Flips in combinatorial pointed pseudo-triangulations with face degree at most four

In this paper we consider the flip operation for combinatorial pointed pseudo-triangulations where faces have size 3 or 4, so-called combinatorial 4-PPTs. We show that every combinatorial 4-PPT is stretchable to a geometric pseudo-triangulation, which in general is not the case if faces may have size larger than 4. Moreover, we prove that the flip graph of combinatorial 4-PPTs with triangular outer face is connected and has diameter O(n2).European Science FoundationAustrian Science FundMinisterio de Ciencia e InnovaciónJunta de Castilla y Leó

Hamiltonicity for convex shape Delaunay and Gabriel graphs

© 2020. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/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 Ram´on 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 Sk lodowska-Curie grant agreement No 734922.Peer ReviewedPostprint (author's final draft

Hamiltonicity for convex shape Delaunay and Gabriel graphs

We study Hamiltonicity for some of the most general variants of Delaunay and Gabriel graphs. Let S be a point set in the plane. The k-order Delaunay graph of S, denoted k-DGC(S), has vertex set S and edge pq provided that there exists some homothet of 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-GGC(S) is defined analogously, except for the fact that the homothets considered are restricted to be smallest homothets of C with p and q on its boundary. We provide upper bounds on the minimum value of k for which k-GGC(S) is Hamiltonian. Since k-GGC(S) ¿ k-DGC(S), all results carry over to k-DGC(S). In particular, we give upper bounds of 24 for every C and 15 for every point-symmetric 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 = 10).Peer ReviewedPostprint (published version