Mizan Rahman
November 23, 2012
67

# A Survey paper on Planar drawings in the minimum number of layers

This survey paper was prepared as it was required for my undergraduate level computational geometry course

## Mizan Rahman

November 23, 2012

## Transcript

1. ### A Survey paper on Planar Drawings in the minimum number

of Layers Mizan Rahman Sadi Anwar Chowdhury Muhammad Rashed Alam
2. ### Definition • A layered drawing of a graph G is

a planar straight-line drawing of G,where each vertex of G is placed on a set of horizontal lines called layers. • Such a drawing of G is also called a k-layer drawing of G when the number of layers in the drawing is k.
3. ### common variants of layered drawing proper upright short 4-layer planar

not proper, not upright, short 3-layer planar not proper, upright, not short, 4-layer planar.

graph
5. ### Previous Results in Layered Drawings • Cornelsen et al. [CSW04]

has given necessary and sufficient conditions for recognizing graphs which are a 2-layer planar and 3-layer planar. • Heath and Rosenberg [HR92] have shown that recognizing graphs with proper and planar drawing on layers is NP-complete. • Sugiyama et al. developed a simple method for drawing layered networks in 1979. • Dujmovic et al. [DFH+01] show that it can be solved in polynomial time when number of layers k is bounded by a constant
6. ### Previous Results in Layered Drawings(contd.) • Kaufmann [FK97] have recognized

graphs that have proper and planar drawings on three layers • Vida, Michael & Matthew employ bounded pathwidth techniques On the Parameterized Complexity of Layered Graph Drawing?
7. ### Previous Results in Layered Drawings(contd.) • Sohaee et al prove

that the minimum number of layers for layered upward embedding of a st-graph is one more than the length of longest path from source node s to sink node t. • genetic algorithms (GAs) with the problem of drawing of level planar graph or hierarchical planar graph, and explore the potential use of GAs to solve this particular problem.
8. ### Previous Results in Layered Drawings(contd.) • Jawaherul Alam et al

gave a linear-time algorithm to obtain an upward drawing of a given tree T on the minimum number of layers. • They also gave a linear-time algorithm to check whether a given biconnected graph G admits an upright drawing on three layers and in case it admits one, our algorithm obtains such an upright drawing of G on three layers.
9. ### Algorithms • Sugiyama framework for layered drawings, which in itself

is the most followed approach in this field. • algorithms for 2-layer drawings of planar graphs • algorithm for 3-layer proper drawing of biconnected graph
10. ### Sugiyama framework • Four phases. 1. Remove all cycles from

the input graph 2. graph is layered with objective to minimize the number of layers or the number of vertices within a layer [DETT99]. 3. the vertices within each layer are permuted to reduce crossings among edges, typically using a layer-by-layer sweep algorithm [STT81] 4. assigns coordinates to the vertices [BJL01].

12. ### Sugiyama framework • disadvantage : – layer assignments are not

changed during the crossing minimization process in the second phase. • The worst-case running-time O(|V ||E| log |E|) requiring O(|V ||E|) memory. • Mutzel96 showed an way able to keep the number of dummy vertices and edges linear in the size of the graph without increasing the number of crossings. • They reduce the worst-case time complexity to O((|V | + |E|) log |E|) requiring O(|V | + |E|) space.
13. ### Two-layer Drawings • Trees: – A tree T has a

planar two layer drawing if and only if it contains a spine i.e., a path S such that T- S is collection of paths. • Biconnected Graphs – A biconnected graph G has a planar two layer drawing if and only if G is outer planar • Bipartite Graphs – A connected bipartite graph G is two layer (proper) drawable if and only if G is a caterpillar.
14. ### Three-layer Drawings • Kaufmann [FK97] et al have recognized graphs

that have proper planar 3-layer drawings • Findings: – If a graph G admits a 3-layer proper drawing , then G isbipartite. – If a graph G admits a 3-layer drawing, then there is no separator vertex v in G such that it has at least three neighbors with non-caterpillar components.
15. ### Problems yet to be solved: • The problem of determining

whether a graph is k-layer planar for any given value of k is yet to be solved and till now, the results in [CSW04] has been the best known result for unrestricted layered drawings of general planar graphs. • To develop efficient algorithms to obtain a planar straight-line drawings of a given tree on the minimum number of layers.
16. ### Problems yet to be solved: • To provide necessary and

sufficient conditions for upright drawings of biconnected graphs on a fixed number of layers greater than three. • To obtain necessary and sufficient conditions for upright drawings of general graphs. • To address the layered drawability problems of general graphs for both the unrestricted case and various restricted cases including upright drawings.
17. ### Reference • [DFH+01] V. Dujmovic, M. R. Fellows et al

On the parameterized complexity of layered graph drawing. • [CSW04] S. Cornelsen, T. Schank, D. Wagner. Drawing Graphs on Two and Three Lines. • [DETT99] R.Tamassia and I. G. Tollis, Graph Drawing: Algorithms for the Visualization of Graphs • [FK97] M. Kaufmanne et al , Nice drawings for planar bipartite graphs • Upright drawings of graphs on three Layers M. Jawaherul Alam, Md. Mashfiqui Rabbi, Md. Saidur Rahman And Md. Rezaul Karim