Question

2) Let G ME) be an undirected Graph. A node cover of G is a subset U of the vertex set V such that every edge in E is incident to at least one vertex in U. A minimum node cover MNC) is one with the lowest number of vertices. For example {1,3,5,6is a node cover for the following graph, but 2,3,5} is a min node cover Consider the following Greedy algorithm for this problem: Algorithm NodeCover (V,E) Uempty While E <> empty Find a vertex v of maximum degree v=v-v E = E - la.x} for all {Χ.Χ} such that x = v or y = v a) b) Give a graph example (other than the above graph) where this algorithm generates a MNC Give a graph counter-example to prove that this algorithm does not always generate the MNC.

Answer 1

a) Consider the graph with 4 vertices, where . Then the algorithm will simply choose as the minimal set cover, which is indeed correct, as all edges are incident on .

b) Consider the graph with 7 vertices where .

In this case the algorithm will pick first, after which the remaining edges will be . Now 3 more vertices need to be picked for the vertex cover, which means a total of 4 vertices in the cover.

While, the optimal solution is picking , which is of size 3, better than the one chosen by the greedy algorithm.

Comment in case of any doubts.