The Strong Kings of Tournaments

• Main Authors: An-Hang Chen, 陳恩航
• Other Authors: Yue-Li Wang
• Format: Others
• Language: en_US
• Published: 2007
• Online Access: http://ndltd.ncl.edu.tw/handle/87070173320073028372

博士 === 國立臺灣科技大學 === 資訊管理系 === 96 === Abstract A tournament Tn is a complete oriented simple graph with n vertices, in which every pair of vertices is joined by exactly one arc. If the arc joining vertices x and y is directed from x to y, then x is said to dominate y and is denoted by x → y. The score of a vertex x is the number of vertices dominated by x in Tn. The score vector of Tn is the list of scores of vertices of Tn in a nondecreasing order. A king x in a tournament Tn is a vertex who dominates any other vertex y directly or indirectly through a third vertex z (i.e., x → z and z → y). For x, y ? Tn, let b(x, y) denote the number of third vertices through which x dominates y indirectly. A vertex x is said to be a strong king if the following condition is fulfilled: b(x, y) > b(y, x) whenever y → x. In this dissertation, we determine whether there exists a tournament Tn with exactly k strong kings where 1 ≦ j ≦ n. A result shows that the answer is affirmative with the following exceptions: k = n - 1 when n is odd, and k = n when n is even. On the other hand, we further determine the uniqueness of tournaments, where a tournament is said to be unique if no other tournament with the same number of vertices (barring isomorphic ones) shares the same number of strong kings. As a result, we can completely determine the uniqueness of tournaments. Moreover, we propose an upper bound and a lower bound on the number of strong kings in tournaments under a score vector is given. Next, we deal with the topic on interchange graphs. Let T (S) be the collection of tournaments that realize a given score vector S. A Δ-interchange is a transformation which reverses the directions of the arcs in a 3-cycle of a digraph. An interchange graph is an undirected graph whose vertices are the tournaments in T (S) and an edge joins two vertices (tournaments) if they can be transformed to each other by a Δ-interchange. In this dissertation, we show that a set of speci#c score vectors of tournaments, said S = (1, 1, 2, . . . , n - 2, n - 2), whose corresponding interchange graphs form a class of graphs called hypercubes. Moreover, we also present some properties of an interchange graph of bipartite tournaments. Based on a list of score vectors provided by Bagga et al. [6] for completely determining the uniquely realizable, we derive formulae for computing the regularity of regular interchange graphs of bipartite tournaments. Finally, we study the topic of king sets. Landau [53] gave a result that implies that every tournament has at least a king. Maurer [65] proved that every tournament without a source has at least three kings. Neither of these results can be extended to arbitrary digraphs. Hence, Bowser and Cable [19] generalized the notion of kings to that of king sets which can be applied to arbitrary digraphs. In fact, the concept of king sets is the same idea of semi-kernel that was #rst introduced by Chvatal and Lovasz [26]. In this dissertation, we derive formulae for counting the number of king sets of digraphs whose underlying graphs are caterpillars with regular legs.