Discrete Mathematics with Graph Theory (3rd Edition) 340

Discrete Mathematics with Graph Theory (3rd Edition) 340 -...

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338 Solutions to Exercises on edges of the subgraph are all less than the weights on all other edges. In either case, Kruskal's algorithm selects first the edges of the sub graph since these have weights smaller than other edges and there are no circuits among them. 9. (a) If the graph is unweighted, put a weight of 1 on edge e = uv and 2 on every other edge. If the graph is weighted, ensure (by temporarily changing weights if necessary) that the weights of the edges different from e are all larger than the weight of e. In either case, if Prim's algorithm starts at u or v, it will select e first and, if it starts at any other vertex, then as soon as the first of u, v is included in one of the sequence of growing trees (to be specific, suppose this is u and the tree is 'T), edge e will be selected since it is eligible for selection ('T U {e} is also a tree) and e is the edge of least weight in the graph. (b) If the graph is unweighted, put weights of 1 on the edges of the subgraph and 2 on all other edges.
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