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 1≦x≦d 2,b 1≦Ax≦b 2 is obtained by adding the inequalitiescx≦|δ|, wherec is an integral vector andcx≦δ holds for each solution ofd 1≦x≦d 2,b 1≦Ax≦b 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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References
M. Boulala andJ. P. Uhry, Polytope des indépendants d’un graphe série-parallèle,Discrete Mathematics 27 (1979), 225–243.
V. Chvátal, Edmonds polytopes and a hierarchy of combinatorial problems,Discrete Mathematics 4 (1973), 305–337.
V. Chvátal, On certain polytopes associated with graphs,Journal of Combinatorial Theory (B) 18 (1975), 138–154.
J. Edmonds, Maximum matching and a polyhedron with 0, 1 vertices,Journal of Research of the National Bureau of Standards (B) 69 (1965), 125–130.
J. Edmonds andE. L. Johnson, Matching: a well-solved class of integer linear programs,in: Combinatorial Structures and Their Applications (R. K. Guy, et al., eds.), Gordon and Breach, New York, 1970, 89–92.
J. Edmonds andE. L. Johnson, Euler tours and the Chinese postman,Mathematical Programming 5 (1973), 88–124.
J. Fonlupt andJ. P. Uhry, Transformations which preserve perfectness andh-perfectness of graphs,Annals of Discrete Mathematics 16 (1982), 83–95.
M. Grötschel, L. Lovász andA. Schrijver, The ellipsoid method and its consequences in combinatorial optimization,Combinatorica 1 (1981), 169–197.
N. Sbihi andJ. P. Uhry, A class ofh-perfect graphs,Discrete Mathematics 51 (1984), 191–205.
P. D. Seymour, The matroids with the max-flow min-cut property,Journal of Combinatorial Theory (B) 23 (1977), 189–222.
K. Truemper,Max-flow min-cut matroids: polynomial testing and polynomial algorithms for maximum flow and shortest routes, preprint, University of Texas at Dallas, 1984.
F. T. Tseng andK. Truemper,A decomposition of the matroids with the max-flow min-cut property, preprint, University of Texas at Dallas, 1984.
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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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DOI: https://doi.org/10.1007/BF02579262