Outer 1-Planar Graphs

A graph is outer 1-planar ( o1p ) if it can be drawn in the plane such that all vertices are in the outer face and each edge is crossed at most once. o1p graphs generalize outerplanar graphs, which can be recognized in linear time, and specialize 1-planar graphs, whose recognition is N P -hard. We e...

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Vydáno v:	Algorithmica Ročník 74; číslo 4; s. 1293 - 1320
Hlavní autoři:	Auer, Christopher, Bachmaier, Christian, Brandenburg, Franz J., Gleißner, Andreas, Hanauer, Kathrin, Neuwirth, Daniel, Reislhuber, Josef
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.04.2016 Springer Nature B.V
Témata:	Algorithm Analysis and Problem Complexity Algorithms Apexes Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Graph theory Graphs Mathematics of Computing Recognition Theory of Computation Visibility Graph parameters Embeddings and drawings Density Planar and outerplanar graphs 1-Planarity
ISSN:	0178-4617, 1432-0541
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Abstract	A graph is outer 1-planar ( o1p ) if it can be drawn in the plane such that all vertices are in the outer face and each edge is crossed at most once. o1p graphs generalize outerplanar graphs, which can be recognized in linear time, and specialize 1-planar graphs, whose recognition is N P -hard. We explore o1p graphs. Our first main result is a linear-time algorithm that takes a graph as input and returns a positive or a negative witness for o1p . If a graph G is o1p , then the algorithm computes an embedding and can augment G to a maximal o1p graph. Otherwise, G includes one of six minors, which is detected by the recognition algorithm. Secondly, we establish structural properties of o1p graphs. o1p graphs are planar and are subgraphs of planar graphs with a Hamiltonian cycle. They are neither closed under edge contraction nor under subdivision. Several important graph parameters, such as treewidth, colorability, stack number, and queue number, increase by one from outerplanar to o1p graphs. Every o1p graph of size n has at most 5 2 n - 4 edges and there are maximal o1p graphs with 11 5 n - 18 5 edges, and these bounds are tight. Finally, every o1p graph has a straight-line grid drawing in O ( n 2 ) area with all vertices in the outer face, a planar visibility representation in O ( n log n ) area, and a 3D straight-line drawing in linear volume, and these drawings can be constructed in linear time.
AbstractList	A graph is outer 1-planar (o1p) if it can be drawn in the plane such that all vertices are in the outer face and each edge is crossed at most once. o1p graphs generalize outerplanar graphs, which can be recognized in linear time, and specialize 1-planar graphs, whose recognition is NP-hard. We explore o1p graphs. Our first main result is a linear-time algorithm that takes a graph as input and returns a positive or a negative witness for o1p. If a graph G is o1p, then the algorithm computes an embedding and can augment G to a maximal o1p graph. Otherwise, G includes one of six minors, which is detected by the recognition algorithm. Secondly, we establish structural properties of o1p graphs. o1p graphs are planar and are subgraphs of planar graphs with a Hamiltonian cycle. They are neither closed under edge contraction nor under subdivision. Several important graph parameters, such as treewidth, colorability, stack number, and queue number, increase by one from outerplanar to o1p graphs. Every o1p graph of size n has at most 52n-4 edges and there are maximal o1p graphs with 115n-185 edges, and these bounds are tight. Finally, every o1p graph has a straight-line grid drawing in O(n2) area with all vertices in the outer face, a planar visibility representation in O(nlogn) area, and a 3D straight-line drawing in linear volume, and these drawings can be constructed in linear time. A graph is outer 1-planar ( o1p ) if it can be drawn in the plane such that all vertices are in the outer face and each edge is crossed at most once. o1p graphs generalize outerplanar graphs, which can be recognized in linear time, and specialize 1-planar graphs, whose recognition is N P -hard. We explore o1p graphs. Our first main result is a linear-time algorithm that takes a graph as input and returns a positive or a negative witness for o1p . If a graph G is o1p , then the algorithm computes an embedding and can augment G to a maximal o1p graph. Otherwise, G includes one of six minors, which is detected by the recognition algorithm. Secondly, we establish structural properties of o1p graphs. o1p graphs are planar and are subgraphs of planar graphs with a Hamiltonian cycle. They are neither closed under edge contraction nor under subdivision. Several important graph parameters, such as treewidth, colorability, stack number, and queue number, increase by one from outerplanar to o1p graphs. Every o1p graph of size n has at most 5 2 n - 4 edges and there are maximal o1p graphs with 11 5 n - 18 5 edges, and these bounds are tight. Finally, every o1p graph has a straight-line grid drawing in O ( n 2 ) area with all vertices in the outer face, a planar visibility representation in O ( n log n ) area, and a 3D straight-line drawing in linear volume, and these drawings can be constructed in linear time.
Author	Bachmaier, Christian Hanauer, Kathrin Neuwirth, Daniel Brandenburg, Franz J. Reislhuber, Josef Auer, Christopher Gleißner, Andreas
Author_xml	– sequence: 1 givenname: Christopher surname: Auer fullname: Auer, Christopher organization: University of Passau – sequence: 2 givenname: Christian surname: Bachmaier fullname: Bachmaier, Christian email: bachmaier@fim.uni-passau.de organization: University of Passau – sequence: 3 givenname: Franz J. surname: Brandenburg fullname: Brandenburg, Franz J. organization: University of Passau – sequence: 4 givenname: Andreas surname: Gleißner fullname: Gleißner, Andreas organization: University of Passau – sequence: 5 givenname: Kathrin surname: Hanauer fullname: Hanauer, Kathrin organization: University of Passau – sequence: 6 givenname: Daniel surname: Neuwirth fullname: Neuwirth, Daniel organization: University of Passau – sequence: 7 givenname: Josef surname: Reislhuber fullname: Reislhuber, Josef organization: University of Passau
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Cites_doi	10.1137/S0097539794280736 10.1137/0221055 10.1016/0020-0190(79)90075-9 10.1016/0166-218X(89)90031-0 10.1007/s00454-010-9310-z 10.1002/jgt.3190050304 10.1016/j.disc.2005.11.056 10.1016/j.dam.2012.11.019 10.1016/0095-8956(79)90021-2 10.1137/S0097539793251219 10.1002/jgt.21630 10.1002/jgt.3190120306 10.1215/S0012-7094-37-00336-3 10.1016/j.tcs.2013.09.029 10.1007/BF01215922 10.1137/S0097539702416141 10.1142/S021819591250015X 10.1007/s00453-007-9117-3 10.1016/j.ipl.2013.01.013 10.1137/130908051 10.1016/j.jctb.2005.09.009 10.1145/1634.322451 10.1137/120872310 10.1016/j.comgeo.2010.03.007 10.1016/0095-8956(80)90083-0 10.1007/BF02996313 10.1007/BF02122694 10.7155/jgaa.00347 10.1007/978-3-319-03841-4_7 10.1007/BF02941309 10.1007/978-3-642-40104-6_9
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Snippet	A graph is outer 1-planar ( o1p ) if it can be drawn in the plane such that all vertices are in the outer face and each edge is crossed at most once. o1p... A graph is outer 1-planar (o1p) if it can be drawn in the plane such that all vertices are in the outer face and each edge is crossed at most once. o1p graphs...
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Apexes Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Graph theory Graphs Mathematics of Computing Recognition Theory of Computation Visibility
Title	Outer 1-Planar Graphs
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