(Total) Vector domination for graphs with bounded branchwidth

Given a graph G=(V,E)G=(V,E) of order nn and an nn-dimensional non-negative vector d=(d(1),d(2),…,d(n))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⊆VS⊆V such that every vertex vv in V∖SV∖S (resp., in VV) has at least d(v)d(v) neighbors in SS. The (total) vector domination is a generalization of many dominating set type problems, e.g., the dominating set problem, the kk-tuple dominating set problem (this kk 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 kk, where kk is the size of solution.