3,013 research outputs found
Two problems on independent sets in graphs
Let denote the number of independent sets of size in a graph
. Levit and Mandrescu have conjectured that for all bipartite the
sequence (the {\em independent set sequence} of ) is
unimodal. We provide evidence for this conjecture by showing that is true for
almost all equibipartite graphs. Specifically, we consider the random
equibipartite graph , and show that for any fixed its
independent set sequence is almost surely unimodal, and moreover almost surely
log-concave except perhaps for a vanishingly small initial segment of the
sequence. We obtain similar results for .
We also consider the problem of estimating for
in various families. We give a sharp upper bound on the number of
independent sets in an -vertex graph with minimum degree , for all
fixed and sufficiently large . Specifically, we show that the
maximum is achieved uniquely by , the complete bipartite
graph with vertices in one partition class and in the
other.
We also present a weighted generalization: for all fixed and , as long as is large enough, if is a graph on
vertices with minimum degree then with equality if and only if
.Comment: 15 pages. Appeared in Discrete Mathematics in 201
On the unimodality of independence polynomials of some graphs
In this paper we study unimodality problems for the independence polynomial
of a graph, including unimodality, log-concavity and reality of zeros. We
establish recurrence relations and give factorizations of independence
polynomials for certain classes of graphs. As applications we settle some
unimodality conjectures and problems.Comment: 17 pages, to appear in European Journal of Combinatoric
Segmentation and classification of individual tree crowns
By segmentation and classification of individual tree crowns in high spatial resolution aerial images, information about the forest can be automatically extracted. Segmentation is about finding the individual tree crowns and giving each of them a unique label. Classification, on the other hand, is about recognising the species of the tree. The information of each individual tree in the forest increases the knowledge about the forest which can be useful for managements, biodiversity assessment, etc. Different algorithms for segmenting individual tree crowns are presented and also compared to each other in order to find their strengths and weaknesses. All segmentation algorithms developed in this thesis focus on preserving the shape of the tree crown. Regions, representing the segmented tree crowns, grow according to certain rules from seed points. One method starts from many regions for each tree crown and searches for the region that fits the tree crown best. The other methods start from a set of seed points, representing the locations of the tree crowns, to create the regions. The segmentation result varies from 73 to 95 % correctly segmented visual tree crowns depending on the type of forest and the method. The former value is for a naturally generated mixed forest and the latter for a non-mixed forest. The classification method presented uses shape information of the segments and colour information of the corresponding tree crown in order to decide the species. The classification method classifies 77 % of the visual trees correctly in a naturally generated mixed forest, but on a forest stand level the classification is over 90 %
Improved Protocols and Hardness Results for the Two-Player Cryptogenography Problem
The cryptogenography problem, introduced by Brody, Jakobsen, Scheder, and
Winkler (ITCS 2014), is to collaboratively leak a piece of information known to
only one member of a group (i)~without revealing who was the origin of this
information and (ii)~without any private communication, neither during the
process nor before. Despite several deep structural results, even the smallest
case of leaking one bit of information present at one of two players is not
well understood. Brody et al.\ gave a 2-round protocol enabling the two players
to succeed with probability and showed the hardness result that no
protocol can give a success probability of more than~.
In this work, we show that neither bound is tight. Our new hardness result,
obtained by a different application of the concavity method used also in the
previous work, states that a success probability better than 0.3672 is not
possible. Using both theoretical and numerical approaches, we improve the lower
bound to , that is, give a protocol leading to this success
probability. To ease the design of new protocols, we prove an equivalent
formulation of the cryptogenography problem as solitaire vector splitting game.
Via an automated game tree search, we find good strategies for this game. We
then translate the splits that occurred in this strategy into inequalities
relating position values and use an LP solver to find an optimal solution for
these inequalities. This gives slightly better game values, but more
importantly, it gives a more compact representation of the protocol and a way
to easily verify the claimed quality of the protocol.
These improved bounds, as well as the large sizes and depths of the improved
protocols we find, suggests that finding good protocols for the
cryptogenography problem as well as understanding their structure are harder
than what the simple problem formulation suggests
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