Inner Dual, Excised Internal Structure, Branching Graph

1 Inner Dual, Excised Internal Structure, Branching Graph 

A large number of graphs has been associated with benzenoid systems. Here we mention three of them, neither of which characterizing the benzenoid system up to isomorphism. 

The vertices of the inner dual (ID) represent the hexagons of a benzenoid system. Vertices corresponding to adjacent hexagons are adjacent. For example, ID(1) is the inner dual of the systems 1, 2 or 3 (see Fig. 1) whereas ID (9) is the inner dual of the system ) 9  

A remarkable property of the inner dual is summarized below. 

Theorem 3 [14, 15]. If ID is the inner dual of a benzenoid system B, then 

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