Aravind for his constant support in my thesis and related research, for his patience, motivation and immense knowledge of theory. His guidance helped me think like a researcher, he also reviewed presentations and this thesis, which led to writing better content for readers. I also thank the professors for their valuable comments and suggestions during the presentations.
Finally, I would like to express my deep gratitude to my parents for providing me immense support throughout my research work. The most important part of this thesis is chapter 5, where the implementation of the planar 3-Tree is given. We have implemented the queue layout of 2-layer planar 3-Tree using two queues and then generalized this experiment to any arbitrary number of levels.
Interactive and scientific presentation of data using a computer-aided tool so that the user can understand it better is called information visualization. With the help of a graph, the human visual system can capture a large amount of data (which is impossible without graphical visualization) and also find a pattern in the data. A limitation of the human visual system is that it can only work with image data.
The data structure for artificial intelligence based production of geometric representations of interrelated information is a graph where vertices and edges represent the entities and the relationships between them respectively.
Area
Aspect Ratio
Sub-graph Separation
Closest Leaf
Farthest leaf
Size
Edge Length
Angular Resolution
Symmetry
Graph Drawing Techniques
Orthogonal Drawing
Poly Sub-tree Drawing
Upward and Non Upward Drawing
Planar Drawing
Grid Drawing
Application
Topology Shape Metrics
Topology
Shape
Metrics
A brief summary of upward quasi-planarity, 3D orthogonal boxplots, visualization of interconnected data, algorithms for drawing metric graphs, plane verification, 3D orthogonal plotting using the splash and push approach, geometric thickness of graphs are also covered in the book.
Planar Graph
Tree Drawing
Hierarchical layout algorithms
Orthogonal Drawing
Stack and Queue layout
Track layout
According to Dujmovicet al.[19] a (K, T)-trace layout of the graph G consists of a proper vertex T' coloring of G, a total order of each vertex class, and an improper edge K' coloring such that between each pair of color classes there are no two monochrome edges cross each other. We first capture the linear set of vertices of the graph in a hierarchical manner, so that no two vertices at the next level have the same color. The number of colors and tracks is inversely proportional to each other. At-Track Layout for the graph G is the minimum number of tracks in G when the number of colors is minimized.
The chromatic number of G, denoted by χ (G), is the minimum number such that G has a vertex color. Let there be two levels dj and two edges and cds such that vertices a,c are at level i and b, dare at level j. Letibe any integer if we place vertices at (i, i2, i3) then neither edge will produce X - intersection.
For any two ends to intersect or intersect, all four endpoints must be in the same plane. Two edges will intersect if they are in the same plane, but the opposite can always be true.
Layouts of Fixed Order
There are some edges in Graph G, which can cause problems when we need to reduce stack/queue number. K rainbow creates a problem for Queue Layout because it creates nested edges that cannot be placed. K-turning creates a problem for stack layout because it creates cutting edges that cannot be placed in the same stack.
The largest size of k-rainbow and k-twist in a graph G sets the queue number and stack number of G, respectively.
Graph with Queue number 1 and stack number 1
Queue number and Stack number Trade off
Queue number and Stack number in a nutshell
Terminology
Procedure
Algorithm
Each extrinsic planar graph can only be bounded by a triangle, so adding a vertex v to the already existing extrinsic planar graph G will not affect the extrinsic planarity.
Track layout of outer-planar Graph
A graph is called planar such that the edges in the drawn graph lie in the same plane so that they do not intersect each other. The unbounded region is called the outer face, and the bounded region is called the inner face. Vertices that lie on exterior faces are called exterior vertices, while vertices that lie inside interior faces are called interior vertices.
A maximal outer-plane graph is internally triangulated outer-plane graph with the maximum number of edges [30].
Implementation of planar 3-tree
Terminology
Layer: Layer of a level 3 tree is the set of vertices that are at the boundary after all edges that are level i−1 have been removed. Binding edges: Binding edges are those edges that connect two layers of planar 3-tree. Flat edges: Edges used to connect the vertices of the same layer are called flat edges.
Anchor: Let< u, v, w >is a triangle with neighboring vertices having a maximum vertical height of no more than 2. Letvandware upper and lower vertices, then vertexuis is called the anchor vertex of the face < u, v, w > .
Queue layout of 2-level planar 3-tree
For each layer, the order of vertices will be determined by the property of the triangulated outer-plane graph. Each triangulated graph in the outer plane can be converted to a graph such that neighboring vertices differ by vertical height at most 2. If there are multiple connected components at any level, then each connected component either precedes or follows.
If L1 has more than one connected component, say c1 and c2, then find the sequence between these two and then insert into the queues. If all edges of ∆ are placed before every edge of ∆0, this means that there will be no nesting edges.
Implementation
These connected components can be bi-connected, so add extra edges to make them bi-connected. Edges connecting vertices of the same layer are called congruent edges and connected edges connecting the vertices of successive layers are called connecting edges. For each level pairLi andLi+1 we first place the vertices of each layer linearly so that they will follow total order.
When we are done with layer pairs Li and Li+1, the same procedure will apply to layer pairs Li+1 and Li+2. Here the queue used for level edges of layer can be reused for level edges of layer i+ 2.
Result
Converting into levels
Queue layout
The implementation of an external planar graph was the basis for finding the next representation of planar 3-trees. Any maximal outer-planar graph can be transformed into an outer-planar graph with neighboring vertices maintained at a vertical distance of no more than two. We reduce the upper bound to 5, and there is a planar 3-tree whose lower bound is 4.
So far, the lower bound of the planar graph of the queue layout is 4 and the upper bound is O(logn). There is paper showing that if the tree width of the graph is bounded by a constant, then the requirement of the number of queues in the planar queuing layout is also bounded by a constant. Comparing queues and stacks as graph formatting machines.SIAM Journal on Discrete Mathematics.
X− Crossing
Edge Wrapping into the 3−tracks
Maximal Outer Planar Graph
Upper envelop and Lower envelop
Graph with height of neighboring vertices differ by at most 2
Planar 3-tree
Planar 3−Tree and level split
Level wise placement of vertices of planar 3-tree
Queue layout of Planar 3-Tree
