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
respectto k, where k is the size of solution.Comment: 16 page