12,450 research outputs found
The extremal genus embedding of graphs
Let Wn be a wheel graph with n spokes. How does the genus change if adding a
degree-3 vertex v, which is not in V (Wn), to the graph Wn? In this paper,
through the joint-tree model we obtain that the genus of Wn+v equals 0 if the
three neighbors of v are in the same face boundary of P(Wn); otherwise,
{\deg}(Wn + v) = 1, where P(Wn) is the unique planar embedding of Wn. In
addition, via the independent set, we provide a lower bound on the maximum
genus of graphs, which may be better than both the result of D. Li & Y. Liu and
the result of Z. Ouyang etc: in Europ. J. Combinatorics. Furthermore, we obtain
a relation between the independence number and the maximum genus of graphs, and
provide an algorithm to obtain the lower bound on the number of the distinct
maximum genus embedding of the complete graph Km, which, in some sense,
improves the result of Y. Caro and S. Stahl respectively
Learning Local Metrics and Influential Regions for Classification
The performance of distance-based classifiers heavily depends on the
underlying distance metric, so it is valuable to learn a suitable metric from
the data. To address the problem of multimodality, it is desirable to learn
local metrics. In this short paper, we define a new intuitive distance with
local metrics and influential regions, and subsequently propose a novel local
metric learning method for distance-based classification. Our key intuition is
to partition the metric space into influential regions and a background region,
and then regulate the effectiveness of each local metric to be within the
related influential regions. We learn local metrics and influential regions to
reduce the empirical hinge loss, and regularize the parameters on the basis of
a resultant learning bound. Encouraging experimental results are obtained from
various public and popular data sets
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