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An edge $e$ is removable if $G-e$ is also matching covered; furthermore, $e$ is $b$-invariant if $b(G-e)=1$, and $e$ is quasi-$b$-invariant if $b(G-e)=2$. (Each edge of the Petersen graph is quasi-$b$-invariant.)\n  A brick $G$ is near-bipartite if it has a pair of edges $\\{e,f\\}$ so that $G-e-f$ is matching covered and bipartite; such a pair $\\{e,f\\}$ is a removable doubleton. 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