Matrices with the edmonds—Johnson property

Abstract

A matrixA=(a ij ) has theEdmonds—Johnson property if, for each choice of integral vectorsd 1,d 2,b 1,b 2, the convex hull of the integral solutions ofd 1xd 2,b 1Axb 2 is obtained by adding the inequalitiescx≦|δ|, wherec is an integral vector andcx≦δ holds for each solution ofd 1xd 2,b 1Axb 2. We characterize the Edmonds—Johnson property for integral matricesA which satisfy\(\mathop \Sigma \limits_j |a_{ij} | \leqq 2\) for each (row index)i. A corollary is that ifG is an undirected graph which does not contain any homeomorph ofK 4 in which all triangles ofK 4 have become odd circuits, thenG ist-perfect. This extends results of Boulala, Fonlupt, Sbihi and Uhry.

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First author’s research supported by the Netherlands Organization for the Advancement of Pure Research (Z.W.O.).

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Gerards, A.M.H., Schrijver, A. Matrices with the edmonds—Johnson property. Combinatorica 6, 365–379 (1986). https://doi.org/10.1007/BF02579262

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