On optimizing discrete Morse functions

Hersh, Patricia

doi:10.1016/j.aam.2005.04.001

Cited by 36 publications

(38 citation statements)

References 16 publications

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“…Examples of lexicographic discrete Morse functions for several other posets appear in [15] and [17]. All of these examples use very natural edge-labelings, and they lead us to believe that the most natural of labelings will often yield Morse functions which may be transformed by gradient path reversal into perfect Morse functions, i.e.…”

Section: Definition 12mentioning

confidence: 99%

Discrete Morse functions from lexicographic orders

Babson

Hersh

2004

Trans. Amer. Math. Soc.

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Abstract. This paper shows how to construct a discrete Morse function with a relatively small number of critical cells for the order complex of any finite poset with0 and1 from any lexicographic order on its maximal chains. Specifically, if we attach facets according to the lexicographic order on maximal chains, then each facet contributes at most one new face which is critical, and at most one Betti number changes; facets which do not change the homotopy type also do not contribute any critical faces. Dimensions of critical faces as well as a description of which facet attachments change the homotopy type are provided in terms of interval systems associated to the facets. As one application, the Möbius function may be computed as the alternating sum of Morse numbers.The above construction enables us to prove that the poset Πn/S λ of partitions of a set {1 λ 1 , . . . , k λ k } with repetition is homotopy equivalent to a wedge of spheres of top dimension when λ is a hook-shaped partition; it is likely that the proof may be extended to a larger class of λ and perhaps to all λ, despite a result of Ziegler (1986) which shows that Πn/S λ is not always Cohen-Macaulay.

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Section: Definition 12mentioning

confidence: 99%

Discrete Morse functions from lexicographic orders

Babson

Hersh

2004

Trans. Amer. Math. Soc.

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“…This is done by simply observing that u/u À bðuÞ is an isomorphism U-Û; the inverse is the canonical projection function A"B"U-U restricted toÛ: Combining (9) and (10), we obtain (8). &…”

Section: Discrete Morse Theory On Complexes Of R-modulesmentioning

confidence: 99%

“…This theory has proven to be a powerful tool for analyzing the topology of a wide range of different complexes; see [2,4,6,8,10,11] for a few nice examples. For an interesting application of discrete Morse theory to geometry, see [5].…”

Section: Introductionmentioning

confidence: 99%

On the topology of simplicial complexes related to 3-connected and Hamiltonian graphs

Jönsson

2003

Journal of Combinatorial Theory, Series A

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Using techniques from Robin Forman's discrete Morse theory, we obtain information about the homology and homotopy type of some graph complexes. Specifically, we prove that the simplicial complex D 3 n of not 3-connected graphs on n vertices is homotopy equivalent to a wedge of ðn À 3Þ Á ðn À 2Þ!=2 spheres of dimension 2n À 4; thereby verifying a conjecture by Babson, Bjo¨rner, Linusson, Shareshian, and Welker. We also determine a basis for the corresponding nonzero homology group in the CW complex of 3-connected graphs. In addition, we show that the complex G n of non-Hamiltonian graphs on n vertices is homotopy equivalent to a wedge of two complexes, one of the complexes being the complex D 2 n of not 2-connected graphs on n vertices. The homotopy type of D 2 n has been determined, independently, by the five authors listed above and by Turchin. While G n and D 2 n are homotopy equivalent for small values on n; they are nonequivalent for n ¼ 10: r

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“…Optimal discrete Morse functions has been widely studied in the literature [6,7]. However, this question is not usually considered as an optimization problem in terms of obtaining discrete Morse functions with as less critical simplices as possible.…”

Section: Introductionmentioning

confidence: 99%

“…First, it can be applied to discrete objects more general than manifolds. Second, it is more suitable in the digital context on topics like pattern recognition, shape classification and recognition, thinning 2D-objects where usually discretized functions are used.Optimal discrete Morse functions has been widely studied in the literature [6,7]. However, this question is not usually considered as an optimization problem in terms of obtaining discrete Morse functions with as less critical simplices as possible.…”

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confidence: 99%

Perfect discrete Morse functions on 2-complexes

Fernández-Ternero

Vilches

2012

Pattern Recognition Letters

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A bst r ac t This paper is focused on the study of perfect discrete Morse functions on a 2-simplicial complex. These are those discrete Morse functions such that the number of critical i-simplices coincides with the i-th Betti number of the complex. In particular, we establish conditions under which a 2-complex admits a perfect discrete Morse function and conversely, we get topological properties of a 2-complex admitting such kind of functions. This approach is more general than the known results in the literature [7], since our study is not restricted to surfaces. These results can be considered as a first step in the study of perfect discrete Morse functions on 3-manifolds.K e y wor ds perfect discrete Morse function, Betti number, simplicial complex. I nt roductionSince it was introduced, Morse theory has been a powerful tool in the study of smooth manifolds by means of differential geometry techniques. Basically, it allows us to describe the topology of a manifold in terms of the cellular decomposition generated by the critical points of a scalar smooth map defined on it.At the end of the last century, R. Forman [4] developed a discrete version of Morse theory that turned out to be a fruitful and efficient method for the study of the topology of discrete objects, such as simplicial and cellular complexes, which play a central role in many different fields of pure and applied mathematics.Essentially, a discrete Morse function on a simplicial complex is a way to assign a real number to each simplex of a complex, without any continuity, in such a way that for each simplex the natural order given by the dimension simplices is respected, except at most in one (co)face of the given simplex. As in the smooth setting, changes in the topology of the level subcomplexes are deeply related to the presence of critical simplices of the function, and the analysis of the evolution of the homology of these complexes can be a very useful tool in computer vision to deal with shape recognition problems by means of topological shape descriptors. In our opinion, there are many advantages of using Forman's theory. First, it can be applied to discrete objects more general than manifolds. Second, it is more suitable in the digital context on topics like pattern recognition, shape classification and recognition, thinning 2D-objects where usually discretized functions are used.Optimal discrete Morse functions has been widely studied in the literature [6,7]. However, this question is not usually considered as an optimization problem in terms of obtaining discrete Morse functions with as less critical simplices as possible. On the contrary, this problem is mainly settled as a problem of perfect discrete Morse functions, that is, those functions satisfyingwhere m i ( f ) is indicating the number of i-critical simplices of f on K and b i ( K ; F ) is the i-th Betti number of M . In this setting is interesting to point out that there are complexes which do not admit perfect discrete Morse functions. It can be explained by two main reasons:...

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On optimizing discrete Morse functions

Cited by 36 publications

References 16 publications

Discrete Morse functions from lexicographic orders

Discrete Morse functions from lexicographic orders

On the topology of simplicial complexes related to 3-connected and Hamiltonian graphs

Perfect discrete Morse functions on 2-complexes

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