### Mathematics Research Projects

## Project 10: Triangulation of Plane Polygons

**Problems**

a. Can every plane (non-self-intersecting) polygon be triangulated
to form a graph where every vertex has valence at most s? (Or,
can you find a family of polygons P_{t} such that for every triangulation
of P_{t} there is a vertex of valence at least t?)

b. Can you find a family of plane (non-self-intersecting) polygons
Pt which have exactly t ways to triangulate the polygon?

c. Explore the possibility of some connection between the number
of guards of a polygon and number of different ways to triangulate
the polygon. (The number of guards for a polygon is the minimum
number of points in the polygon from which all of the polygon
is collectively visible from one or more of the points.)

(See Project 1.)

**References**

1. O'Rourke, J., Art Gallery Theorems and Algorithms, Oxford University
Press, New York, 1987.

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**Joseph Malkevitch**

*Department of Mathematics and Computing*

York College (CUNY)

Jamaica, New York 11451-0001

email: malkevitch@york.cuny.edu

(Comments and results related to the project above are welcome.)

**Acknowledgements**

Some of this work was prepared with partial support from the National
Science Foundation (Grant Number: DUE 9555401) to the Long Island
Consortium for Interconnected Learning (administered by SUNY at
Stony Brook, Alan Tucker, Project Director).