Matroids and Graphs with Few Non-Essential Elements

An essential element of a 3–connected matroid M is one for which neither the deletion nor the contraction is 3–connected. Tutte’s Wheels and Whirls Theorem proves that the only 3–connected matroids in which every element is essential are the wheels and whirls. In an earlier paper, the authors showed that a 3–connected matroid with at least one non-essential element has at least two such elements. This paper completely determines all 3–connected matroids with exactly two non-essential elements. Furthermore, it is proved that every 3–connected matroid M for which no single-element contraction is 3–connected can be constructed from a similar such matroid whose rank equals the rank in M of the set of elements e for which the deletion M\e is