### Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 80-89 |

Number of pages | 10 |

Journal | Discrete Applied Mathematics |

Volume | 207 |

DOIs | |

Publication status | Published - Jul 10 2016 |

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### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Discrete Applied Mathematics*,

*207*, 80-89. https://doi.org/10.1016/j.dam.2016.03.002

**(Total) Vector domination for graphs with bounded branchwidth.** / Ishii, Toshimasa; Ono, Hirotaka; Uno, Yushi.

Research output: Contribution to journal › Article

*Discrete Applied Mathematics*, vol. 207, pp. 80-89. https://doi.org/10.1016/j.dam.2016.03.002

}

TY - JOUR

T1 - (Total) Vector domination for graphs with bounded branchwidth

AU - Ishii, Toshimasa

AU - Ono, Hirotaka

AU - Uno, Yushi

PY - 2016/7/10

Y1 - 2016/7/10

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84979457604&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84979457604&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2016.03.002

DO - 10.1016/j.dam.2016.03.002

M3 - Article

AN - SCOPUS:84979457604

VL - 207

SP - 80

EP - 89

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

ER -