Bases-cobases graphs and polytopes of matroids

Abstract

LetM be ablock matroid (i.e. a matroid whose ground setE is the disjoint union of two bases). We associate withM two objects:

  1. 1.

    Thebases-cobases graph G=G(M,M *) having as vertices the basesB ofM for which the complementE\B is also a base, and as edges the unordered pairs (B,B′) of such bases differing exactly by two elements.

  2. 2.

    Thepolytope of the bases-cobases K=K(M,M *) whose extreme points are the incidence vectors of the bases ofM whose complement is also a base.

We prove that, ifM is graphic (or cographic), the distance between any two vertices ofG corresponding to disjoint bases is equal to the rank ofM (generalizing a result of [10]).

Concerning the polytope we prove thatK is an hypercube if and only if dim(K)=rank(M). A constructive characterization of the class of matroids realizing this equality is given.

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Partially supported by a grant from I. N. I. C.

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Cordovil, R., Moreira, M.L. Bases-cobases graphs and polytopes of matroids. Combinatorica 13, 157–165 (1993). https://doi.org/10.1007/BF01303201

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AMS subject classification code (1991)

  • 05 B 35
  • 05 C 38