# vb.net code to generate barcode Figure 15.19 Strongly connected and weakly connected components in Java Encoder EAN-13 Supplement 5 in Java Figure 15.19 Strongly connected and weakly connected components

Figure 15.19 Strongly connected and weakly connected components
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WEIGHTED DIGRAPHS AND GRAPHS A weighted digraph is a pair (V, w) where V is a finite set of vertices and w is a function that assigns to each pair (x, y) of vertices either a positive integer or (infinity). The function w is called the weight function, and its value w(x, y) can be interpreted as the cost (or time or distance) for moving directly from x to y. The value w(x, y) = indicates that there is no edge from x to y. A weighted graph is a weighted digraph (V,w) whose weight function w is symmetric, that is, w(y,x) w(x,y) for all x,y V. Just as every digraph has an embedded graph, every weighted digraph has an embedded weighted graph (V, w) and an embedded (unweighted) digraph. The weight function for the embedded weighted graph can be defined as w (x, y) = min{w(x,y), w(y,x)}, where w is the weight function of the weighted digraph. The vertex set for the embedded digraph can be defined as E = {(x,y) : w(x,y) < }. The properties described above for digraphs and graphs apply to weighted digraphs and weighted graphs. In addition there are some extended properties that depend upon the underlying weight function in the obvious manner. For example, the weighted path length is the sum of the weights of the edges along the path. And the shortest distance from x to y would be the minimum weighted path length among all the paths from x to y. EXAMPLE 15.21 A Weighted Digraph and Its Embedded Structures
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Figure 15.20 shows a weighted digraph together with its embedded weighted graph, its embedded digraph, and its embedded graph. The weights are shown on the edges.
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Figure 15.20 Embedded graphs
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In graph G1 the weighted path length of the path cabd is |cabd| = 2 + 3 + 2 = 7, and the shortest distance from c to d is 6 (along the path cad). But in graph G2 that shortest distance is 1 (along the path cd). Note that graph G3 is the same as that in Example 15.13 on page 292, and graph G4 is the same as that in Example 15.1 on page 285. Figure 15.21 shows the adjacency matrix, the incidence matrix, and the adjacency list for graph G1.
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EULER PATHS AND HAMILTONIAN CYCLES An euler path in a graph is a walk that includes each edge exactly once. An euler cycle is a closed walk that includes each edge exactly once. An eulerian graph is a graph that has an euler cycle. Note that euler paths and cycles need not have distinct vertices, so they are not strict paths. EXAMPLE 15.22 Euler Paths and Cycles
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In the graph in Figure 15.22, the closed walk acedabefdbcfa is an euler cycle. So this is an eulerian graph. Note that every vertex in this graph has degree 4, and its 12 edges are partitioned into three circles. As the Theorem 15.8 reports, each of these two properties will always guarantee that the graph is eulerian.
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Figure 15.21 Adjacency matrix, incidence matrix, and adjacency list
Theorem 15.8 Eulerian Graphs If G is a connected graph, then the following conditions are equivalent: 1. G is eulerian. 2. The degree of each vertex is even. 3. The set of all edges of G can be partitioned into cycles. A hamiltonian path in a graph is a path that includes each vertex exactly once. A hamiltonian cycle is a cycle that includes each vertex exactly once. A hamiltonian graph is a graph that has a hamiltonian cycle. Unfortunately, there is no simple characterization like Theorem 15.8 for hamiltonian graphs. In fact, the problem of finding such a simple characterization is one of the big unsolved problems in computer science. EXAMPLE 15.23 Hamiltonian Graphs
In Figure 15.23, the graph on the left is hamiltonian. The graph on the right is not; it has a hamiltonian path, but no hamiltonian cycle.