Non-Separating Cocircuits and Graphicness in Matroids

Let $M$ be a 3-connected binary matroid and let $Y(M)$ be the set of elements of $M$ avoiding at least $r(M)+1$ non-separating cocircuits of $M$. Lemos proved that $M$ is non-graphic if and only if $Y(M)\mathrm{\ne }\text{emp}$. We generalize this result when by establishing that $Y(M)$ is very large when $M$ is non-graphic and $M$ has no $M\text{s}(K_{3,3}{\mathrm{"}}^{\mathrm{\prime }})$-minor if $M$ is regular. More precisely that $\mathrm{\mid }E(M)-Y(M)\mathrm{\mid }\le 1$ in this case. We conjecture that when $M$ is a regular matroid with an $M\text{s}(K_{3,3})$-minor, then $r{\text{s}}_M(E(M)-Y(M))\le 2$. The proof of such conjecture is reduced to a computational verification.