Abstract
We study Hamiltonicity for some of the most general variants of Delaunay and Gabriel graphs. Instead of defining these proximity graphs using circles, we use an arbitrary convex shape \(\mathcal {C}\). Let S be a point set in the plane. The k-order Delaunay graph of S, denoted k-\({DG}_{\mathcal {C}}(S)\), has vertex set S and edge pq provided that there exists some homothet of \(\mathcal {C}\) with p and q on its boundary and containing at most k points of S different from p and q. The k-order Gabriel graph k-\({GG}_{\mathcal {C}}(S)\) is defined analogously, except for the fact that the homothets considered are restricted to be smallest homothets of \(\mathcal {C}\) with p and q on its boundary. We provide upper bounds on the minimum value of k for which k-\({GG}_{\mathcal {C}}(S)\) is Hamiltonian. Since k-\({GG}_{\mathcal {C}}(S)\) \(\subseteq \) k-\({DG}_{\mathcal {C}}(S)\), all results carry over to k-\({DG}_{\mathcal {C}}(S)\). In particular, we give upper bounds of 24 for every \(\mathcal {C}\) and 15 for every point-symmetric \(\mathcal {C}\). We also improve the bound to 7 for squares, 11 for regular hexagons, 12 for regular octagons, and 11 for even-sided regular t-gons (for \(t \ge 10)\). These constitute the first general results on Hamiltonicity for convex shape Delaunay and Gabriel graphs.
P.B. was partially supported by NSERC. P.C. was supported by CONACyT. M.S. was supported by the Czech Science Foundation, grant number GJ19-06792Y, and by institutional support RVO:67985807. R.S. was supported by MINECO through the Ramón y Cajal program. P.C. and R.S. were also supported by projects MINECO MTM2015-63791-R and Gen. Cat. 2017SGR1640. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 734922.
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- 1.
A graph is 1-tough if removing k vertices from it results in \(\le k\) connected components.
- 2.
Note that this implies that the standard Delaunay triangulation is the 0-DG.
- 3.
According to the definition of k-RNG in [9], they showed Hamiltonicity for 20-RNG.
- 4.
A shape \(\mathcal {C}\) is point-symmetric with respect to a point \(x \in \mathcal {C}\) provided that for every point \(p \in \mathcal {C}\) there is a corresponding point \(q \in \mathcal {C}\) such that \(pq \in \mathcal {C}\) and x is the midpoint of pq.
- 5.
A function \(\rho (x)\) is a norm if: (a) \(\rho (x)=0\) if and only if \(x=\bar{o}\), (b) \(\rho (\lambda x)=|\lambda |\rho (x)\) where \(\lambda \in \mathbb {R}\), and (c) \(\rho (x+y) \le \rho (x)+\rho (y)\).
- 6.
The Minkowski sum of two sets A and B is defined as \(A\oplus B=\{a+b: a\in A, b\in B \}\).
- 7.
Since S is in general position, only a and b can lie on the boundary of \(\mathcal {C}(a,b)\).
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Bose, P., Cano, P., Saumell, M., Silveira, R.I. (2019). Hamiltonicity for Convex Shape Delaunay and Gabriel Graphs. In: Friggstad, Z., Sack, JR., Salavatipour, M. (eds) Algorithms and Data Structures. WADS 2019. Lecture Notes in Computer Science(), vol 11646. Springer, Cham. https://doi.org/10.1007/978-3-030-24766-9_15
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