1396 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1957 



shortest connection network <-^ shortest spanning subtree 



SCN ^ SSS 

 It will be useful and worthwhile to carry over the concepts of "fragment" 

 and "nearest neighbor" into the graph theoretic framework. PI and P2 

 can now be regarded as construction principles for finding a shortest 

 spanning subtree of a labelled graph. 



In the originating context of connection networks, the graphs from 

 which a shortest spanning subtree is to be extracted are complete graphs ; 

 that is, graphs having an edge between every pair of vertices. It is 

 natural, now, to generalize the original problem by seeking shortest 

 spanning subtrees for arbitrary connected labelled graphs. Consider, for 

 example, the labelled graph of Fig. 6(a) which is derived from that of 

 Fig. 5(a) by deleting some of the edges. (In the connection network 

 context, this is equivalent to barring direct connections between certain 

 terminal pairs.) It is easily verified that NCI and NC2, and hence PI 

 and P2, are valid also in these more general cases. For the example of 

 Fig. 6(a), they yield readily the SSS of Fig. 6(b). 



As a further generalization, it is not at all necessary for the validity 

 of PI and P2 that the edge "lengths" in the given labelled graph be 

 derived, as were those of Figs. 4-6, from the inter-point distances of 

 some point set in the plane. PI and P2 will provide a SSS for any con- 

 nected labelled graph with any set of real edge "'lengths." The "lengths" 

 need not even be positive, or of the same sign. See, for example, the 

 labelled graph of Fig. 7(a) and its SSS, Fig. 7(b). It might be noted in 

 passing that this degree of generality is sufficient to include, among 

 other things, shortest connection networks in an arbitrary number of 

 dimensions. 



A further extension of the range of problems solved b}' PI and P2 

 follows trivially from the obser\'ation that the maximum of a set of 



C6)0 



C6)0 



(a) (b) 



Fig. 6 — Example of a shortest spanning subtree of an incomplete labelled graph. 



