A well-known result of Kupitz from 1982 asserts that the maximal number of
edges in a convex geometric graph (CGG) on n vertices that does not contain
k+1 pairwise disjoint edges is kn (provided n>2k). For k=1 and
k=n/2β1, the extremal examples are completely characterized. For all other
values of k, the structure of the extremal examples is far from known: their
total number is unknown, and only a few classes of examples were presented,
that are almost symmetric, consisting roughly of the kn "longest possible"
edges of CK(n), the complete CGG of order n.
In order to understand further the structure of the extremal examples, we
present a class of extremal examples that lie at the other end of the spectrum.
Namely, we break the symmetry by requiring that, in addition, the graph admit
an independent set that consists of q consecutive vertices on the boundary of
the convex hull. We show that such graphs exist as long as qβ€nβ2k and
that this value of q is optimal.
We generalize our discussion to the following question: what is the maximal
possible number f(n,k,q) of edges in a CGG on n vertices that does not
contain k+1 pairwise disjoint edges, and, in addition, admits an independent
set that consists of q consecutive vertices on the boundary of the convex
hull? We provide a complete answer to this question, determining f(n,k,q) for
all relevant values of n,k and q.Comment: 17 pages, 9 figure