By Rudolf Kingslake
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Extra resources for Applied optics and optical engineering,Vol.II
1 we may further assume that G satisfies m G 3n; otherwise G is nonplanar. The “st-numbering” plays a crucial role in the testing algorithm. A numbering of the vertices of G by 1, 2, . . , n is called an st-numbering if the two vertices “1” and ‘‘n” are necessarily adjacent and eachj of the other vertices is adjacent to two vertices i and k such that i < j < k . The vertex “1” is called a source and is denoted by s, while the vertex “n” is called a sink and is denoted by t. Fig. l(a) illustrates an st-numbering of a graph.
We next present a function PATH. Initially, the two vertices s and t together with the first searched edge (s, t ) are marked “old” and all the other vertices and edges are marked “new”. Note that DFS(s) = 2 and DFS(t) = 1. PATH(v) takes as the value a path going from v to an “old” vertex. The procedure PATH(v) is as follows: Case 1: there is a “new” back edge (v, w). mark (v, w) “old”; PATH := VW; return. Case 2: there is a “new”tree edge (v, w). let wo( = w)wI w, . W~; mark all the vertices and edges on the path “ o l d ; return.
A graph may be represented in many ways. For example we can associate with a graph G = ( V , E) its n X n adjacency matrixA = [a,] such that a, = 1 if (v,, v , ) E E , and a, = 0 otherwise. Fig. 2 illustrates a graph (a) and its adjacency matrix (b). By definition the main diagonal of A is all zeros, and A is symmetric. EHm - vertex v 4 -. vertex v 5 1 * II - 4T€+El (c) Fig. 2. Representation of a graph: (a) graph, (b) adjacency matrix, and (c) adjacency lists. vertices. It is not economical when a graph is sparse, that is, the number m of edges is far less than n(n - 1)/2.