Pants decomposition graph ($V=2g-2,E=3g-3$)

The graph of a pants decomposition has one vertex per pair of pants and one edge per glued cuff. It is a connected trivalent multigraph, possibly with loops. A closed surface of genus $g$ gives $2g-2$ vertices and $3g-3$ edges, so its first Betti number is $g$. The graph records a decomposition rather than the homeomorphism class alone, and omits cuff lengths and twists.