Introduction to graph theory by West D.

By West D.

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K' which we conveniently order so that if i < i, then is last visited in a depth-first traversal of G before'l is last visited. , r,-l. In the same way that we defined the parameter P(v) to help in the computational discovery of articulation points in undirected graphs, we define a parameter Q(v) to help in the computational identification of the roots of the strongly connected components of a digraph. Q(v) is defined as follows: r, Q(v) = min ({DFl(v)} U {DFl(v')l(x, v') is in B1 or C, x is a descendant of v and the root, r, of the strongly connected component containing v'is an ancestor of v}) The value of this definition lies in the following theorem.

We remove i1 from 1 and Sl from S and the process is repeated with the new S and the new 1. Finally, S contains no elements and the last edge to be added to T is that defined by the remaining pair of labels in I. , n). The • number of such words is n",,-2 and so the theorem follows. We come now to the general problem of counting the number of spanning trees for an arbitrary multi-graph G.

20 to check whether v'is on the stack or not in one step. This avoids unnecessary enhancement of the complexity through a search of the stack in line 11. Introducing data structures and depth-first searching 31 Fig. 21. 20. (a) /,,--, 3 I 4 , I I I ,~ • , ~ I 8 \ ' ,\ 2 7 I , / ~ I ~ 4/ I \ '" " 7 / 4 \ \ \ 6 r~-7 I I I I / 1/ 5 1 / I / / v DFI 1 I 2 3 3 4 4 5 6 7 8 2 5 6 7 8 Q I I 4 4 4 7 8 1 Vertices inducing a strongly connected component [1,2,8] (ii) 6 5 4 --+ 7 (iii) 3 [6] 3 3 5 4 5 4 -i (iV)EE-§ [7] [3,4, 5] (i) Occurs within DFSSCC(I) after completion of DFSSCC(8) which contains a nested call of DFSSCC(2).

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