Local Out-Tournaments with Upset Tournament Strong Components I: Full and Equal {0,1}-Matrix Ranks

A digraph D is a local out-tournament if the outset of every vertex is a tournament. Here, we use local out-tournaments, whose strong components are upset tournaments, to explore the corresponding ranks of the adjacency matrices. Of specific interest is the out-tournament whose adjacency matrix has boolean, nonnegative integer, term, and real rank all equal to the number of vertices, n. Corresponding results for biclique covers and partitions of the digraph are provided

The Geometry of Random Tournaments

A tournament is an orientation of a graph. Each edge is a match, directed towards the winner. The score sequence lists the number of wins by each team. In this article, by interpreting score sequences geometrically, we generalize and extend classical theorems of Landau (Bull. Math. Biophys. 15, 143–148 (1953)) and Moon (Pac. J. Math. 13, 1343–1345 (1963)), via the theory of zonotopal tilings

Ranking tournaments with no errors I: Structural description

In this series of two papers we examine the classical problem of ranking a set of players on the basis of a set of pairwise comparisons arising from a sports tournament, with the objective of minimizing the total number of upsets, where an upset occurs if a higher ranked player was actually defeated by a lower ranked player. This problem can be rephrased as the so-called minimum feedback arc set problem on tournaments, which arises in a rich variety of applications and has been a subject of extensive research. In this series we study this NP-hard problem using structure-driven and linear programming approaches. Let T=(V,A) be a tournament with a nonnegative integral weight w(e) on each arc e. A subset F of arcs is called a feedback arc set if T\F contains no cycles (directed). A collection C of cycles (with repetition allowed) is called a cycle packing if each arc e is used at most w(e) times by members of C. We call T cycle Mengerian (CM) if, for every nonnegative integral function w defined on A, the minimum total weight of a feedback arc set is equal to the maximum size of a cycle packing. The purpose of these two papers is to show that a tournament is CM iff it contains none of four Möbius ladders as a subgraph; such a tournament is referred to as Möbius-free. In this first paper we present a structural description of all Möbius-free tournaments, which relies heavily on a chain theorem concerning internally 2-strong tournaments

Applications of network optimization

Includes bibliographical references (p. 41-48).Ravindra K. Ahuja ... [et al.]

Out-Tournament Adjacency Matrices with Equal Ranks

Much work has been done in analyzing various classes of tournaments, giving a partial characterization of tournaments with adjacency matrices having equal and full real, nonnegative integer, Boolean, and term ranks. Relatively little is known about the corresponding adjacency matrix ranks of local out-tournaments, a larger family of digraphs containing the class of tournaments. Based on each of several structural theorems from Bang-Jensen, Huang, and Prisner, we will identify several classes of out-tournaments which have the desired adjacency matrix rank properties. First we will consider matrix ranks of out-tournament matrices from the perspective of the structural composition of the strong component layout of the adjacency matrix. Following that, we will consider adjacency matrix ranks of an out-tournament based on the cycles that the out-tournament contains. Most of the remaining chapters consider the adjacency matrix ranks of several classes of out-tournaments based on the form of their underlying graphs. In the case of the strong out-tournaments discussed in the final chapter, we examine the underlying graph of a representation that has the strong out-tournament as its catch digraph

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application

Mark Sequences In Digraphs

In Chapter 1, we present a brief introduction of digraphs and some def- initions. Chapter 2 is a review of scores in tournaments and oriented graphs. Also we have obtained several new results on oriented graph scores and we have given a new proof of Avery's theorem on oriented graph scores. In chap- ter 3, we have introduced the concept of marks in multidigraphs, non-negative integers attached to the vertices of multidigraphs. We have obtained several necessary and su cient conditions for sequences of non-negative integers to be mark sequences of some r-digraphs. We have derived stronger inequalities for these marks. Further we have characterized uniquely mark sequences in r-digraphs. This concept of marks has been extended to bipartite multidi- graphs and multipartite multidigraphs in chapter 4. There we have obtained characterizations for mark sequences in these types of multidigraphs and we have given algorithms for constructing corresponding multidigraphs. Chap- ter 5 deals with imbalances and imbalance sequences in digraphs. We have generalized the concept of imbalances to oriented bipartite graphs and have obtained criteria for a pair of integers to be the pair of imbalance sequences of some oriented bipartite graph. We have shown the existence of an oriented bipartite graph whose imbalance set is the given set of integers

Applications of network optimization

Includes bibliographical references (p. 41-48).Ravindra K. Ahuja ... [et al.]