Figure 12
Illustration of directed paths and the FTC. The first transitive component is coloured red in (b, c). Since the graph in (a) is strongly connected, its first transitive component is the whole graph. (a) Illustration of directed paths. The purple arrows form a directed path from the purple node to the green node. The green arrows form a directed path in the other direction. (b) If a directed path exists from a node A to a node B, then we say that A ≪ B, which is a partial ordering among the nodes in a graph. In this graph, A ≪ B is true for each pair of blue nodes, and also for each pair of red nodes. It is also true that A r ≪ B b for each red node A r and each blue node B b . However, it is not true that A b ≪ B r for any blue node A b and red node B r . The red nodes, thus, form a minimum for the partial ordering and are the FTC. (c) The first transitive component may also be disconnected. The definition of the FTC ensures that it is either strongly connected as in (a, b) or disconnected as is the case in (c).