On minimally 3-connected binary matroids (Q2777894)

\textit{R. Halin} [Math. Ann. 182, 175-188 (1969; Zbl 0175.50503)] proved that a minimally \(3\)-connected graph with \(n\) vertices has at least \(\frac{2n+ 6}{5}\) vertices of degree 3. The three edges meeting a degree-3 vertex of a \(3\)-connected graph \(G\) form a triad, a 3-element cocircuit, in its cycle matroid \(M(G)\). Moreover, the deletion of this triad from \(M(G)\) leaves a connected matroid. Such a triad is called a vertex triad. The reviewer [Trans. Am. Math. Soc. 265, No. 1, 47-58 (1981; Zbl 0481.05021)] proved an analogue of Halin's result by showing that every minimally \(3\)-connected matroid \(M\) with at least four elements has at least \(\frac{1}{2} r(M^*)+ 1\) triads. In this paper, the authors give short proofs of this and several other related results. They also improve this result for minimally \(3\)-connected binary matroids by showing that, in every such matroid \(M\) of rank at least six, at least \(\frac{3}{2} r(M^*)+ 3\) elements belong to some vertex triad. The main tool in the proof of this result is another generalization of results of Halin and Oxley, namely that for each element \(e\) of a minimally \(3\)-connected binary matroid \(M\), either \(e\) is in a vertex triad of \(M\), or \(M/e\) is minimally \(3\)-connected.