Geodesic
Optimization problems over unit-distance representations of graphs
Marcel Kenji de Carli Silva1 ; Levent Tunçel1
1University of Waterloo
The electronic journal of combinatorics, Tome 20 (2013) no. 1
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We study the relationship between unit-distance representations and the Lovász theta number of graphs, originally established by Lovász. We derive and prove min-max theorems. This framework allows us to derive a weighted version of the hypersphere number of a graph and a related min-max theorem. Then, we connect to sandwich theorems via graph homomorphisms. We present and study a generalization of the hypersphere number of a graph and the related optimization problems. The generalized problem involves finding the smallest ellipsoid of a given shape which contains a unit-distance representation of the graph. Arbitrary positive semidefinite forms describing the ellipsoids yield NP-hard problems.
DOI : 10.37236/3128
Classification : 05C62, 90C35, 90C22
Mots-clés : geometric representation, semidefinite programming, unit distance representation, Lovasz theta number

Marcel Kenji de Carli Silva  1   ; Levent Tunçel  1

1 University of Waterloo 
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Marcel Kenji de Carli Silva; Levent Tunçel. Optimization problems over unit-distance representations of graphs. The electronic journal of combinatorics, Tome 20 (2013) no. 1. doi: 10.37236/3128

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