Realizability of Score Sequence Pair of an (r1l, r12, r22)-Tournament

Let G be any directed graph 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 with S as the prescribed sequence(s) of outdegrees of the vertices. Let G be the property satisfying the following (1) and (2): (1) G has two disjoint vertex sets A and B. (2) For every vertex pair u, visin G (u ne v), G satisfies |{uv}| + |{vu}| = { r₁₁ if u, visin A, r₁₂ if uisin A, visin B, r₂₂ if u, visin B, Then G is called an (r₁₁, r₁₂, r₂₂)-tournament ("tournament", for short). When G is a "tournament," the prescribed degree sequence problem is called the score sequence pair problem of a "tournament", and S is called a score sequence pair of a "tournament" (or S is realizable) if the answer is "yes." The paper proposes the characterizations of a "tournament" and an algorithm for determining in linear time whether a pair of two integer sequences is realizable or not