Alternating path in a matching

An alternating path in a matching $M$ is a graph path whose successive edges alternate between membership in $M$ and membership outside $M$. Here graph paths have no repeated vertices. Alternation alone does not make a graph path an augmenting path in a matching: its endpoints must both be unmatched vertices in a matching. For example, relative to $M=\{(2,3),(4,5)\}$ in the six-vertex path graph, $2,3,4,5$ alternates but has matched endpoints; toggling just those three edges loses one selected edge.