Hamiltonian-with-Handles Graphs and the k-Spine Drawability Problem

A planar graph G is k-spine drawable, k >= 0, if there exists a planar drawing of G in which each vertex of G lies on one of k horizontal lines, and each edge of G is drawn as a polyline consisting of at most two line segments. In this paper we: (i) Introduce the notion of hamiltonian-with-handles graphs and show that a planar graph is 2-spine drawable if and only if it is hamiltonian-with-handles. (ii) Give examples of planar graphs that are/are not 2-spine drawable and present linear-time drawing techniques for those that are 2-spine drawable. (iii) Prove that deciding whether or not a planar graph is 2-spine drawable is NP-Complete. (iv) Extend the study to k-spine drawings for k >= 2, provide examples of non-drawable planar graphs, and show that the k-drawability problem remains NP-Complete for each fixed k >= 2. The authors would like to thank Sue Whitesides for the useful discussion about the topic of this paper. Research supported in part by ldquoProgetto ALINWEB: Algoritmica per Internet e per il Webrdquo, MIUR Programmi di Ricerca Scientifica di Rilevante Interesse Nazionale

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version