On the Matroid Isomorphism Problem

Abstract

Let M to be a matroid defined on a finite set E and L ⊂ E. L is locked in M if M |L and M * |(E\L) are 2-connected, and min{r(L), r * (E\L)} ≥ 2. Given a positive integer k, M is k-locked if the number of its locked subsets is O(|E| k ). L k is the class of k-locked matroids (for a fixed k). In this paper, we give a new axiom system for matroids based on locked subsets. We deduce that the matroid isomorphism problem (MIP) for L k is polynomially time reducible to the graph isomorphism problem (GIP). L k is closed under 2-sums and contains the class of uniform matroids, the Vámos matroid and all the excluded minors of 2-sums of uniform matroids. MIP is coNP-hard even for linear matroids.