Walks, Paths and Connectivity · Superchallenge problem

A connected graph on $10$ vertices must stay connected after removing ANY $2$ of its vertices (that is, $\kappa(G) \ge 3$). What is the fewest edges such a graph can have, and does the Petersen graph (outer $5$-cycle, inner pentagram, and $5$ spokes joining them: $3$-regular on $10$ vertices) achieve it?