Collision Free Motion Planning on Graphs

Farber, Michael

doi:10.1007/10991541_10

Cited by 27 publications

(44 citation statements)

References 7 publications

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“…By Proposition 4.5(1) & (3), these equivalence classes correspond to the only elements of the standard basis for H 1 (U D 4 T min ; Z/2Z) which might have non-trivial cup products. In fact, checking labels of edges, it is not difficult to see that cases (1)-(5) and(8) are all (distinct) critical cells, and thus correspond to linearly independent elements of the standard basis for H 2 (U D 4 T min ; Z/2Z). We now consider the remaining edges.…”

mentioning

confidence: 98%

On the cohomology rings of tree braid groups

Farley

Sabalka

2008

Journal of Pure and Applied Algebra

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Let $\Gamma$ be a finite connected graph. The (unlabelled) configuration space $UC^n \Gamma$ of $n$ points on $\Gamma$ is the space of $n$-element subsets of $\Gamma$. The $n$-strand braid group of $\Gamma$, denoted $B_n\Gamma$, is the fundamental group of $UC^n \Gamma$. We use the methods and results of our paper "Discrete Morse theory and graph braid groups" to get a partial description of the cohomology rings $H^*(B_n T)$, where $T$ is a tree. Our results are then used to prove that $B_n T$ is a right-angled Artin group if and only if $T$ is linear or $n<4$. This gives a large number of counterexamples to Ghrist's conjecture that braid groups of planar graphs are right-angled Artin groups.Comment: 25 pages, 7 figures. Revised version, accepted by the Journal of Pure and Applied Algebr

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confidence: 98%

On the cohomology rings of tree braid groups

Farley

Sabalka

2008

Journal of Pure and Applied Algebra

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“…The second part was claimed in [4,Theorems 9,10]. Both parts easily follow from our local step-by-step computations.…”

Section: Two-point Braid Groups Of Graphs In the Unordered Casementioning

confidence: 56%

Computing braid groups of graphs with applications to robot motion planning

Kurlin

2012

Homology, Homotopy and Applications

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An algorithm is designed to write down presentations of graph braid groups. Generators are represented in terms of actual motions of robots moving without collisions on a given connected graph. A key ingredient is a new motion planning algorithm whose complexity is linear in the number of edges and is quadratic in the number of robots. The computing algorithm implies that 2-point braid groups of all light planar graphs have presentations where all relators are commutators.

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“…vertices with valence at least 3). Farber showed that TC(Conf n (T )) = 2|V ≥3 | whenever n ≥ 2|V ≥3 |; see [Far05] and also Farber's survey article [Far17]. In particular, TC(Conf n (T )) doesn't depend on n within that range.…”

Section: Introductionmentioning

confidence: 99%

Topological Complexity of Configuration Spaces of Fully Articulated Graphs and Banana Graphs

Lütgehetmann

Recio-Mitter

2019

Discrete Comput Geom

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In this paper we determine the topological complexity of configuration spaces of graphs which are not necessarily trees, which is a crucial assumption in previous results. We do this for two very different classes of graphs: fully articulated graphs and banana graphs.We also complete the computation in the case of trees to include configuration spaces with any number of points, extending a proof of Farber.At the end we show that an unordered configuration space on a graph does not always have the same topological complexity as the corresponding ordered configuration space (not even when they are both connected). Surprisingly, in our counterexamples the topological complexity of the unordered configuration space is in fact smaller than for the ordered one.Definition 1.1. The topological complexity of X, denoted TC(X), is defined to be the minimal k such that X × X admits a cover by k + 1 open sets U 0 , U 1 , . . . , U k , on each of which there exists a local section of p X (that is, a continuous mapNote that here we use the reduced version of TC(X), which is one less than the original definition by Farber.

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Collision Free Motion Planning on Graphs

Cited by 27 publications

References 7 publications

On the cohomology rings of tree braid groups

On the cohomology rings of tree braid groups

Computing braid groups of graphs with applications to robot motion planning

Topological Complexity of Configuration Spaces of Fully Articulated Graphs and Banana Graphs

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