Elements belonging to triangles in 3-connected matroids

An element e of a 3-connected matroid M is said to be superfluous provided M/e is 3-connected. In this paper, we show that a 3-connected matroid M with exactly k superfluous elements has at leastmax5|E(M)|+30-15k9,5r*(M)+30-10k7elements covered by triangles. For each k, an infinite family of matroids that attain this bound is constructed.