# The Stony Brook Algorithm Repository

## `INPUT OUTPUT`

Input Description: A graph G.

Problem: Give a drawing of graph G which accurately reflects its structure.

Excerpt from The Algorithm Design Manual: Drawing graphs nicely is a problem that constantly arises in applications, such as displaying file directory trees or circuit schematic diagrams.Yet it is inherently ill-defined. What exactly does nicely mean? We seek an algorithm that shows off the structure of the graph so the viewer can best understand it. We also seek a drawing that looks aesthetically pleasing. Unfortunately, these are ``soft'' criteria for which it is impossible to design an optimization algorithm. Indeed, it is possible to come up with two or more radically different drawings of certain graphs and have each be most appropriate in certain contexts.

Several ``hard'' criteria can partially measure the quality of a drawing:

• Crossings -- We seek a drawing with as few pairs of crossing edges as possible, since they are distracting.
• Area -- We seek a drawing that uses as little paper as possible, while ensuring that no pair of vertices are placed too close to each other.
• Edge Length -- We seek a drawing that avoids long edges, since they tend to obscure other features of the drawing.
• Aspect Ratio -- We seek a drawing whose aspect ratio (width/height) reflects that of the desired output medium (typically a computer screen at 4/3) as close as possible.

Unfortunately, these goals are mutually contradictory, and the problem of finding the best drawing under any nonempty subset of them will likely be NP-complete.

## Recommended Books

 Handbook of Graph Drawing and Visualization by R. Tamassia Exploratory Social Network Analysis with Pajek by W. Nooy and A. Mrvar and V. Batagelj Drawing Graphs: Methods and Models by M. Kaufmann and D. Wagner Graph Drawing: Algorithms for the Visualization of Graphs by Giuseppe Di Battista, Peter Eades, Roberto Tamassia, and Ionnis G. Tollis Computational Discrete Mathematics: Combinatorics and Graph Theory with Mathematica by S. Pemmaraju and S. Skiena

## Related Problems

 ` ` Drawing Trees ` ` Planarity Detection and Embedding