vb.net barcode printing Figure 15.13 A digraph in Java

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Figure 15.13 A digraph
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Theorem 15.6 If G is a digraph with m edges, then the sum of all outdegrees equals m and the sum of all indegrees equals m. Each edge contributes 1 to the total of all outdegrees and 1 to the total of all indegrees. So each total must be m. The complete digraph a the digraph that has a (directed) edge from every vertex to every other vertex. EXAMPLE 15.14 The Complete Digraph on Six Vertices
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The graph shown in Figure 15.14 is the complete digraph on six vertices. It has 15 double-directed edges, so the total number of (one-way) edges is 30, which is n(n 1) = 6(6 1) = 6(5) = 30.
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Theorem 15.7 The number of edges in the complete digraph on n vertices is n(n 1). By Theorem 15.2 on page 286, there are n(n 1)/2 undirected edges on the corresponding complete undirected graph. That makes n(n 1)/2 double-directed edges, so the total number of (one-way) directed edges must be twice that number.
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Figure 15.14 A complete digraph
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Corollary 15.2 The number of edges in any digraph on n vertices is m
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n(n 1).
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GRAPHS
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Every digraph has an embedded graph, obtained by converting each directed edge into an undirected edge and then removing duplicate edges and loops. Mathematically, this amounts to converting each ordered pair (x, y) of vertices in E into the set {x, y} and then removing all sets of size one (i.e., singletons). EXAMPLE 15.15 The Embedded Graph of a Digraph
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The embedded graph of the digraph in Figure 15.13 is the graph shown in Figure 15.15.
An adjacency matrix for a digraph (V, E) is a two-dimensional boolean array
boolean[][] a;
Figure 15.15 An embedded graph
obtained by ordering the vertices V = {v0 , v1, . . . , vn 1} and then assigning true to a[i][j] if and only if there exists an edge emanating from vertex vi and terminating at vertex vj. EXAMPLE 15.16 An Adjacency Matrix for a Digraph
Figure 15.16 shows the adjacency matrix for the graph in Figure 15.13 on page 292.
Note that the number of true entries in an adjacency matrix for a digraph is equal to the number of edges. Also, as with undirected graphs, different orderings of the vertex set V will result in different adjacency matrices for the same digraph.
Figure 15.16 An adjacent matrix
An incidence matrix for a digraph (V, E) is a two-dimensional integer array
int[][] a;
obtained by ordering the vertices V = {v0 , v1, . . . , vn 1} and the edges E = {e0 , e1, . . . , em 1} and then assigning 1 to a[i][j] and 1 to a[j][i] if there exists an edge emanating from vertex vi and terminating at vertex vj, and assigning 0 everywhere else. EXAMPLE 15.17 An Incidence Matrix for a Digraph
Figure 15.17 shows an incidence matrix for the digraph in Figure 15.13 on page 292. The first row indicates that two edges emanate from vertex a and one edge terminates there. The last 1 is in the row for vertex d and the last column. The only other nonzero entry in that column is the -1 in the row for vertex c, meaning that this edge emanates from vertex d and terminates at vertex c.
Figure 15.17 An incidence matrix
An adjacency list for a digraph (V, E) is a list that contains one element for each vertex in the graph and in which each vertex list element contains a list of the edges that emanate from that vertex. This is the same as the adjacency list for a graph, except that the links are not duplicated unless there are edges going both ways between a pair of vertices.
GRAPHS
[CHAP. 15
EXAMPLE 15.18 An Adjacency List for a Digraph
Figure 15.18 shows the adjacency list for the digraph in Figure 15.13 on page 292. The edge list for vertex a has two elements, one for each of the two edges that emanate from a: ab and ad.
PATHS IN A DIGRAPH A walk from vertex a to vertex b in a digraph is a sequence of edges (a0 a1, a1 a2, . . . , ak 1 ak) where a0 = a and ak = b. As with undirected paths in an undirected graphs, directed paths are usually abbreviated by their Figure 15.18 An adjacency list vertex string: p = a0 a1 a2 . . . ak 1 ak. Either way, we say that the path emanates from (or starts at) vertex a and terminates at (or ends at) vertex b. A walk is closed if it terminates at the same vertex from which it emanates. A path is a walk with all distinct vertices. A cycle is a closed walk with all interval vertices distinct. EXAMPLE 15.19 Directed Paths
In the digraph of Figure 15.13, adcabdc is a walk of length 6 which is not closed. The walk abdcacda is closed, but it is not a cycle because d (and c) are repeated internal vertices. The walk dcab is a path, which is not closed. The walk cabdc is a cycle of length 4, and the walk dcad is a cycle of length 3.
Note that different cycles may traverse the same vertices. For example, adca and cadc are different cycles in the digraph in Figure 15.13. A digraph is strongly connected if there is a path between every pair of vertices. A digraph is weakly connected if its embedded graph is connected. A digraph that is not weakly connected is said to be disconnected. EXAMPLE 15.20 Strongly Connected and Weakly Connected Digraphs
In Figure 15.19, digraph G1 is strongly connected (and therefore also weakly connected). Digraph G2 is weakly connected, but not strongly connected because there is no path that terminates at vertex x. Digraph G3 is disconnected.
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