An Algorithm for Determining Whether a given Binary Matroid Is Graphic

1. Introduction. In a recent series of papers [l-4] on graphs and matroids I used definitions equivalent to the following. A binary chain-group N on a. finite set M is a class of subsets of M forming a group under mod 2 addition. These subsets are the chains of N. A chain of N is elementary if it is non-null and has no other non-null chain of AT as a subset. A binary matroid is the class of elementary chains of a binary chain-group. As an example of a binary chain-group we may take the class of all cuts of a given finite graph G. A cut of G is determined by a partition of its set of vertices into two disjoint subsets U and V, and is defined as the set of all edges having one end in U and the other in V. I have called the corresponding binary matroid the bond-matroid of G. In the above-mentioned series of papers I obtained necessary and sufficient conditions for a given binary matroid to be graphic, that is representable as the bond-matroid of a graph. On several occasions it has been pointed out to me that these results are of interest to electrical engineers,1 but that a practical method for deciding whether or not a given binary matroid was graphic would be still more interesting. In what follows I present an algorithm which I hope will be of some use in this connection. This algorithm is described in §3 and the theorems needed to justify