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Given a graph $G=(V,E)$ of order $n$ and an $n$-dimensional non-negativevector $d=(d(1),d(2),\ldots,d(n))$, called demand vector, the vector domination(resp., total vector domination) is the problem of finding a minimum$S\subseteq V$ such that every vertex $v$ in $V\setminus S$ (resp., in $V$) hasat least $d(v)$ neighbors in $S$. The (total) vector domination is ageneralization of many dominating set type problems, e.g., the dominating setproblem, the $k$-tuple dominating set problem (this $k$ is different from thesolution size), and so on, and its approximability and inapproximability havebeen studied under this general framework. In this paper, we show that a(total) vector domination of graphs with bounded branchwidth can be solved inpolynomial time. This implies that the problem is polynomially solvable alsofor graphs with bounded treewidth. Consequently, the (total) vector dominationproblem for a planar graph is subexponential fixed-parameter tractable withrespectto $k$, where $k$ is the size of solution.

(Total) Vector Domination for Graphs with Bounded Branchwidth