Construction of an (r_(11), r_(12), r_(22))-Tournament from a Score Sequence Pair

摘要

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 E G (u not= v), G satisfies |{uv}| + |{vu}| velence r_(11) if u, v E A |{uv}| + |{vu}| velence r_(12) if u E A, v E B |{uv}| + |{vu}| velence r_(22) if u, v E 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))-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." We proposed the characterizations of a "tournament" and an algorithm for determining in linear time whether a pair of two integer sequences is realizable or not [5]. In this paper, we propose an algorithm for constructing a "tournament" from such a score sequence pair.