NP-completeness of minimum spanner problems
Discrete Applied Mathematics
SIAM Journal on Discrete Mathematics
Tree spanners in planar graphs
Discrete Applied Mathematics - Special issue on international workshop of graph-theoretic concepts in computer science WG'98 conference selected papers
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
The Geometric Dilation of Finite Point Sets
Algorithmica
Minimum weight triangulation is NP-hard
Proceedings of the twenty-second annual symposium on Computational geometry
Geometric Spanner Networks
On spanners of geometric graphs
SWAT'06 Proceedings of the 10th Scandinavian conference on Algorithm Theory
Sparse geometric graphs with small dilation
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
Computing a minimum-dilation spanning tree is NP-hard
CATS '07 Proceedings of the thirteenth Australasian symposium on Theory of computing - Volume 65
Optimal Embedding into Star Metrics
WADS '09 Proceedings of the 11th International Symposium on Algorithms and Data Structures
Improved multi-criteria spanners for ad-hoc networks under energy and distance metrics
INFOCOM'10 Proceedings of the 29th conference on Information communications
Near-optimal multicriteria spanner constructions in wireless ad hoc networks
IEEE/ACM Transactions on Networking (TON)
Improved multicriteria spanners for Ad-Hoc networks under energy and distance metrics
ACM Transactions on Sensor Networks (TOSN)
Minimum weight Euclidean t-spanner is NP-hard
Journal of Discrete Algorithms
Approximated algorithms for the minimum dilation triangulation problem
Journal of Heuristics
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Consider a geometric graph G, drawn with straight lines in the plane. For every pair a, b of vertices of G, we compare the shortest-path distance between a and b in G (with Euclidean edge lengths) to their actual Euclidean distance in the plane. The worst-case ratio of these two values, for all pairs of vertices, is called the vertex-to-vertex dilation of G. We prove that computing a minimum-dilation graph that connects a given n-point set in the plane, using not more than a given number m of edges, is an NP-hard problem, no matter if edge crossings are allowed or forbidden. In addition, we show that the minimum dilation tree over a given point set may in fact contain edge crossings.