(Total) Vector Domination for Graphs with Bounded Branchwidth
Given a graph G = (V,E) of order n and an n-dimensional non-negative vector d = (d(1),d(2),…,d(n)), called demand vector, the vector domination (resp., total vector domination) is the problem of finding a minimum S ⊆ V such that every vertex v in V ∖ S (resp., in V) has at least d(v) neighbors in S. The (total) vector domination is a generalization of many dominating set type problems, e.g., the dominating set problem, the k-tuple dominating set problem (this k is different from the solution size), and so on, and its approximability and inapproximability have been studied under this general framework. In this paper, we show that a (total) vector domination of graphs with bounded branchwidth can be solved in polynomial time. This implies that the problem is polynomially solvable also for graphs with bounded treewidth. Consequently, the (total) vector domination problem for a planar graph is subexponential fixed-parameter tractable with respect to k, where k is the size of solution.
KeywordsPlanar Graph Time Algorithm Dynamic Programming Algorithm Minimum Vector Solution Size
Unable to display preview. Download preview PDF.
- 6.Chapelle, M.: Parameterized complexity of generalized domination problems on bounded tree-width graphs. arXiv preprint arXiv:1004.2642 (2010)Google Scholar
- 13.Demaine, E.D., Fomin, F.V., Hajiaghayi, M., Thilikos, D.M.: Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs. Journal of the ACM (JACM) 52, 866–893 (2005)Google Scholar
- 17.Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness, Cornell University, Mathematical Sciences Institute (1992)Google Scholar
- 23.Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Domination in graphs: advanced topics, vol. 40. Marcel Dekker (1998)Google Scholar
- 24.Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of domination in graphs. Marcel Dekker (1998)Google Scholar
- 29.Robertson, N., Seymour, P.D.: Graph minors. XIII. the disjoint paths problem, Journal of Combinatorial Theory, Series B 63, 65–110 (1995)Google Scholar