5,997 research outputs found
Discrete Morse theory on graphs
AbstractWe characterize the topology of a graph in terms of the critical elements of a discrete Morse function defined on it. Besides, we study the structure and some properties of the gradient vector field induced by a discrete Morse function defined on a graph. Finally, we get results on the number of non-homologically equivalent excellent discrete Morse functions defined on some kind of graphs
The number of excellent discrete Morse functions on graphs
AbstractIn Nicolaescu (2008) [7] the number of non-homologically equivalent excellent Morse functions defined on S2 was obtained in the differentiable setting. We carried out an analogous study in the discrete setting for some kinds of graphs, including S1, in Ayala et al. (2009) [1]. This paper completes this study, counting excellent discrete Morse functions defined on any infinite locally finite graph
Counting excellent discrete Morse functions on compact orientable surfaces
We obtain the number of non-homologically equivalent excellent discrete Morse functions defined on compact orientable surfaces. This work is a continuation of the study which has been done in [2, 4] for graphs
Local Equivalence and Intrinsic Metrics between Reeb Graphs
As graphical summaries for topological spaces and maps, Reeb graphs are
common objects in the computer graphics or topological data analysis
literature. Defining good metrics between these objects has become an important
question for applications, where it matters to quantify the extent by which two
given Reeb graphs differ. Recent contributions emphasize this aspect, proposing
novel distances such as {\em functional distortion} or {\em interleaving} that
are provably more discriminative than the so-called {\em bottleneck distance},
being true metrics whereas the latter is only a pseudo-metric. Their main
drawback compared to the bottleneck distance is to be comparatively hard (if at
all possible) to evaluate. Here we take the opposite view on the problem and
show that the bottleneck distance is in fact good enough {\em locally}, in the
sense that it is able to discriminate a Reeb graph from any other Reeb graph in
a small enough neighborhood, as efficiently as the other metrics do. This
suggests considering the {\em intrinsic metrics} induced by these distances,
which turn out to be all {\em globally} equivalent. This novel viewpoint on the
study of Reeb graphs has a potential impact on applications, where one may not
only be interested in discriminating between data but also in interpolating
between them
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