### Mathematics Research Projects

## Project 1: Triangulated Polygons

Figure 1 shows a plane polygon with 7 sides.

Figure 1

By adding diagonals, it becomes possible to triangulate the polygon,
that is, to have all the regions after the diagonals are added
consist of triangles (except for the unbounded region that surrounds
the polygon, which will be a triangle only for the case when the
original polygon has n = 3 sides).

One way of doing this for the polygon in Figure 1 is shown in
Figure 2.

Figure 2

One can now count the number of line segments that meet at each
vertex of the polygon. The resulting numbers for the triangulated
polygon in Figure 2 are shown in Figure 3.

Figure 3

If the resulting collection of numbers is arranged in decreasing
order one obtains what will be called the valence sequence of
the triangulated polygon. The valence sequence for the triangulated
polygon in Figure 3 is:

5, 4, 4, 3, 2, 2, 2.

**Problem**

1. Determine those integer sequences which can arise from some
triangulated polygon.

Comment: Triangulated polygons are a special case of a class of
graphs that can be drawn in the plane which are called outerplanar
graphs. These are graphs for which all the vertices of the graph
lie on a single face of the graph.

**References**:

1. Ore, O., Graphs and their Uses, Mathematical Association of
America, Washington, 1963. (There is a new revised edition prepared
by R. Wilson.)

2. West, D., Graph Theory, Prentice-Hall, Englewood Cliffs, 1996.

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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).