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Two directed graphs are said to be isomorphic if (1) their associated undirected graphs are isomorphic, and (2) the directions of their corresponding edges are preserved for some correspondences of (1). In other words, if Gd and Gd are two isomorphic fl-node directed graphs, then we can label their nodes 1, 2 , . . , η and Γ , 2 ' , . . , n', respectively, so that for any i and j , the edge (i,j) is in Gd if, and only if, the edge {i'J') is in Gd9 including the parallel edges if they exist. As an example, consider the directed graphs of fig.

18. T h e geometric diagram of a directed graph. Fig. 19. A sectional subgraph of the directed graph o f fig. 18. The terms subgraph, sectional subgraph, rank, nullity, the complement of a subgraph, and other graph operations outlined in § 3 can similarly be defined for Gd. The only difference is that we use Gd instead of G. F o r example, in fig. 18 the sectional subgraph Gd[FJ defined by the node set Vs= {1, 2, 3, 6, 7} is presented in fig. 19. 25: Associated undirected graph. To every directed graph Gd, there is an associated undirected graph Gu whose node and edge sets are the same as those in Gd except that the directions of the edges (orders in the pairs of Gd are removed.

28: Directed-edge train. If all the edges appearing in a directededge sequence are distinct, the directed-edge sequence is called a directed-edge train. Like the concepts of a path and a circuit, if in addition we require that aJl the nodes in a directed-edge train except the initial and terminal nodes be distinct, we have the concepts of a directed path and a directed circuit. 29: Directed path. 9 Jfc-1. 30: Directed circuit. , are distinct, in this case i1 = ik, is called a directed circuit of length k—l.

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