TY - GEN

T1 - Beyond Outerplanarity

AU - Chaplick, Steven

AU - Kryven, Myroslav

AU - Liotta, Giuseppe

AU - Löffler, Andre

AU - Wolff, Alexander

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 - 2017

Y1 - 2017

N2 - We study straight-line drawings of graphs where the vertices are placed in convex position in the plane, i.e., convex drawings. We consider two families of graph classes with nice convex drawings: outer k-planar graphs, where each edge is crossed by at most k other edges; and, outer k-quasi-planar graphs where no k edges can mutually cross.we show that the outer k-planar graphs are \((\lfloor \sqrt{4k+1}\rfloor +1)\)-degenerate, and consequently that every outer k-planar graph can be \((\lfloor \sqrt{4k+1}\rfloor +2)\)-colored, and this bound is tight. We further show that every outer k-planar graph has a balanced separator of size at most \(2k+3\). For each fixed k, these small balanced separators allow us to test outer k-planarity in quasi-polynomial time, i.e., none of these recognition problems are np-hard unless eth fails.for the outer k-quasi-planar graphs we discuss the edge-maximal graphs which have been considered previously under different names. We also construct planar 3-trees that are not outer 3-quasi-planar.finally, we restrict outer k-planar and outer k-quasi-planar drawings to closed drawings, where the vertex sequence on the boundary is a cycle in the graph. For each k, we express closed outer k-planarity and closed outer k-quasi-planarity in extended monadic second-order logic. Thus, since outer k-planar graphs have bounded treewidth, closed outer k-planarity is linear-time testable by courcelle’s theorem.

AB - We study straight-line drawings of graphs where the vertices are placed in convex position in the plane, i.e., convex drawings. We consider two families of graph classes with nice convex drawings: outer k-planar graphs, where each edge is crossed by at most k other edges; and, outer k-quasi-planar graphs where no k edges can mutually cross.we show that the outer k-planar graphs are \((\lfloor \sqrt{4k+1}\rfloor +1)\)-degenerate, and consequently that every outer k-planar graph can be \((\lfloor \sqrt{4k+1}\rfloor +2)\)-colored, and this bound is tight. We further show that every outer k-planar graph has a balanced separator of size at most \(2k+3\). For each fixed k, these small balanced separators allow us to test outer k-planarity in quasi-polynomial time, i.e., none of these recognition problems are np-hard unless eth fails.for the outer k-quasi-planar graphs we discuss the edge-maximal graphs which have been considered previously under different names. We also construct planar 3-trees that are not outer 3-quasi-planar.finally, we restrict outer k-planar and outer k-quasi-planar drawings to closed drawings, where the vertex sequence on the boundary is a cycle in the graph. For each k, we express closed outer k-planarity and closed outer k-quasi-planarity in extended monadic second-order logic. Thus, since outer k-planar graphs have bounded treewidth, closed outer k-planarity is linear-time testable by courcelle’s theorem.

U2 - 10.1007/978-3-319-73915-1_42

DO - 10.1007/978-3-319-73915-1_42

M3 - Conference article in proceeding

T3 - Lecture Notes in Computer Science

SP - 546

EP - 559

BT - Graph Drawing and Network Visualization. GD 2017

A2 - Frati, F.

A2 - Ma, K.L.

PB - Springer, Cham

ER -