Decomposition Of Width Research Articles

Computing the Number of k-Component Spanning Forests of a Graph with Bounded Treewidth

Let G be a graph on n vertices with bounded treewidth. We use $$f_k(G)$$ to denote the number of spanning forests of G with k components. Given a tree decomposition of width at most p of G, we present an algorithm to compute $$f_k(G)$$ for $$k = 1,2, \cdots , n$$ . The running time of our algorithm is $$O((B(p+1))^3pn^3)$$ , where $$B(p+1)$$ is the $$(p+1)$$ -th Bell number.

On the vertex cover $$P_3$$ P 3 problem parameterized by treewidth

Consider a graph G. A subset of vertices, F, is called a vertex cover \(P_t\) (\(VCP_t\)) set if every path of order t contains at least one vertex in F. Finding a minimum \(VCP_t\) set in a graph is is NP-hard for any integer \(t\ge 2\) and is called the \(MVCP_3\) problem. In this paper, we study the parameterized algorithms for the \(MVCP_3\) problem when the underlying graph G is parameterized by the treewidth. Given an n-vertex graph together with its tree decomposition of width at most p, we present an algorithm running in time \(4^{p}\cdot n^{O(1)}\) for the \(MVCP_3\) problem. Moreover, we show that for the \(MVCP_3\) problem on planar graphs, there is a subexponential parameterized algorithm running in time \(2^{O(\sqrt{k})}\cdot n^{O(1)}\) where k is the size of the optimal solution.

Efficient and Constructive Algorithms for the Pathwidth and Treewidth of Graphs

In this paper we give, for all constantsk,l, explicit algorithms that, given a graphG=(V,E) with a tree-decomposition ofGwith treewidth at mostl, decide whether the treewidth (or pathwidth) ofGis at mostk, and, if so, find a tree-decomposition (or path-decomposition) ofGof width at mostk, and that useO(|V|) time. In contrast with previous solutions, our algorithms do not rely on non-constructive reasoning and are single exponential inkandl. This result can be combined with a result of B. Reed [in“Proceedings of the 24th Annual Symposium on Theory of Computing,” pp. 221–228, 1992], yielding explicitO(nlogn) algorithms for the problem, given a graphG, to determine whether the treewidth (or pathwidth) ofGis at mostk, and, if so, to find a tree- (or path-) decomposition of width at mostk(kconstant). Also, H. L. Bodlaender [in“Proceedings of the 25th Annual Symposium on Theory of Computing,” pp. 226–234, 1993] has used the result of this paper to obtain linear time algorithms for these problems. We also show that for all constantsk, there exists a polynomial time algorithm that, when given a graphG=(V,E) with treewidth ≤k, computes the pathwidth ofGand a path-decomposition ofGof minimum width.