The Combinatorial Complexity of Polygon Overlay

The number of elementary connected regions arising from polygon overlay of two or more map layers is an important value to have in planning for data stoiage and in making processing time estimates for overlay applications. That number may be computed di rectly from the line graphs of the two (or more) layers and from the intersection graph(s) of those line graphs. A formula for that computation is derived using tools of algebraic and combinatorial topology which relate the connectivity of a union of sets to the con nectivity of the sets themselves and their intersection. The result and the formula may be stated as follows: Suppose X is the line graph (1-skeleton) of a map. Regard X as embedded in the plane. Let r(X) be the number of regions of the plane separated by X. Then r(X) is the number of connected components in the planar complement of A"; r(X) is also one more than the maximum number of independent cycles in the graph A"; and r(A') is easily computed using standard graph traversal techniques for counting independent cycles. Let c(X) be the number of connected components of A'. If A and B are the line graphs of maps to be overlaid, then A U B is the line graph of the overlay; and: r(A UJ5) = r (A)-c(A) + r(B)-c(B)r(A n B) + c(A n B} + c(A U B) All of the values on the right hand side of the equation can be readily computed using standard graph traversal and line intersection algorithms to obtain the desired value. r(AU B). the number of regions after overlaying.