# Simultaneous Orthogonal Planarity

• Patrizio Angelini
• Steven Chaplick
• Sabine Cornelsen
• Giordano Da Lozzo
• Giuseppe Di Battista
• Peter Eades
• Philipp Kindermann
• Jan Kratochvíl
• Fabian Lipp
• Ignaz Rutter
Conference paper

DOI: 10.1007/978-3-319-50106-2_41

Part of the Lecture Notes in Computer Science book series (LNCS, volume 9801)
Cite this paper as:
Angelini P. et al. (2016) Simultaneous Orthogonal Planarity. In: Hu Y., Nöllenburg M. (eds) Graph Drawing and Network Visualization. GD 2016. Lecture Notes in Computer Science, vol 9801. Springer, Cham

## Abstract

We introduce and study the $${\textsc {OrthoSEFE}\text {-}{k}}$$ problem: Given k planar graphs each with maximum degree 4 and the same vertex set, do they admit an OrthoSEFE, that is, is there an assignment of the vertices to grid points and of the edges to paths on the grid such that the same edges in distinct graphs are assigned the same path and such that the assignment induces a planar orthogonal drawing of each of the k graphs? We show that the problem is NP-complete for $$k \ge 3$$ even if the shared graph is a Hamiltonian cycle and has sunflower intersection and for $$k \ge 2$$ even if the shared graph consists of a cycle and of isolated vertices. Whereas the problem is polynomial-time solvable for $$k=2$$ when the union graph has maximum degree five and the shared graph is biconnected. Further, when the shared graph is biconnected and has sunflower intersection, we show that every positive instance has an OrthoSEFE with at most three bends per edge.

## Copyright information

© Springer International Publishing AG 2016

## Authors and Affiliations

• Patrizio Angelini
• 1
• Steven Chaplick
• 2
• Sabine Cornelsen
• 3
• Giordano Da Lozzo
• 4
• Giuseppe Di Battista
• 4
• Peter Eades
• 5
• Philipp Kindermann
• 6
• Jan Kratochvíl
• 7
• Fabian Lipp
• 2
• Ignaz Rutter
• 8
1. 1.Universität TübingenTübingenGermany
2. 2.Universität WürzburgWürzburgGermany
3. 3.Universität KonstanzKonstanzGermany
4. 4.Roma Tre UniversityRomeItaly
5. 5.The University of SydneySydneyAustralia
6. 6.FernUniversität in HagenHagenGermany
7. 7.Charles UniversityPragueCzech Republic
8. 8.Karlsruhe Institute of TechnologyKarlsruheGermany