Minimal Networks: A Review

Ivanov, A O; Tuzhilin, A. A.

doi:10.1007/978-3-319-40673-2_4

Cited by 9 publications

(4 citation statements)

References 42 publications

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“…The study of geodesic nets bears some relation to minimal networks and the Steiner Problem as studied by Ivanov and Tuzhilin (for a survey, see [IT16]). In fact, Corollary 1.1 in Chapter 3 of [IT94] states that geodesic nets are local minima (with respect to the length of all edges) of immersed parametric networks.…”

Section: Introductionmentioning

confidence: 99%

Geodesic nets with three boundary vertices

Parsch¹

2018

Preprint

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We prove that a geodesic net with three boundary (= unbalanced) vertices on a non-positively curved plane has at most one balanced vertex. We do not assume any a priori bound for the degrees of unbalanced vertices.The result seems to be new even in the Euclidean case.We demonstrate by examples that the result is not true for metrics of positive curvature on the plane, and that there are no immediate generalizations of this result for geodesic nets with four unbalanced vertices.

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Section: Introductionmentioning

confidence: 99%

Geodesic nets with three boundary vertices

Parsch¹

2018

Preprint

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“…The locally minimal geodesic nets in Euclidean spaces and, more generally, Riemannian manifolds were extensively investigated by A. Ivanov and V. Tuzhilin (cf. [IT94], [IT16]). (Note that although general geodesic net are not locally minimal with respect to this condition, they are locally minimal in the following less restrictive sense: For each point p on the net and all sufficiently small r the intersection of the net with the ball of radius r provides the global minimum of the length among all trees of the same shape (i.e.…”

mentioning

confidence: 99%

Geodesic Nets: Some Examples and Open Problems

Nabutovsky

Parsch

2020

Experimental Mathematics

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Geodesic nets on Riemannian manifolds form a natural class of stationary objects generalizing geodesics. Yet almost nothing is known about their classification or general properties even when the ambient Riemannian manifold is the Euclidean plane or the round 2-sphere.In the first half of this paper we survey some results and open questions (old and new) about geodesic nets on Riemannian manifolds. Many of these open questions are about geodesic nets on the Euclidean plane. The second half contains a partial answer for one of these questions, namely, a description of a new infinite family of geodesic nets on the Euclidean plane with 14 boundary (or unbalanced) vertices and arbitrarily many inner (or balanced) vertices of degree ≥ 3.

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“…The problem of forming a minimum-cost proxy tree can be formalized as constructing a minimum Steiner tree [14] that connects a given set of terminals in a graph. Due to the dense deployment of sensor nodes, we can further formalize the problem as constructing a minimum Steiner tree in an Euclidean plane.…”

Section: Centralized Tree Reconfiguration Schemesmentioning

confidence: 99%

“…In ONMST, a new proxy (denoted as P n ) is added to the current proxy tree in two steps: first, P n is added to the current tree via the shortest path that connects the tree and P n ; second, a small subtree that contains P n is optimized based on the locality property of the Voronoi diagram [14]. Specifically, the procedure is described as follows:…”

Section: On-line Schemesmentioning

confidence: 99%