, Assume that d (n ; m)=2. Without loss of generality, we assume that m 1, since otherwise G is simply the complete graph. For each connected component A i , i 2 f 1 : : : m g, w e root a spanning tree T i at any v ertex of A i . L e t n i denote the number of vertices of A i . We n o w construct a shortest path interval routing scheme R = ( L I) o n G. F or i 2 f 1 : : : m g T i , for all j 2 f 1 : : : n i g, G denote the complement graph of G. G is composed of m connected components A 1 : : : A m of order at least two a n d o f d single vertices, vol.1

, For vertices of G with a degree strictly lower than n ; 1, we assign intervals as follows: let i 2 f 1 : : : m g and x be a vertex of A i . For every arc (xx y) o f G

, Hence all vertices of G are labeled and the compactness of R is 1. To prove that R is connected and deene a shortest path routing scheme

. =-k-n, Hence either x and y are adjacent or there is a third vertex z of G such that z is connected to both x and y. I f x and y are adjacent, then we h a ve L(y) 2 I (xxy) in both cases (i) and (ii), and by symmetry L(x) 2 I (yyx)

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