SHORTEST CONNECTION NETWORKS 1393 



between terminals are different, so that each terminal or fragment has 

 a single, uniquely defined, nearest neighbor. This restriction will be 

 removed later. 



Consider first NCI. Suppose there is a SCN for which it is untrue. 

 Then [Fig. 3(a)] some terminal, t, in this network is not directly joined 

 to its nearest neighbor, n. Since the network is connected, t is necessarily 

 joined directly to some one or more terminals other than n — say /i , 

 • • ■ ,fr ■ For the same reason, n is necessarily joined through some chain, 

 C, of one or more links to one of /i , • • • , fr — say to fk . Now if the link 

 t — fh is removed from the network and the link t — n is added [Fig. 3(b)], 

 the connectedness of the network is clearty not destroyed — fk being 

 joined to t through 7i, and C, rather than directly. However, the total 

 length of the network has now been decreased, because, by hypothesis, 

 t — nis shorter than t — fk ■ Hence, contrary to the initial supposition, the 

 network contradicting NCl could not have been the shortest, and the 

 truth of NCI follows. From NCl follows PI, which merely permits the 

 addition of links which NCl shows have to appear in the final SCN. 



Turning now to NC2, the above argument carries over directly if t 

 is thought of as a fragment of the supposed contradictory SCN, rather 

 than as an individual terminal — pro^'ided, of course that the t — n link 

 substituted for t — fk is the shortest link from n to any of the terminals 

 belonging to t. From the validity of NC2 follows P2 — again the links 

 whose addition is permitted by P2 are all necessary, by NC2, in the 

 final SCN. 



The temporary restrictive assumption that no two inter-terminal 

 distances are identical must now be removed. A reappraisal of the 

 proofs of NCl and NC2 shows that they are still valid if n is not the 

 only terminal at distance t — n from /, for in the supposedly contradictory 

 network the distance t — fk must be greater than t — n. What remains to be 

 established is that the length of the final connection network residting 



Fig. 3 — Schematic demonstration of NCl. 



