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

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

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

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Conference article in proceeding › Academic › peer-review

Abstract

We introduce and study the OrthoSEFE-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 >= 3 even if the shared graph is a Hamiltonian cycle and has sunflower intersection and for k >= 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.

Original language	English
Title of host publication	Graph Drawing and Network Visualization. GD 2016
Pages	532-545
DOIs	https://doi.org/10.1007/978-3-319-50106-2_41
Publication status	Published - 2016
Externally published	Yes

Publication series

Series	Lecture Notes in Computer Science
Volume	9801
ISSN	0302-9743

Access to Document

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

Cite this

@inproceedings{0d92b60762d342bc892c3a5455242ef9,

title = "Simultaneous Orthogonal Planarity",

abstract = "We introduce and study the OrthoSEFE-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 >= 3 even if the shared graph is a Hamiltonian cycle and has sunflower intersection and for k >= 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.",

author = "Patrizio Angelini and Steven Chaplick and Sabine Cornelsen and Lozzo, {Giordano Da} and Battista, {Giuseppe Di} and Peter Eades and Philipp Kindermann and Jan Kratochv{\'i}l and Fabian Lipp and Ignaz Rutter",

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doi = "10.1007/978-3-319-50106-2_41",

language = "English",

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T1 - Simultaneous Orthogonal Planarity

AU - Angelini, Patrizio

AU - Chaplick, Steven

AU - Cornelsen, Sabine

AU - Lozzo, Giordano Da

AU - Battista, Giuseppe Di

AU - Eades, Peter

AU - Kindermann, Philipp

AU - Kratochvíl, Jan

AU - Lipp, Fabian

AU - Rutter, Ignaz

N1 - DBLP's bibliographic metadata records provided through http://dblp.org/search/publ/api are distributed under a Creative Commons CC0 1.0 Universal Public Domain Dedication. Although the bibliographic metadata records are provided consistent with CC0 1.0 Dedication, the content described by the metadata records is not. Content may be subject to copyright, rights of privacy, rights of publicity and other restrictions.

PY - 2016

Y1 - 2016

N2 - We introduce and study the OrthoSEFE-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 >= 3 even if the shared graph is a Hamiltonian cycle and has sunflower intersection and for k >= 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.

AB - We introduce and study the OrthoSEFE-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 >= 3 even if the shared graph is a Hamiltonian cycle and has sunflower intersection and for k >= 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.

U2 - 10.1007/978-3-319-50106-2_41

DO - 10.1007/978-3-319-50106-2_41

M3 - Conference article in proceeding

T3 - Lecture Notes in Computer Science

SP - 532

EP - 545

BT - Graph Drawing and Network Visualization. GD 2016

ER -