Planar Octilinear Drawings with One Bend Per Edge M. A. Bekos 1 , M. - PowerPoint PPT Presentation

Planar Octilinear Drawings with One Bend Per Edge M. A. Bekos 1 , M. Gronemann 2 , M. Kaufmann 1 , R. Krug 1 1 Wilhelm Schickard Institut f¨ ur Informatik, Universit¨ at T¨ ubingen, Germany 2 Institut fur Informatik, Universit¨ at zu K¨ oln, Germany 26.09.2014

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion Motivation

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion Previous- and Related Work M. N¨ ollenburg: Automated drawings of metro maps [2005] NP-hard if 0 bends is possible

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion Previous- and Related Work M. N¨ ollenburg: Automated drawings of metro maps [2005] NP-hard if 0 bends is possible B. Keszegh et al.: Drawing planar graphs of bounded degree with few slopes [2013] maxdeg. 8 with 2 bends

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion Previous- and Related Work M. N¨ ollenburg: Automated drawings of metro maps [2005] NP-hard if 0 bends is possible B. Keszegh et al.: Drawing planar graphs of bounded degree with few slopes [2013] maxdeg. 8 with 2 bends E. Di Giacomo et al.: The planar slope number of subcubic graphs [2014] maxdeg. 3 with 0 bends

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion Preliminaries k -planar graph

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion Preliminaries k -planar graph k -connected graph

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion Preliminaries k -planar graph k -connected graph Canonical ordering (for triconnected graphs)

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion Preliminaries k -planar graph k -connected graph Canonical ordering (for triconnected graphs) Partitioning of G into m paths with P 0 = { v 1 , v 2 } and P m = { v n } such that:

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion Preliminaries k -planar graph k -connected graph Canonical ordering (for triconnected graphs) Partitioning of G into m paths with P 0 = { v 1 , v 2 } and P m = { v n } such that: G k is biconnected

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion Preliminaries k -planar graph k -connected graph Canonical ordering (for triconnected graphs) Partitioning of G into m paths with P 0 = { v 1 , v 2 } and P m = { v n } such that: G k is biconnected All neighbors of P k + 1 in G k are on the outer face of G k

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion Preliminaries k -planar graph k -connected graph Canonical ordering (for triconnected graphs) Partitioning of G into m paths with P 0 = { v 1 , v 2 } and P m = { v n } such that: G k is biconnected All neighbors of P k + 1 in G k are on the outer face of G k All vertices of P k have at least one neighbor in a P j with j > k

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion Preliminaries k -planar graph k -connected graph Canonical ordering (for triconnected graphs) Partitioning of G into m paths with P 0 = { v 1 , v 2 } and P m = { v n } such that: G k is biconnected All neighbors of P k + 1 in G k are on the outer face of G k All vertices of P k have at least one neighbor in a P j with j > k | P k | = 1 is called singleton , | P k | > 1 is called chain

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion The Triconnected Case v 1 v 2 Start of the construction

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion The Triconnected Case v 1 v 3 v | P 1 | +2 v 2 First Partition

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion The Triconnected Case v i v j v ′ v ′ i j v 1 v 2 Placing a chain may require stretching

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion The Triconnected Case v i v j v ′ v ′ i j v 1 v 2 Placing a chain

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion The Triconnected Case v i v v 1 v 2 Placing a singleton

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion The Triconnected Case v n v 1 v 2 v 3 Placing of v n step 1

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion The Triconnected Case v 1 v n v 3 v 2 Final layout

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion Results for 4-planar Graphs Theorem There exists an infinite class of 4-planar graphs which do not admit bendless octilinear drawings and if they are drawn with at most one bend per edge, then a linear number of bends is required

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion Results for 4-planar Graphs Theorem There exists an infinite class of 4-planar graphs which do not admit bendless octilinear drawings and if they are drawn with at most one bend per edge, then a linear number of bends is required Theorem Given a triconnected 4-planar graph G , we can compute in O ( n ) time an octilinear drawing of G with at most one bend per edge on an O ( n 2 ) × O ( n ) integer grid.

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion Non-triconnected Graphs Extend to biconnected by using SPQR -trees and the triconnected algorithm for the R -nodes

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion Non-triconnected Graphs Extend to biconnected by using SPQR -trees and the triconnected algorithm for the R -nodes Extend to connected using the BC -tree and the biconnected algorithm

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion The Triconnected Case v 1 v 2 Start of the construction

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion The Triconnected Case v 1 v 3 v | P 1 | +2 v 2 First Partition

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion The Triconnected Case v i v j v ′ v ′ i j v 1 v 2 Placing a chain

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion The Triconnected Case v i v ′ v v 1 v 2 Placing a singleton

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion The Triconnected Case v n v 1 v 2 v 3 Final layout

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion Bad news G n +1 h ( G n ) > w ( G n ) w ( G n + 1 ) ≥ 2 w ( G n ) w ( G n + 1 ) ≥ h ( G n ) G n h ( G n + 1 ) ≥ 2 h ( G n ) v 1 v 2 Super-polynomial area requirement

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion Properties of the 5-planar Algorithm Theorem Given a triconnected 5-planar graph G , we can compute in O ( n 2 ) time an octilinear drawing of G with at most one bend per edge.

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion Non-triconnected Graphs Extend to biconnected by using SPQR -trees and the triconnected algorithm for the R -nodes

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion Non-triconnected Graphs Extend to biconnected by using SPQR -trees and the triconnected algorithm for the R -nodes Extend to connected using the BC -tree and the biconnected algorithm

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion One Bend Per Edge Is Not Always Enough v 1 v 2 v 3 Outer Face that does not admit a one-bend drawing

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion One Bend Per Edge Is Not Always Enough 6-planar triangulation in which each is adjacent to only degree 6 (grey) vertices and at most one degree 5 (black) vertex

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion One Bend Per Edge Is Not Always Enough G aug G 1 1 f ′ f 1 1 G aug G 2 2 f ′ f 2 2 G aug ⊕ G aug f ′ 1 2 1 Γ( G aug ) 2 Construction of an infinite family of graphs

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion One Bend Per Edge Is Not Always Enough G aug G 1 1 f ′ f 1 1 G aug G 2 2 f ′ f 2 2 G aug ⊕ G aug f ′ 1 2 1 Γ( G aug ) 2 Construction of an infinite family of graphs

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion One Bend Per Edge Is Not Always Enough G aug G 1 1 f ′ f 1 1 G aug G 2 2 f ′ f 2 2 G aug ⊕ G aug f ′ 1 2 1 Γ( G aug ) 2 Construction of an infinite family of graphs

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion One Bend Per Edge Is Not Always Enough G aug G 1 1 f ′ f 1 1 G aug G 2 2 f ′ f 2 2 G aug ⊕ G aug f ′ 1 2 1 Γ( G aug ) 2 Construction of an infinite family of graphs

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion Conclusion 4-planar graphs are octilinear drawable with at most one bend per edge in cubic area in linear time

Introduction 4-planar Graphs 5-planar Graphs 6-planar Graphs Conclusion Conclusion 4-planar graphs are octilinear drawable with at most one bend per edge in cubic area in linear time 5-planar graphs are octilinear drawable with at most one bend per edge in super-polynomial area in quadratic time