On the surface duality of linear graphs

TH EOR EM : A 1·1 co rrespo nde nce be twee n the edges of two connec ted gra phs is a uu a lit y with re s pe c t to su me polyhedral surface e mbedding if and onl y if for eac h ve rte x v uf eac h gra ph , the edges which mee t v co rres po nd in the oth e r graph to th e edges of a subgraph G,· whic h is conn ec ted a nd whi c h has a n e ve n numbe r of it s edge·ends to e ac h of it s ve rti ces (w he re if a n ed ge mee ts va t bo th e nd s it s image in G,. is co unt ed twice) . Us ing the Eule r furmula, th e charac te ri s ti c of the surface is de termin ed by th e two graphs. Thu s, th e theore m ge ne rali zes a var ia ti on of the H. Whitn ey conditio n fur a graph to be pl a nar.