Alternating chain

Alternating chain — альтернирующая цепь, чередующаяся цепь.

Given a chain [math]\displaystyle{ P = v_{0},v_{1},\ldots,v_{k} }[/math], [math]\displaystyle{ \,P }[/math] is alternating with respect to a matching [math]\displaystyle{ \,M }[/math] if each edge in [math]\displaystyle{ \,P }[/math] that does not belong to [math]\displaystyle{ \,M }[/math] is followed in [math]\displaystyle{ \,P }[/math] by an edge that belongs to [math]\displaystyle{ \,M }[/math], and vice versa. If [math]\displaystyle{ \,P }[/math] is alternating, the index [math]\displaystyle{ \,k }[/math] is odd, and [math]\displaystyle{ \,v_{0}, v_{k} \not \in V(M) }[/math], then [math]\displaystyle{ \,P }[/math] is augmenting with respect to [math]\displaystyle{ \,M }[/math]. A well known result due to Berge states that [math]\displaystyle{ \,M }[/math] is maximum if and only if [math]\displaystyle{ \,M }[/math] does not admit any augmenting chains.