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 matrix R , where r(i,j) is equal to the specified lower bound on the number of line-disjoint paths required between points v_i and v_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 where r(i,j) > 1 . The result for the general case where some r(i,j) \leq 1 was 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.