By Rudolf Kingslake

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**Extra info for Applied optics and optical engineering,Vol.II**

**Sample text**

The second part of Gabai's result shows that box (Z4)> 1, for Z4 is a graph consisting of 2 independent edges. Hence, box (Z4) = 2, as we have already observed. 3 (Roberts (1969b)). Every graph G of n vertices is the intersection graph of boxes in n-space. FOOD WEBS, NICHE OVERLAP GRAPHS, AND BOXICITY 41 Proof. Note that every box in n-space is defined by giving n intervals, Ji^ /2>' ' ' » J m the projections of the box onto the different coordinate axes. 2. The boxicity of ecological phase space.

Thus, we may restrict ourselves to connected transitively orientable graphs. A set K of vertices of a graph is called partitive if for every x, y in K and every u not in K, there is an edge from x to u if and only if there is an edge from y to u. A set of vertices is independent if there are no edges joining any of the vertices in the set. 10 (Shevrin and Filippov, Trotter, Moore, and Sumner). A connected transitively orientable graph G with n vertices and at least one edge has exactly two transitive orientations (one obtained from the other by reversing direc tions} if and only if every partitive set K with at least two vertices but fewer than n vertices is independent.

For example, the normal healthy environment is determined by a range of values of temperature, of light, of pH, of moisture, and so on. If there are n factors in all, and each defines an interval of values, then the corresponding region in n-space is a box. This box corresponds to what is frequently called in ecology the ecological niche of the species. " For this reason, the n-dimensional Euclidean space defined by the n factors is sometimes called ecological phase space. Recent reviews of the concept of ecological niche are by Miller (1967), Vandermeer (1972), and Pianka (1976).