I am given a graph $T$ with an odd number greater than or equal to 3 of vertices. Its line graph $L(T)$ has exactly one perfect matching.
I need to prove that if we remove any vertex from $T$, the number of even connected components in $T$ will be even.
I need an idea on how to start on this. How exactly does existence of the perfect matching in $L(T)$ influence the original graph?