By Richard S.H. Mah, Howard Brenner, Andreas Acrivos and James E. Bailey (Auth.)
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Hence, it is true for trees with any number of vertices. Although we have chosen to define a tree as a minimally connected graph, it can in fact be defined alternatively by any two of the three following conditions: 1. 2. 3. It is connected; It is circuitiess; It contains Ν -1 edges. Trees are important in graph theory, because they arise naturally in many applications and because tiiey provide die clue to the understanding of more complex graphs and structures. We shall first consider its properties as a graph-theoretic entity and then study its relatioship to the more complex structures.
We must also refine our definition of connectedness. A digraph is said to be strongly connected, if there exists a directed path between every pair of its vertices. It is weakly connectedif the corresponding graph is connected. Each maximally (weakly or strongly) connected subgraph of a digraph will still be referred to as a component. But a maximally strongly connected subgraph of a digraph will be called a strong component or fragment. In Fig. 2-20(a) edges 1 to 6 and associated vertices form a strong component, but the graph as a whole is weakly connected.
Graphs and Dignq>hs Chap. 2 • 37 · (a) (b) Fig. 2-18 A separable gri^h and its biconnected components. A connected graph G may also be disconnected by the removal of one or more vertices and their associated edges. In Fig. 2-16(a) the removal of vertex b and die associated edges 1,2,3 and 4 will disconnected the graph. Such a vertex is called a cut vertex, and a graph which may be disconnected 38 Chemical Process Structures Chap. 2 by the removal of a single vertex (and its associated edges) is said to be separable.