The Klee and Quaife minimum (d, 1, 3)-graphs revisited

After Klee and Qualfe [1], a
(d,c,\upsilon)
-graph
G
is one which is regular of degree
\upsilon
, diameter
d
, and connectivity
c
, so that
c
is necessarily at most v; and G is said to be minimum if it is of minimum order. The cited authors have noted that such a graph is that of a survivable communication network in which the lines of
G
represent the communication channels, its points representing the switching centers (stations) at which messages originate or are received, or through which they are routed. The network remains connected when fewer than c stations (and hence certainly fewer than
c
channels) are incapacitated. They exhibited and classified all minimuim
(d, 1, 3)
-graphs and
(d, 2,3)
-graphs, i.e., all minimum cubic graphs of diameter
d
and connectivity
c \leq 2
, and gave counts of their numbers. The purpose of the present paper is expository: we derive Klee and Qualfe's aesthetic and complete results by a different technique which provides new insights into the general
(d, c,\upsilon)
-graph problem through the use of associated pseudographs to generate nonseparable minimum blocks. We also introduce and use the new concepts of bridge trees and of a cut-diameter graph. We differ from Klee and Qualfe in defining a graph
G
to be
c
-connected if and only if removal of some minimum set of c of its points disconnects
G
, whereas they considered such a graph to be k-connected for all nonzero values of k less than or equal to c, when
c > 0
. And except where otherwise defined, our terminology is Harary's [2].