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, r12, 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.
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