Gray codes for non-crossing partitions and dissections of a convex polygon

Huemer, Clemens; Hurtado, Ferrán; Noy, Marc; Omaña-Pulido, Elsa

doi:10.1016/j.dam.2008.06.018

Cited by 12 publications

(15 citation statements)

References 17 publications

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“…The following Lemmata are immediate from the definitions, and they are a generalization of those presented in [19].…”

Section: The Gray Codesmentioning

confidence: 98%

“…There are several bijections between noncrossing and nonnesting set partitions (see, for example [2,13,17,29,45]), and since in [19] a Gray code for noncrossing partitions is presented, it is tempting to try employing these bijections in order to obtain a Gray code for nonnesting partitions. But, as referred in [48], a Gray code for a combinatorial class is intrinsically bound to the representation of objects in the class, and in the present case, the Gray code is not preserved under bijection.…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

“…One of the oldest and more important tools used to produce a Gray code for general set partitions of [n] is the recursive algorithm described in [21], which is also one of the main ingredients for some of the constructions in [19].…”

Section: The Gray Codesmentioning

confidence: 99%

“…[3,5-7, 9,14-16,21,25,27,30-32,47]. Furthermore, Gray codes for noncrossing partitions have recently been obtained in [19], but so far nothing is known for nonnesting partitions.…”

Section: Introductionmentioning

confidence: 97%

“…In this paper we present Gray codes for the two classes of nonnesting partitions and sparse partitions, therefore extending the results for noncrossing partitions given in [19], and we explicitly design efficient algorithms for lexicographical combinatorial generation of the two sets of nonnesting and sparse nonnesting set partitions with k arcs.…”

Section: Introductionmentioning

confidence: 97%

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Gray codes and lexicographical combinatorial generation for nonnesting and sparse nonnesting set partitions

Conflitti

Mamede

2015

Theoretical Computer Science

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“…The following Lemmata are immediate from the definitions, and they are a generalization of those presented in [19].…”

Section: The Gray Codesmentioning

confidence: 98%

Section: Preliminaries and Notationsmentioning

confidence: 99%

Section: The Gray Codesmentioning

confidence: 99%

“…[3,5-7, 9,14-16,21,25,27,30-32,47]. Furthermore, Gray codes for noncrossing partitions have recently been obtained in [19], but so far nothing is known for nonnesting partitions.…”

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 97%

See 3 more Smart Citations

Gray codes and lexicographical combinatorial generation for nonnesting and sparse nonnesting set partitions

Conflitti

Mamede

2015

Theoretical Computer Science

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Flip Distances Between Graph Orientations

Aichholzer¹,

Cardinal²,

Huynh³

et al. 2019

Graph-Theoretic Concepts in Computer Science

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Flip graphs are a ubiquitous class of graphs, which encode relations induced on a set of combinatorial objects by elementary, local changes. Skeletons of associahedra, for instance, are the graphs induced by quadrilateral ips in triangulations of a convex polygon. For some de nition of a ip graph, a natural computational problem to consider is the ip distance: Given two objects, what is the minimum number of ips needed to transform one into the other?We consider ip graphs on orientations of simple graphs, where ips consist of reversing the direction of some edges. More precisely, we consider so-called α-orientations of a graph G, in which every vertex v has a speci ed outdegree α(v), and a ip consists of reversing all edges of a directed cycle. We prove that deciding whether the ip distance between two α-orientations of a planar graph G is at most two is NP-complete. This also holds in the special case of perfect matchings, where ips involve alternating cycles. This problem amounts to nding geodesics on the common base polytope of two partition matroids, or, alternatively, on an alcoved polytope. It therefore provides an interesting example of a ip distance question that is computationally intractable despite having a natural interpretation as a geodesic on a nicely structured combinatorial polytope.We also consider the dual question of the ip distance between graph orientations in which every cycle has a speci ed number of forward edges, and a ip is the reversal of all edges in a minimal directed cut. In general, the problem remains hard. However, if we restrict to ips that only change sinks into sources, or vice-versa, then the problem can be solved in polynomial time. Here we exploit the fact that the ip graph is the cover graph of a distributive lattice. This generalizes a recent result from Zhang, Qian, and Zhang (Acta. Math. Sin.-English Ser., 2019).

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