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

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.