Topological complexity of configuration spaces of fully articulated graphs and banana graphs

Lütgehetmann, Daniel; Recio-Mitter, David

doi:10.48550/arxiv.1806.00659

Cited by 1 publication

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“…Proof. The approach we take here is essentially the approach taken by Farber in [4] and Lütgehetmann and Recio-Mitter in [15]. Since D n (Γ) ⊂ n i=1 Γ, we have dim(D n (Γ)) ≤ n, so TC(D n (Γ)) ≤ 2n + 1.…”

Section: Topological Complexitymentioning

confidence: 99%

“…Recently, Lütgehetmann and Recio-Mitter extended Farber's result concerning the ordered spaces to include all values of n : Theorem 1.5. [15] Let Γ be a tree with at least one essential vertex. For n = 2, assume Γ is not homeomorphic to the letter Y .…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 1.6. [15] If Γ is fully articulated and Γ is not homeomorphic to the letter Y, and n ≥ 2m(Γ), then…”

Section: Introductionmentioning

confidence: 99%

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Topological complexity of unordered configuration spaces of certain graphs

Scheirer

2019

Preprint

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The unordered configuration space of n points on a graph Γ, denoted here by UC n (Γ), can be viewed as the space of all configurations of n unlabeled robots on a system of one-dimensional tracks, which is interpreted as a graph Γ. The topology of these spaces is related to the number of vertices of degree greater than 2; this number is denoted by m(Γ). We discuss a combinatorial approach to compute the topological complexity of a "discretized" version of this space, UD n (Γ), and give results for certain classes of graphs. In the first case, we show that for a large class of graphs, as long as the number of robots is at least twice the number of essential vertices, then TC(UD n (Γ)) = 2m(Γ) + 1. In the second, we show that as long as the number of robots is at most half the number of vertex-disjoint cycles in Γ, we have TC(UD n (Γ)) = 2n + 1.

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Section: Topological Complexitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%