# Drawing Graphs Nicely

 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.

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