The Minimum-Cost Dominating Set Problem is specified by an undirected graph G = (V, E) and costs c (v) on the nodes v ε V .A subset S ⊂ V is said to be a dominating set if all nodes u ε V – S have an edge (u, v) to a node v in S. (Note the difference between dominating sets and vertex covers: in a dominating set, it is fine to have an edge (u, v) with neither u nor v in the set S as long as both u and v have neighbors in S.)
(a) Give a polynomial-time algorithm for the Dominating Set Problem for the special case in which G is a tree.
(b) Give a polynomial-time algorithm for the Dominating Set Problem for the special case in which G has tree-width 2, and we are also given a tree decomposition of G with width 2.
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