Distance Between α-Orientations of Plane Graphs by Facial Cycle Reversals

Zhang, Wei Juan; Qian, Jian Guo; Zhang, Fu Ji

doi:10.1007/s10114-018-7403-4

Cited by 2 publications

(2 citation statements)

References 12 publications

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“…Zhang et al [60] recently provided a closed formula for this flip distance, which can be turned into a polynomial-time algorithm. We prove the analogous stronger statement for Problem 4.…”

Section: Problemmentioning

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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“…Zhang et al [60] recently provided a closed formula for this flip distance, which can be turned into a polynomial-time algorithm. We prove the analogous stronger statement for Problem 4.…”

Section: Problemmentioning

confidence: 99%

Flip Distances Between Graph Orientations

et al. 2020

View full text Add to dashboard Cite

show abstract

“…Zhang, Qian, and Zhang [ZQZ19] recently provided a closed formula for this ip distance, which can be turned into a polynomial-time algorithm. We prove the analogous stronger statement for Problem 4.…”

Section: Theorem 5 ([Kna08]mentioning

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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Distance Between α-Orientations of Plane Graphs by Facial Cycle Reversals

Cited by 2 publications

References 12 publications

Flip Distances Between Graph Orientations

Flip Distances Between Graph Orientations

Flip Distances Between Graph Orientations

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