Realizability of score sequence pair of an (r11, 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, v is an element of G (u not equal v), G satisfies vertical bar{uv}vertical bar + vertical bar{vu}vertical bar = {r(11) if u, v is an element of A {r(12) if u is an element of A, v is an element of B {r(22) if u, v is an element of B where uv (vu, respectively) means a directed edges from u to v (from v to u). Then G is called an (r(11),r(12),r(22))-tournamenf ("tournament", for short). When G is a "tournament," the prescribed degree sequence problem is called the score sequence pairproblem of a "tournament", and S is called a score sequence pair of a "tournament" (or S is realizable) if the answer is "yes." In this paper, we propose the characterizations of a "tournament" and an algorithm for determining in linear time whether a pair of two integer sequences is realizable or not.