Joint realization of (0,1) matrices revisited

Abstract

Let R and R′ be nonnegative integral vectors with m components, and let S and S′ be nonnegative integral vectors with n components. Let U ( R, S) denote the class of all (0,1) matrices of size m by n with prescribed row sum vector R and column sum vector S. The classes U ( R, S) and U ( R′, S') are jointly realizable provided there exist matrices A ϵ U ( R, S) and A′ ϵ U ( R′, S′) such that A ⩽ A′. The pair ( R, S) is totally joint provided that for all R′ and S′ the nonemptiness of the three classes U ( R, S), U ( R, S), and U( R′ − R, S′ − S) implies that the classes U ( R, S) and U ( R′, S′) are jointly realizable. Chen and Shastri recently obtained sufficient conditions for the pair (R, S) to be totally joint. However, their conditions involve two forbidden configurations in the matrices in the class U ( R, S) and thus may be difficult to verify. In this paper we present relatively simple conditions on R and S that characterize the two forbidden configurations. As a corollary, we improve the main result of Chen and Shastri.