Claw-free minimal matching covered graphs

A matching covered graph $G$ is minimal if for each edge $e$ of $G$, $G-e$ is not matching covered. An edge $e$ of a matching covered graph $G$ is removable if $G-e$ is also matching covered. Thus a matching covered graph is minimal if and only if it is free of removable edges. For bipartite graphs, Lov\'{a}sz and Plummer gave a characterization of bipartite minimal matching covered graphs. For bricks, Lov\'{a}sz showed that the only bricks that are minimal matching covered are $K_4$ and $\overline{C_6}$. In this paper, we present a complete characterization of minimal matching covered graphs that are claw-free. Moreover, for cubic claw-free matching covered graphs that are not minimal matching covered, we obtain the number of their removable edges (with respect to their bricks), and then prove that they have at least 12 removable edges (the bound is sharp).