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vb.net code to generate barcode Figure 15.19 Strongly connected and weakly connected components in Java
Figure 15.19 Strongly connected and weakly connected components Recognizing UPC  13 In Java Using Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications. Encode EAN13 In Java Using Barcode creation for Java Control to generate, create EAN13 image in Java applications. CHAP. 15] Scanning EAN 13 In Java Using Barcode decoder for Java Control to read, scan read, scan image in Java applications. Painting Bar Code In Java Using Barcode generator for Java Control to generate, create bar code image in Java applications. GRAPHS
Bar Code Reader In Java Using Barcode recognizer for Java Control to read, scan read, scan image in Java applications. Create GS1  13 In Visual C#.NET Using Barcode printer for .NET Control to generate, create European Article Number 13 image in .NET applications. 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 EAN13 Generator In .NET Using Barcode drawer for ASP.NET Control to generate, create EAN13 image in ASP.NET applications. European Article Number 13 Encoder In Visual Studio .NET Using Barcode encoder for .NET framework Control to generate, create EAN13 image in .NET applications. 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. Encoding GTIN  13 In Visual Basic .NET Using Barcode encoder for .NET framework Control to generate, create GTIN  13 image in .NET framework applications. Encoding Code 128 Code Set C In Java Using Barcode creator for Java Control to generate, create Code128 image in Java applications. Figure 15.20 Embedded graphs
Barcode Creator In Java Using Barcode generation for Java Control to generate, create barcode image in Java applications. Linear 1D Barcode Maker In Java Using Barcode maker for Java Control to generate, create Linear 1D Barcode image in Java applications. 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. Making I2/5 In Java Using Barcode maker for Java Control to generate, create ITF image in Java applications. Creating Code 128A In ObjectiveC Using Barcode printer for iPad Control to generate, create Code 128B image in iPad applications. 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 EAN / UCC  13 Drawer In None Using Barcode maker for Excel Control to generate, create UCC  12 image in Microsoft Excel applications. Encode Code128 In None Using Barcode generation for Software Control to generate, create Code 128B image in Software applications. 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. UPC A Decoder In C# Using Barcode recognizer for .NET Control to read, scan read, scan image in Visual Studio .NET applications. UPC A Creation In .NET Framework Using Barcode drawer for .NET Control to generate, create UPC A image in Visual Studio .NET applications. GRAPHS
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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.

