help in the Donalds B. Johnson's algorithm, i cannot understand the pseudo code (PART II)

Posted by Pitelk on Stack Overflow See other posts from Stack Overflow or by Pitelk
Published on 2010-05-30T18:50:47Z Indexed on 2010/05/30 18:52 UTC
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Hi all , i cannot understand a certain part of the paper published by Donald Johnson about finding cycles (Circuits) in a graph.

More specific i cannot understand what is the matrix Ak which is mentioned in the following line of the pseudo code :

Ak:=adjacency structure of strong component K with least vertex in subgraph of G induced by {s,s+1,....n};

to make things worse some lines after is mentins " for i in Vk do " without declaring what the Vk is...

As far i have understand we have the following: 1) in general, a strong component is a sub-graph of a graph, in which for every node of this sub-graph there is a path to any node of the sub-graph (in other words you can access any node of the sub-graph from any other node of the sub-graph)

2) a sub-graph induced by a list of nodes is a graph containing all these nodes plus all the edges connecting these nodes. in paper the mathematical definition is " F is a subgraph of G induced by W if W is subset of V and F = (W,{u,y)|u,y in W and (u,y) in E)}) where u,y are edges , E is the set of all the edges in the graph, W is a set of nodes.

3)in the code implementation the nodes are named by integer numbers 1 ... n.

4) I suspect that the Vk is the set of nodes of the strong component K.

now to the question. Lets say we have a graph G= (V,E) with V = {1,2,3,4,5,6,7,8,9} which it can be divided into 3 strong components the SC1 = {1,4,7,8} SC2= {2,3,9} SC3 = {5,6} (and their edges)

Can anybody give me an example for s =1, s= 2, s= 5 what if going to be the Vk and Ak according to the code?

The pseudo code is in my previous question in

and the paper can be found at

thank you in advance

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