  A graph is called diameter-$k$-critical if its diameter is $k$, and the
removal of any edge strictly increases the diameter. In this paper, we prove
several results related to a conjecture often attributed to Murty and Simon,
regarding the maximum number of edges that any diameter-$k$-critical graph can
have. In particular, we disprove a longstanding conjecture of Caccetta and
H\"aggkvist (that in every diameter-2-critical graph, the average edge-degree
is at most the number of vertices), which promised to completely solve the
extremal problem for diameter-2-critical graphs.
  On the other hand, we prove that the same claim holds for all higher
diameters, and is asymptotically tight, resolving the average edge-degree
question in all cases except diameter-2. We also apply our techniques to prove
several bounds for the original extremal question, including the correct
asymptotic bound for diameter-$k$-critical graphs, and an upper bound of
$(\frac{1}{6} + o(1))n^2$ for the number of edges in a diameter-3-critical
graph.
