Rainbow Cycles in Flip Graphs

Felsner, Stefan; Kleist, Linda; Mütze, Torsten; Sering, Leon

doi:10.1137/18m1216456

Cited by 11 publications

(24 citation statements)

References 34 publications

(29 reference statements)

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“…In this section, we assume that the number n of matching edges is even. It was proved in [FKMS20] that in this case the graph H n has at least n − 1 components. We improve upon this considerably, by showing that H n has exponentially many components, and we also provide a fine-grained picture of the component structure of the graph H n (Theorem 9 and Corollary 15).…”

Section: Component Structure For Even Nmentioning

confidence: 99%

“…The main motivation for considering centered flips comes from the study of rainbow cycles in flip graphs, a direction of research that was initiated in a recent paper by Felsner, Kleist, Mütze, and Sering [FKMS20]. Roughly speaking, along a rainbow cycle in G n all possible lengths of quadrilateral edges that are involved in flip operations must appear equally often, which leads to non-centered flips becoming unusable, so we may restrict our attention to the subgraph H n given by centered flips only.…”

Section: Introductionmentioning

confidence: 99%

“…In this work we investigate the structure of the graph H n ⊆ G n . In particular, we solve some of the open questions and conjectures from the aforementioned paper [FKMS20] on rainbow cycles in flip graphs. For several graph parameter, we observe an intriguing dichotomy between the cases where n is odd or even, i.e., the graph H n has an entirely different structure in those two cases.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On flips in planar matchings

Milich

Mütze

Pergel

2021

Discrete Applied Mathematics

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Section: Component Structure For Even Nmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On flips in planar matchings

Milich

Mütze

Pergel

2021

Discrete Applied Mathematics

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“…Other flip graphs include the flip graph on plane spanning trees [2], the flip graph of non-crossing partitions of a point set or dissections of a polygon [28], the mutation graph of simple pseudoline arrangements [53], the Eulerian tour graph of an Eulerian graph [59], and many others. There is also a vast collection of interesting flip graphs for nongeometric objects, such as bitstrings, permutations, combinations, and partitions [22].…”

Section: Introductionmentioning

confidence: 99%

Flip Distances Between Graph Orientations

et al. 2020

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Flip graphs are a ubiquitous class of graphs, which encode relations on a set of combinatorial objects by elementary, local changes. Skeletons of associahedra, for instance, are the graphs induced by quadrilateral flips in triangulations of a convex polygon. For some definition of a flip graph, a natural computational problem to consider is the flip distance: Given two objects, what is the minimum number of flips needed to transform one into the other? We consider flip graphs on orientations of simple graphs, where flips consist of reversing the direction of some edges. More precisely, we consider so-called-orientations of a graph G, in which every vertex v has a specified outdegree (v) , and a flip consists of reversing all edges of a directed cycle. We prove that deciding whether the flip distance between two-orientations of a planar graph G is at most two is NP-complete. This also holds in the special case of perfect matchings, where flips involve alternating cycles. This problem amounts to finding geodesics on the common base polytope of two partition matroids, or, alternatively, on an alcoved polytope. It therefore provides an interesting example of a flip 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 flip distance between graph orientations in which every cycle has a specified number of forward edges, and a flip is the reversal of all edges in a minimal directed cut. In general, the problem remains hard. However, if we restrict to flips 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 flip graph is the cover graph of a distributive lattice. This generalizes a recent result from Zhang et al.

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“…Other ip graphs include the ip graph on plane spanning trees [AAHV07], the ip graph of non-crossing partitions of a point set or dissections of a polygon [HHNOP09], the mutation graph of simple pseudoline arrangements [Rin57], the Eulerian tour graph of a Eulerian graph [ZG87], and many others. There is also a vast collection of interesting ip graphs for non-geometric objects, such as bitstrings, permutations, combinations, and partitions [FKMS18]. In essence, a ip graph provides the considered family of combinatorial objects with an underlying structure that reveals interesting properties about the objects.…”

Section: Introductionmentioning

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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Rainbow Cycles in Flip Graphs

Cited by 11 publications

References 34 publications

On flips in planar matchings

On flips in planar matchings

Flip Distances Between Graph Orientations

Flip Distances Between Graph Orientations

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