We have been given tree lets say T with vertices set V, now try giving an algorithm to find the minimum cardinality subset of vertices W. Given every vertex in the set V has an edge with at least one vertex in set W.
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This can be solved in linear time by a recursive algorithm. Start by selecting a root for the tree, and building a list of children for each vertex. We say that a set of vertices "covers" a subtree if every vertex in the subtree (except possibly the root of the subtree) is either in the set or adjacent to a member of the set.
The algorithm takes as input a vertex v in this tree, and returns a tuple of three numbers, which are the minimum cardinalities of subsets covering v's subtree which respectively (a) include the vertex v, (b) do not include v but do include at least one of v's children, and (c) include neither v nor any of v's children.
The base case of the algorithm is to return the tuple (1, 0, 0) when the input v is a leaf node. In the recursive case, the tuple (a, b, c) can be computed from the results of recursively calling the algorithm on v's children. Rather than solve the whole problem for you, I will leave it to you to figure out how to do this.
The final answer is then min(a, b, c+1) where a, b, c are the results from calling the algorithm on the root node.
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