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vb.net barcode printing Figure 15.19 Strongly connected and weakly connected components in Java
Figure 15.19 Strongly connected and weakly connected components Scan Data Matrix In Java Using Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications. Data Matrix 2d Barcode Encoder In Java Using Barcode creator for Java Control to generate, create DataMatrix image in Java applications. CHAP. 15] Recognize Data Matrix 2d Barcode In Java Using Barcode reader for Java Control to read, scan read, scan image in Java applications. Encoding Bar Code In Java Using Barcode maker for Java Control to generate, create bar code image in Java applications. GRAPHS
Scan Barcode In Java Using Barcode decoder for Java Control to read, scan read, scan image in Java applications. DataMatrix Generator In Visual C#.NET Using Barcode maker for VS .NET Control to generate, create Data Matrix ECC200 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 Create ECC200 In .NET Using Barcode encoder for ASP.NET Control to generate, create Data Matrix 2d barcode image in ASP.NET applications. Generate DataMatrix In Visual Studio .NET Using Barcode encoder for Visual Studio .NET Control to generate, create Data Matrix 2d barcode 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. Make Data Matrix 2d Barcode In Visual Basic .NET Using Barcode generation for .NET Control to generate, create Data Matrix image in VS .NET applications. Drawing 1D In Java Using Barcode maker for Java Control to generate, create Linear 1D Barcode image in Java applications. Figure 15.20 Embedded graphs
Barcode Creation In Java Using Barcode generator for Java Control to generate, create bar code image in Java applications. ANSI/AIM Code 39 Encoder In Java Using Barcode maker for Java Control to generate, create Code 39 Extended 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. Create UPCE Supplement 2 In Java Using Barcode encoder for Java Control to generate, create UPC  E1 image in Java applications. Drawing 2D Barcode In Visual Basic .NET Using Barcode generator for .NET framework Control to generate, create Matrix Barcode image in VS .NET 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 Printing Bar Code In VB.NET Using Barcode generator for .NET Control to generate, create bar code image in .NET applications. GTIN  12 Printer In .NET Using Barcode encoder for ASP.NET Control to generate, create GTIN  12 image in ASP.NET 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. Code 128 Generator In .NET Using Barcode drawer for ASP.NET Control to generate, create ANSI/AIM Code 128 image in ASP.NET applications. Print Data Matrix ECC200 In Java Using Barcode printer for Android Control to generate, create DataMatrix image in Android 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.

