Abstract
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 \(\mathcal{NP}\)-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 \(\mathcal{NP}\)-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 “Progetto ALINWEB: Algoritmica per Internet e per il Web”, MIUR Programmi di Ricerca Scientifica di Rilevante Interesse Nazionale.
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Di Giacomo, E., Didimo, W., Liotta, G., Suderman, M. (2005). Hamiltonian-with-Handles Graphs and the k-Spine Drawability Problem. In: Pach, J. (eds) Graph Drawing. GD 2004. Lecture Notes in Computer Science, vol 3383. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31843-9_27
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DOI: https://doi.org/10.1007/978-3-540-31843-9_27
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