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Extra resources for Network programming

Example text

In each step, one new node is added to the set X, if it is j, its immediate predecessor is an adjacent node which is already in X, and the line joining it to j is added as a new arc to the set A(X). This implies that (X, A(X)) is always connected and that |A(X)| = |X|− 1. 6, (X, A(X)) is always a tree spanning the nodes in X. Also, every node in X has a unique immediate predecessor, except the node p which has no predecessor, so, these predecessor labels make (X, A(X)) a rooted tree with node p as the root node.

18. 6 may not be apparent, but taking the partition N1 = {S1 , S2 , W1 , W2 , W3 }, N2 = {P1 , P2 , P3 } it can easily be verified to be so. 18: A bipartite network. Here ({1,2,3}, {4,5,6,7}) is the bipartition. 2 A network G = (N , A) is bipartite iﬀ it contains no odd cycles. Proof Clearly a network is bipartite iﬀ each of its connected components is. Without any loss of generality we assume that G is connected, because otherwise the proof can be repeated for each connected component separately.

Otherwise, select a node from X to scan. Step 2 Scanning a node Let i be the node to be scanned, delete it from X. Find J = {j : j ∈ Y and j is joined to i by an in-tree arc }. Nodes in J are the sons or children or immediate successors of i, and i is their parent or immediate predecessor. If J = ∅, i has no children, define its successor index S(i) = ∅. If J = ∅, arrange the nodes in J in some order, say j1 , . . , jr . Then j1 is the eldest child of i. jp is an elder brother of jq (and jq is an younger brother of jp ) if p < q.