Graphic Matroids

Matroid theory was first formalized in 1935 by Whitney [5] who introduced the notion as an attempt to study the properties of vector spaces in an abstract manner. Since then, matroids have proven to have numerous applications in a wide variety of fields including combinatorics and graph theory. Today we will briefly survey matroid representation and then discuss some problems in matroid optimization and the corresponding applications. The tools we develop will help us answer the following puzzle: Puzzle: A game is played on a graph G(V,E) and has two players, George and Ari. Ari’s moves consist of “fixing” edges e ∈ E. George’s moves consist of deleting any unfixed edge. The game ends when every edge has been either fixed or deleted. Ari wins if the graph at the end of the game is connected (i.e. if the fixed edges form a spanning tree). Otherwise George wins. Supposing George moves first, characterize the graphs in which George has a winning strategy.