Graph

For a graphic matroid we draw a graph whose cycle matroid it is: element $i$ is the edge labeled $i$, the circuits are the cycles, and the bases are the spanning trees. Loops are drawn as self-loops and parallel elements as parallel edges.

We find the graph by searching over edge endpoints, checking the rank of every set of edges (number of vertices touched minus number of components) against the matroid. It is connected with $\mathrm{rank}+1$ vertices. By Whitney's 2-isomorphism theorem the graph is unique only up to gluing and splitting at cut vertices and twisting at 2-vertex cuts, so a matroid that is not 3-connected may have other, non-isomorphic graphs. We choose the layout among polygon placements and spring embeddings to minimize edge crossings, so it is not guaranteed to be planar even when the graph is.