Intersection Graphs of Families of Convex Sets with Distinguished Points.

by W. F. Ogden, Fred S. Roberts

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Collection of intersection graphs C prime (n) is defined as all finite graphs (V, E) in which there is an assignment to each x in V of a set C(x) in a family of convex sets so that for x not equal to y, (x, y) is a member of E if and only if f(x) is a member of C(y). A graph in C prime (1) is defined as an indifference graph if each C(x) can be taken as a closed unit interval and f(x) is its midpoint. Given these definitions, two theorems are proven: first, that C prime (1) equals the class of indifference graphs, which equals the class of unit interval graphs and, second, that every graph is in C prime (2). 4 pp. Refs. (KB)

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