We calculate a bound for the fan-out of X \D 1 as follows: we observe that all the endpoints in E(X \ D 1 ) must be either endpoints of X or endpoints of But we also observe that E(X \ D 1 ) cannot contain any of the endpoints where D 1 is adjacent to D 2 (i.e., the members of E(D 1 ) ∩ E(D 2 )), since X \ D 1 does not contain D 1 or D 2 .
we can say that any endpoint of X \ D 1 is either a mem- the number of endpoints in E(X \ D 1 ) cannot exceed the sum of the number of endpoints in these two sets, which, according to the reasonings above, is at most 4 − f (D 1 ) + f (D 1 ) = 4.