Given a rank-r binary matroid we construct a system of O(r) linear equations in O(r) variables that has a solution over GF(2) if and only if the matroid is graphic.

We consider the matroid median problem (MMP), which is defined as follows. As in the uncapacitated facility location problem, we are given a set of facilities F and a set of clients D. Each facility i has an opening cost of fi. Each client j ∈ D has demand dj and assigning client j to facility i incurs an assignment cost of djcij proportional to the distance between i and j. Further, we are giv...

One of the foundations of oriented matroid theory is the topological representation theorem of Folkman and Lawrence [8]. It says that an oriented (simple) matroid can be realized uniquely as an arrangement of pseudospheres. That there is no similar interpretation for the class of all matroids has been taken for granted. For instance, “A non-coordinatizable matroid of abstract origin may be thou...

Even-cycle matroids are elementary lifts of graphic and even-cut cographic matroids. We present a polynomial algorithm to check if binary matroid is an even-cycle we matroid. These two algorithms rely on (to be described in pair follow-up papers) pinch-graphic.