In graph theory, can I use the same vertex twice when I do matching? For example, in the following graph, is matching possible?


The graph doesn't have a perfect matching because every perfect matching must contain the edge $ab$ (or else the vertex $a$ is not in the matching) and so the edges $bc$ and $bh$ can't be part of the perfect matching (the vertex $b$ was already contained in $ab$). Also for the same reasons the edge $de$ must be part of the perfect matching. But now no edge that contains the vertex $c$ can be part of the matching (because neither one of $bc$ or $dc$ is in the perfect matching).
