18 research outputs found
On Convex Geometric Graphs with no Pairwise Disjoint Edges
A well-known result of Kupitz from 1982 asserts that the maximal number of
edges in a convex geometric graph (CGG) on vertices that does not contain
pairwise disjoint edges is (provided ). For and
, the extremal examples are completely characterized. For all other
values of , 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 "longest possible"
edges of , the complete CGG of order .
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 consecutive vertices on the boundary of
the convex hull. We show that such graphs exist as long as and
that this value of is optimal.
We generalize our discussion to the following question: what is the maximal
possible number of edges in a CGG on vertices that does not
contain pairwise disjoint edges, and, in addition, admits an independent
set that consists of consecutive vertices on the boundary of the convex
hull? We provide a complete answer to this question, determining for
all relevant values of and .Comment: 17 pages, 9 figure
Characterization of co-blockers for simple perfect matchings in a convex geometric graph
Consider the complete convex geometric graph on vertices, ,
i.e., the set of all boundary edges and diagonals of a planar convex -gon
. In [C. Keller and M. Perles, On the Smallest Sets Blocking Simple Perfect
Matchings in a Convex Geometric Graph], the smallest sets of edges that meet
all the simple perfect matchings (SPMs) in (called "blockers") are
characterized, and it is shown that all these sets are caterpillar graphs with
a special structure, and that their total number is . In this
paper we characterize the co-blockers for SPMs in , that is, the
smallest sets of edges that meet all the blockers. We show that the co-blockers
are exactly those perfect matchings in where all edges are of odd
order, and two edges of that emanate from two adjacent vertices of
never cross. In particular, while the number of SPMs and the number of blockers
grow exponentially with , the number of co-blockers grows
super-exponentially.Comment: 8 pages, 4 figure