Derivation of the unitary case of minimum multigraphs with prescribed lower bounds on local line-connectivities

To allow for alternate routes in the event of a line failure, a reliable communication network must provide for more than one path between certain pairs of points. We consider here the case where the network is modeled by a multigraph, and path requirements are stated generally by a symmetric reliability matrixR, wherer(i,j)is equal to the specified lower bound on the number of line-disjoint paths required between pointsv_iandv_j, in the multigraph. In a recent paper [1] a new, simple algorithm was derived for constructing minimum size multigraphs which satisfy path requirements specified in a reliability matrix wherer(i,j) > 1. The result for the general case where somer(i,j) \leq 1was merely stated. In many communication networks, the number of required paths between certain pairs of points may be very low, i.e.,r(i, j) \leq 1. In this paper we derive the complete algorithm and proof for this important case.