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8. Figure 11.10 shows an undirected graph representing a section of a department store. The vertices indicate where cashiers are located; the edges denote unblocked aisles between cashiers. The department store wants to set up a security system where (plainclothes) guards are placed at certain cashier locations so that each cashier either has a guard at his or her location or is only one aisle away from a cashier who has a guard. What is the smallest number of guards needed?

I think {a}→{a,d,e,b},{i}→{i,h,e,j},{g}→{f,c,k,g}, where {x}→{T} is meant that we put a guard at x that covers all elements in T. Now we have to prove that we cant get what we desire with a smaller number of guards. Therefore we take 2 guards, and if not possible with 2 its not with one, as well. To cover the elements in the set {d,e,h} we requires 2 guards from {d,e,h,I,a}. If it is the 2nd case then no guards from this set can cover for instance g, Therefore we would have once again at least 3 guards.

11. Let G be a graph that satisfies the condition in Exercise 10. (a) Must G be loop-free? (b) Could G be a multigraph? (c) If G has n vertices, can we determine how many edges it has?

Eliminating an edge from the loop will still lead to a connected graph, therefore it is a necessity. It cant be since eliminating from a vertex with degree more than 1 will still result in a connected graph. It can be verified that these sorts of graphs are tress, therefore the number of edges is n-1.

15. For the undirected graph in Fig. 11.12, find and solve a recurrence relation for the number of closed v-v walks of length n ≥ 1, if we allow such a walk, in this case, to contain or consist of one or more loops.

Beginning with v we can continue to v on the cirle once again or to go to w and then back to v. Therefore beginning with v then we have an arbitrary number of

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