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A new characterization of graphic matroids. (English) Zbl 1170.05020
A graph is planar if and only if the dual of its cycle matroid is graphic. A result of Tutte establishes a necessary condition for a matroid to be graphic, that any cocircuit has bipartite avoidance graph. J.C. Fournier [J. Comb. Theory, Ser. B 16, 181-190 (1974; Zbl 0271.05010] proved that a matroid is graphic if and only if, for any three cocircuits $$X_1$$, $$X_2$$, $$X_3$$ with a common element, one of them separates the other two, e.g., $$X_2 \setminus X_1$$ and $$X_3 \setminus X_1$$ are contained in different components of $$M \setminus X_1.$$ (Necessity for cographic matroids is a consequence of the Jordan Curve Theorem.) In the paper under review, the author combines these two results to obtain necessary and sufficient conditions for a (binary) matroid to be graphic that depend only on properties of fundamental cocircuits relative to a single basis $$B$$ of $$M=M(E).$$ This yields a more efficient test for planarity of graphs. The precise result is: a binary matroid is graphic if and only if every fundamental cocircuit has bipartite avoidance graph and, for any three fundamental cocircuits with a point in common, one of them separates the other two. The proof is by induction, involving a careful analysis of the fundamental (bipartite) graph of $$M$$ relative to $$B,$$ having vertex set $$E$$ and $$e \in E - B$$ adjacent to $$f \in B$$ when $$e$$ and $$f$$ lie in a circuit in $$B \cup e$$.

##### MSC:
 05B35 Combinatorial aspects of matroids and geometric lattices
##### Keywords:
binary matroid; fundamental graph; graphic matroid
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##### References:
 [1] Bixby, R.; Cunningham, W.H., Matroids, graphs and 3-connectivity, (), 91-103 [2] Fournier, J.C., Une relation de separation entre cocircuits d’un matroide, J. combin. theory ser. B, 12, 181-190, (1974) · Zbl 0271.05010 [3] Mighton, J., Computing the Jones polynomial on bipartite graphs, J. knot theory, 10, 5, 703-710, (2001) · Zbl 0998.57026 [4] Oxley, J.G., Matroid theory, (1992), Oxford Univ. Press · Zbl 0784.05002 [5] Tutte, W.T., An algorithm for determining whether a given binary matroid is graphic, Proc. amer. math. soc., 11, 905-917, (1960) · Zbl 0097.38905 [6] Tutte, W.T., Lectures on matroids, J. res. nat. bur. standards sect. B, 69, 1-47, (1965) · Zbl 0151.33801
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