# Score sequence pair problems of (r11, r12, r 22)-tournaments - Determination of realizability

Masaya Takahashi, Takahiro Watanabe, Takeshi Yoshimura

Research output: Contribution to journalArticle

### Abstract

Let G be any graph with property P (for example, general graph, directed graph, etc.) and S be nonnegative and non-decreasing integer sequence(s). The prescribed degree sequence problem is a problem to determine whether there is a graph G having S as the prescribed sequence(s) of degrees or outdegrees of the vertices. From 1950's, P has attracted wide attentions, and its many extensions have been considered. Let P be the property satisfying the following (1) and (2): G is a directed graph with two disjoint vertex sets A and B. There are r11 (r22, respectively) directed edges between every pair of vertices in A(B), and r12 directed edges between every pair of vertex in A and vertex in B. Then G is called an (r11, r 12, r22) -tournament ("tournament", for short). The problem is called the score sequence pair problem of a " tournament" (realizable, for short). S is called a score sequence pair of a "tournament" if the answer of the problem is "yes." In this paper, we propose the characterizations of a score sequence pair of a "tournament" and an algorithm for determining in linear time whether a pair of two integer sequences is realizable or not.

Original language English 440-447 8 IEICE Transactions on Information and Systems E90-D 2 https://doi.org/10.1093/ietisy/e90-d.2.440 Published - 2007 Feb

Directed graphs

### Keywords

• Algorithm
• Graph theory
• Prescribed degrees
• Score sequence
• Tournament

### ASJC Scopus subject areas

• Information Systems
• Computer Graphics and Computer-Aided Design
• Software

### Cite this

Score sequence pair problems of (r11, r12, r 22)-tournaments - Determination of realizability. / Takahashi, Masaya; Watanabe, Takahiro; Yoshimura, Takeshi.

In: IEICE Transactions on Information and Systems, Vol. E90-D, No. 2, 02.2007, p. 440-447.

Research output: Contribution to journalArticle

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