Chapter 10: Q15RE (page 737)
Give examples of at least two problems that can be solved by finding the shortest path in a weighted graph
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Suppose that a connected graph \(G\) has \(n\) vertices and vertex connectivity \(\kappa \left( G \right) = k\). Show that \(G\) must have at least \(\left\lceil {\frac{{kn}}{2}} \right\rceil \) edges.
How many nonisomorphic subgraphs does \({{\rm{K}}_{\rm{3}}}\)have?
Show that g(5)= 1. That is, show that all pentagons can be guarded using one point. (Hint:Show that there are either 0, 1, or 2 vertices with an interior angle greater than 180 degrees and that in each case, one guard suffices.
Construct the dual graph for the map shown:
Then find the number of colors needed to color the map so that no two adjacent regions have the same color.
Extend Dijkstra’s algorithm for finding the length of a shortest path between two vertices in a weighted simple connected graph so that the length of a shortest path between the vertex a and every other vertex of the graph is found.
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