphs

For digraphs D and H , a mapping f : V ( D ) → V ( H ) is a homomorphism of D to H if u v ∈ A ( D ) implies f ( u ) f ( v ) ∈ A ( H ) . For a fixed directed or undirected graph H and an input graph D , the problem of verifying whether there exists a homomorphism of D to H has been studied in a large number of papers. We study an optimization version of this decision problem. Our optimization problem is motivated by a real-world problem in defence logistics and was introduced recently by the authors and M. Tso. Suppose we are given a pair of digraphs D , H and a cost c i ( u ) for each u ∈ V ( D ) and i ∈ V ( H ) . The cost of a homomorphism f of D to H is ∑ u ∈ V ( D ) c f ( u ) ( u ) . Let H be a fixed digraph. The minimum cost homomorphism problem for H , MinHOMP( H ), is stated as follows: For input digraph D and costs c i ( u ) for each u ∈ V ( D ) and i ∈ V ( H ) , verify whether there is a homomorphism of D to H and, if it does exist, find such a homomorphism of minimum cost. In our previous paper we obtained a dichotomy classification of the time complexity of MinHOMP ( H ) when H is a semicomplete digraph. In this paper we extend the classification to semicomplete k -partite digraphs, k ≥ 3 , and obtain such a classification for bipartite tournaments.

Monochromatic paths and quasi-monochromatic cycles in edge-coloured bipartite tournaments

Monochromatic paths and quasi-monochromatic cycles in edge-coloured bipartite tournaments

A sufficient condition for Hamiltonian cycles in bipartite tournaments

In this paper, we present a new sufficient condition on degrees for a bipartite tournament to be Hamiltonian, that is, if an n × n bipartite tournament T satisfies the condition W(n − 3), then T is Hamiltonian, except for four exceptional graphs. This result is shown to be best possible in a sense.

Feedback arc set problem in bipartite tournaments

In this paper we give ratio 4 deterministic and randomized approximation algorithms for the Feedback Arc Set problem in bipartite tournaments. We also generalize these results to give a factor 4 deterministic approximation algorithm for Feedback Arc Set problem in multipartite tournaments.

Improved FPT algorithm for feedback vertex set problem in bipartite tournament

We present an O ( 3 k n 2 + n 3 ) time FPT algorithm for the feedback vertex set problem in a bipartite tournament on n vertices with integral weights. This improves the previously best known O ( 3.12 k n 4 ) time FPT algorithm for the problem.

Feedback arc set in bipartite tournaments is NP-complete

The Feedback Arc Set problem asks whether it is possible to delete at most k arcs to make a directed graph acyclic. We show that Feedback Arc Set is NP-complete for bipartite tournaments, that is, directed graphs that are orientations of complete bipartite graphs.

On the 3-kings and 4-kings in multipartite tournaments

Koh and Tan gave a sufficient condition for a 3-partite tournament to have at least one 3-king in [K.M. Koh, B.P. Tan, Kings in multipartite tournaments, Discrete Math. 147 (1995) 171–183, Theorem 2]. In Theorem 1 of this paper, we extend this result to n-partite tournaments, where n ⩾ 3 . In [K.M. Koh, B.P. Tan, Number of 4-kings in bipartite tournaments with no 3-kings, Discrete Math. 154 (1996) 281–287, K.M. Koh, B.P. Tan, The number of kings in a multipartite tournament, Discrete Math. 167/168 (1997) 411–418] Koh and Tan showed that in any n-partite tournament with no transmitters and 3-kings, where n ⩾ 2 , the number of 4-kings is at least eight, and completely characterized all n-partite tournaments having exactly eight 4-kings and no 3-kings. Using Theorem 1, we strengthen substantially the above result for n ⩾ 3 . Motivated by the strengthened result, we further show that in any n-partite tournament T with no transmitters and 3-kings, where n ⩾ 3 , if there are r partite sets of T which contain 4-kings, where 3 ⩽ r ⩽ n , then the number of 4-kings in T is at least r + 8 . An example is given to justify that the lower bound is sharp.

Acyclically pushable bipartite permutation digraphs: An algorithm

Given a digraph D = ( V , A ) and an X ⊆ V , D X denotes the digraph obtained from D by reversing those arcs with exactly one end in X. A digraph D is called acyclically pushable when there exists an X ⊆ V such that D X is acyclic. Huang, MacGillivray and Yeo have recently characterized, in terms of two excluded induced subgraphs on 7 and 8 nodes, those bipartite permutation digraphs which are acyclically pushable. We give an algorithmic proof of their result. Our proof delivers an O ( m 2 ) time algorithm to decide whether a bipartite permutation digraph is acyclically pushable and, if yes, to find a set X such that D X is acyclic. (Huang, MacGillivray and Yeo's result clearly implies an O ( n 8 ) time algorithm to decide but the polynomiality of constructing X was still open.) We define a strongly acyclic digraph as a digraph D such that D X is acyclic for every X. We show how a result of Conforti et al [Balanced cycles and holes in bipartite graphs, Discrete Math. 199 (1–3) (1999) 27–33] can be essentially regarded as a characterization of strongly acyclic digraphs and also provides linear time algorithms to find a strongly acyclic orientation of an undirected graph, if one exists. Besides revealing this connection, we add simplicity to the structural and algorithmic results first given in Conforti et al [Balanced cycles and holes in bipartite graphs, Discrete Math. 199 (1–3) (1999) 27–33]. In particular, we avoid decomposing the graph into triconnected components. We give an alternate proof of a theorem of Huang, MacGillivray and Wood characterizing acyclically pushable bipartite tournaments. Our proof leads to a linear time algorithm which, given a bipartite tournament as input, either returns a set X such that D X is acyclic or a proof that D is not acyclically pushable.

Pushing Vertices and the Strong Connectivity of Bipartite Tournaments

Pushing Vertices and the Strong Connectivity of Bipartite Tournaments

Hamiltonian paths containing a given arc, in almost regular bipartite tournaments

A tournament is an orientation of a complete graph, and in general a multipartite or c-partite tournament is an orientation of a complete c-partite graph. If x is a vertex of a digraph D, then we denote by d +( x) and d −( x) the outdegree and indegree of x, respectively. The global irregularity of a digraph D is defined by i g( D)=max{ d +( x), d −( x)}−min{ d +( y), d −( y)} over all vertices x and y of D (including x= y). If i g( D)⩽1, then D is called almost regular, and if i g( D)=0, then D is regular. More than 10 years ago, Amar and Manoussakis and independently Wang proved that every arc of a regular bipartite tournament is contained in a directed Hamiltonian cycle. In this paper, we prove that every arc of an almost regular bipartite tournament T is contained in a directed Hamiltonian path if and only if the cardinalities of the partite sets differ by at most one and T is not isomorphic to T 3,3, where T 3,3 is an almost regular bipartite tournament with three vertices in each partite set. As an application of this theorem and other results, we show that every arc of an almost regular c-partite tournament D with the partite sets V 1, V 2,…, V c such that | V 1|=| V 2|=⋯=| V c |, is contained in a directed Hamiltonian path if and only if D is not isomorphic to T 3,3.

On monochromatic paths and monochromatic 4-cycles in edge coloured bipartite tournaments

We call the digraph D an m-coloured digraph if the arcs of D are coloured with m colours. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike. A set N⊆ V( D) is said to be a kernel by monochromatic paths if it satisfies the following two conditions: (i) For every pair of different vertices u, v∈ N, there is no monochromatic directed path between them. (ii) For every vertex x∈( V( D)− N), there is a vertex y∈ N such that there is an xy-monochromatic directed path. In this paper it is proved that if D is an m-coloured bipartite tournament such that every directed cycle of length 4 is monochromatic, then D has a kernel by monochromatic paths.

Cycle-pancyclism in bipartite tournaments I

Cycle-pancyclism in bipartite tournaments I

A representation for the modules of a graph and applications

We describe a simple representation for the modules of a graph G. We show that the modules of G are in one-to-one correspondence with the ideals of certain posets. These posets are characterized and shown to be layered posets, that is, transitive closures of bipartite tournaments. Additionaly, we describe applications of the representation. Employing the above correspondence, we present methods for solving the following problems: (i) generate all modules of G, (ii) count the number of modules of G, (iii) find a maximal module satisfying some hereditary property of G and (iv) find a connected non-trivial module of G.

Schttes Tournament Problem and Intersecting Families of Sets

In this paper we consider a bipartite version of Schütte's well-known tournament problem. A bipartite tournament . We achieve this goal by reformulating the problem in terms of intersecting set families and applying probabilistic as well as constructive methods. Intriguing connections with some famous problems of this area have emerged in this way, leading to new open questions.

The number of 2-edge-colored complete graphs with unique hamiltonian alternating cycle

Let G be a 2-edge-colored complete graph of even order n. A cycle of G is alternating if any two successive edges differ in color. We prove that there is a one-to-one correspondence between the set of bipartite tournaments of order n admitting a unique hamiltonian cycle and the set of 2-edge-colored complete graphs of order n admitting a unique alternating hamiltonian cycle.

Maximum Arc-integrity of Tournaments and Bipartite Tournaments

Abstract In this paper we extend the graphical concept of edge-integrity to directed graphs. After some general results we focus on the maximum arc-integrity which can be obtained by orienting the edges of Kn and Km, n for m ≤ 4.

A Min-Max Theorem on Feedback Vertex Sets

We establish a necessary and sufficient condition for the linear system {x : Hx ≥ e, x ≥ 0} associated with a bipartite tournament to be totally dual integral, where H is the cycle-vertex incidence matrix and e is the all-one vector. The consequence is a min-max relation on packing and covering cycles, together with strongly polynomial time algorithms for the feedback vertex set problem and the cycle packing problem on the corresponding bipartite tournaments. In addition, we show that the feedback vertex set problem on general bipartite tournaments is NP-complete and approximable within 3.5 based on the min-max theorem.

Algebraic Multiplicity of the Eigenvalues of a Bipartite Tournament Matrix

Let $\mathcal{B}$ be a bipartite tournament, and let $A(\mathcal{B})$ be its adjacency matrix. $A(\mathcal{B})$ is called a bipartite tournament matrix. In this paper, we mainly discuss the spectral properties of a bipartite tournament matrix, especially for the algebraic multiplicity of the eigenvalue 0 and the number of distinct eigenvalues.

Path numbers of balanced bipartite tournaments

A path decomposition of a digraph D is a partition of its edge set into edge disjoint simple paths. The minimal number of paths necessary to form a path decomposition is called the path number of D and denoted by pn(D) . A bipartite tournament T(A,B) with partition sets A and B is balanced if |A|=|B|=n⩾1. We prove the following: (a) if n is odd and k is any odd integer from the interval [n,n 2] or (b) if n is even and k is any even integer from the interval [n/2+1,n 2] , then there exists a balanced bipartite tournament T(A,B), |A|=|B|=n, with pn(T(A,B))=k .