Lattice Structures from Planar Graphs

Felsner, Stefan

doi:10.37236/1768

Cited by 87 publications

(127 citation statements)

References 14 publications

(10 reference statements)

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“…We refer to these as facial ips. Felsner [Fel04] considered distributive lattices induced by facial ips. The following computational problem is a special case of Problem 4.…”

Section: Problems and Main Resultsmentioning

confidence: 99%

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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“…We refer to these as facial ips. Felsner [Fel04] considered distributive lattices induced by facial ips. The following computational problem is a special case of Problem 4.…”

Section: Problems and Main Resultsmentioning

confidence: 99%

Flip Distances Between Graph Orientations

Aichholzer¹,

Cardinal²,

Huynh³

et al. 2019

Graph-Theoretic Concepts in Computer Science

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show abstract

“…Remark 2.4. In recent attempts to extend orientations on maps of higher genus, the notion of α-orientations, due to Felsner [18], has been used (e.g. in [20]).…”

Section: Structure Of Orientations Of a Graphmentioning

confidence: 99%

Blossoming bijection for higher-genus maps

Lepoutre

2019

Journal of Combinatorial Theory, Series A

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In 1997, Schaeffer described a bijection between Eulerian planar maps and some trees. In this work we generalize his work to a bijection between bicolorable maps on a surface of any fixed genus and some unicellular maps with the same genus. An important step of this construction is to exhibit a canonical orientation for maps, that allows to apply the same local opening algorithm as Schaeffer.As an important byproduct, we obtain the first bijective proof of a result of Bender and Canfield from 1991, when they proved that the generating series of maps in higher genus is a rational function of the generating series of planar maps.

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“…The case k = 1. Let us go back to System (27). In the first equation, replace U 0 by (U ε − 1)/2 to obtain a single equation involving only U x ε = U ε (x) and U ε = U ε (1).…”

Section: 4mentioning

confidence: 99%

“…The condition is in fact sufficient (such maps even admit an Eulerian circuit [37]). One analogy with the above two classes is that the set of Eulerian orientations of a given planar map can be equipped with a lattice structure [52,27]. Moreover, Eulerian maps (equivalently, Eulerian maps equipped with their minimal Eulerian orientation) have rich combinatorial properties: not only are they counted by simple numbers (see (1)), but they are equinumerous with several other families of objects, like certain trees [15] and permutations [8,32].…”

Section: Introductionmentioning

confidence: 99%

On the number of planar Eulerian orientations

Bousquet-Mélou

Dorbec

Pennarun

2017

European Journal of Combinatorics

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The number of planar Eulerian maps with n edges is well-known to have a simple expression. But what is the number of planar Eulerian orientations with n edges? This problem appears to be difficult. To approach it, we define and count families of subsets and supersets of planar Eulerian orientations, indexed by an integer k, that converge to the set of all planar Eulerian orientations as k increases. The generating functions of our subsets can be characterized by systems of polynomial equations, and are thus algebraic. The generating functions of our supersets are characterized by polynomial systems involving divided differences, as often occurs in map enumeration. We prove that these series are algebraic as well. We obtain in this way lower and upper bounds on the growth rate of planar Eulerian orientations, which appears to be around 12.5.

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Lattice Structures from Planar Graphs

Cited by 87 publications

References 14 publications

Flip Distances Between Graph Orientations

Flip Distances Between Graph Orientations

Blossoming bijection for higher-genus maps

On the number of planar Eulerian orientations

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