Thin Edges in Braces

The bicontraction of a vertex $v$ of degree two in a graph, with precisely two neighbours $v_1$ and $v_2$, consists of shrinking the set $\{v_1,v,v_2\}$ to a single vertex. The retract of a matching covered graph $G$, denoted by $\widehat{G}$, is the graph obtained from $G$ by repeatedly bicontracting vertices of degree two. Up to isomorphism, the retract of a matching covered graph $G$ is unique. If $G$ is a brace on six or more vertices, an edge $e$ of $G$ is thin if $\widehat{G-e}$ is a brace. A thin edge $e$ in a simple brace $G$ is strictly thin if $\widehat{G-e}$ is a simple brace. Theorems concerning the existence of strictly thin edges have been used (implicitly by McCuaig (Pólya's Permanent Problem, Electron. J. of Combin., 11, 2004) and explicitly by the authors (On the Number of Perfect Matchings in a Bipartite Graph, SIAM J. Discrete Math., 27, 940-958, 2013)) as inductive tools for establishing properties of braces.Let $G$ and $J$ be two distinct braces, where $G$ is of order six or more and $J$ is a simple matching minor of $G$. It follows from a theorem of McCuaig (Brace Generation, J. Graph Theory, 38, 124-169, 2001) that $G$ has a thin edge $e$ such that $J$ is a matching minor of $G-e$. In Section 2, we give an alternative, and simpler proof, of this assertion. Our method of proof lends itself to proving stronger results concerning thin edges.Let ${\cal G}^+$ denote the family of braces consisting of all prisms, all Möbius ladders, all biwheels, and all extended biwheels. Strengthening another result of McCuaig on brace generation, we show that every simple brace of order six or more which is not a member of ${\cal G}^+$ has at least two strictly thin edges. We also give examples to show that this result is best possible.