An extension of Boesch-Thomas optimality in communication nets

A
\xi
-disjoint simultaneous-flow communications net is one whose graph has nonintersecting paths joining (at most) any
\xi
pairs of its vertices, and is said to be optimal if it also has the least possible number of edges. Such a graph is optimally invulnerable to disconnection in a sense defined previously by Boesch and Thomas, in that its connectivity is equal to the average of the degrees of its vertices. Not all communication nets that are optimal in the Boesch-Thomas sense, however, are optimal in the
\xi
-disjoint simultaneous-flow sense.