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.

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

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?

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.

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.

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

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].

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.

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.

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.

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.

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.

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