Problem Set 11 Solutions

We will remove vertices from a given graph G of treewidth k in a good elimination ordering, and we can assume w.l.o.g. that G is connected. Some cliques in a current graph will be active cliques. For each active clique C and for each subset S of vertices in C, we store information if there exists a vertex cover of the original graph G that chooses exactly vertices in S out of the vertices in C. This is possible if and only if for each edge (in G) between a pair of vertices in C, at least one of them is chosen. Moreover, if this is possible, we store as well a specific subset of some removed vertices (details follow). Note that the size of any clique during the elimination is bounded by k + 1 (actually active cliques are always of size at most k).