## Algorithmic aspects of domination in graphs (1998)

Venue: | in: Handbook of Combinatorial Optimization (D.-Z. Du and P. M. Pardalos eds |

Citations: | 6 - 2 self |

### BibTeX

@INPROCEEDINGS{Chang98algorithmicaspects,

author = {Gerard J. Chang},

title = {Algorithmic aspects of domination in graphs},

booktitle = {in: Handbook of Combinatorial Optimization (D.-Z. Du and P. M. Pardalos eds},

year = {1998},

pages = {339--405}

}

### OpenURL

### Abstract

Graph theory was founded by Euler [78] in 1736 as a generalization to the solution of the famous problem of the Könisberg bridges. From 1736 to 1936, the same concept as graph, but under different names, was used in various scientific fields as models of real world problems, see the historic book by Biggs, Lloyd and Wilson [19]. This chapter intents to survey the domination problem in graph theory, which is a natural model for many location problems in operations research, from an algorithmic point of view. 1

### Citations

2857 |
Dynamic Programming
- Bellman
- 1957
(Show Context)
Citation Context ...ue minimum independent dominating set. 3.3 Dynamic programming for trees Dynamic programming is a powerful method for solving many discrete optimization problems; see the books by Bellman and Dreyfus =-=[11]-=-, Dreyfus and Law [74] and Nemhauser [147]. The main idea of the dynamic programming approach for domination is to turn the ‘bottom-up’ labeling method into ‘topdown’. Now a specific vertex u is chose... |

158 |
Linear time algorithms for NP-hard problems restricted to partial k-trees
- Arnborg, Proskurowski
- 1989
(Show Context)
Citation Context ...m a k-tree by adding a new vertex adjacent to a clique of k vertices is a k-tree. Partial k-trees are subgraphs of k-trees. For results on variations of domination in k-trees and partial k-trees, see =-=[1, 56, 83, 153, 159, 169, 170]-=-. 4 Interval graphs 4.1 Interval orderings of interval graphs Recall that an interval graph is the intersection graph of a family of intervals in the real line. In many papers, people design algorithm... |

83 | Distancehereditary graphs
- Bandelt, Mulder
- 1989
(Show Context)
Citation Context ... same distance in every connected induced subgraph. Distance-hereditary graphs were introduced by Howorka [93]. The characterization and recognition of distance-hereditary graphs have been studied in =-=[8, 67, 68, 92, 93]-=-. Distance-hereditary graphs are parity graphs [32] and contains all cographs [57, 64]. The hanging hu of a connected graph G = (V, E) at a vertex u ∈ V is the collection of sets L0(u), L1(u), . . . ,... |

56 |
Graph Theory 1736-1936
- Biggs, Lloyd, et al.
- 1976
(Show Context)
Citation Context ...dges. From 1736 to 1936, the same concept as graph, but under different names, was used in various scientific fields as models of real world problems, see the historic book by Biggs, Lloyd and Wilson =-=[19]-=-. This chapter intents to survey the domination problem in graph theory, which is a natural model for many location problems in operations research, from an algorithmic point of view. 1 Introduction D... |

25 | Dually chordal graphs - Brandstädt, Chepoi, et al. - 1998 |

24 | The algorithmic use of hypertree structure and maximum neighbourhood orderings - Brandstädt, Chepoi, et al. - 1998 |

22 |
Dominating sets in chordal graphs
- Booth, Johnson
- 1982
(Show Context)
Citation Context ...on to a bipartite graph. Corollary 5.5 The total and the connected domination problems are NP-complete for bipartite graphs. There are many other NP-complete results for variations of domination, see =-=[9, 14, 21, 56, 63, 66, 84, 102, 110, 145, 180, 181]-=- and [86, 115, 116]. Most of the proofs are more or less similar to the above two. There are only very few are proved by different ways. As an example, Booth and Johnson [21] proved that the dominatio... |

17 |
Efficient dominating sets in graphs
- Bange, Barkauskas, et al.
- 1988
(Show Context)
Citation Context ...j) + γ(vi); γ(vj) ← min{γ 1 (vj), γ 0 (vj)}; end do; γ(T, w, w) ← γ(vn). The dynamic programming method was also used in several papers for solving variations of the domination problems in trees, see =-=[9, 84, 102, 107, 162, 180]-=-. 3.4 Primal-dual approach for trees The most beautiful method used in domination may be the primal-dual approach. In this method, besides the original domination problem we also consider the followin... |

14 | An Efficient Algorithm for Maxdominance with Applications,” Algorithmica - Atallah, Kosaraju - 2000 |

8 |
Total domination in interval graphs
- Bertossi
- 1986
(Show Context)
Citation Context ... larger class of graphs, strongly chordal graphs (see Section 6.3). 4.3 Weighted independent domination in interval graphs There are many algorithms for variants of domination in interval graphs, see =-=[15, 17, 40, 42, 44, 155, 156]-=-. Among them, Ramalingam and Pandu Rangan [156] gave a unified approach to the weighted independent domination, the weighted domination, the weighted total domination and the weighted connected domina... |

8 |
Total domination and irredundance in weighted interval graphs
- Bertossi, Gori
- 1988
(Show Context)
Citation Context ... larger class of graphs, strongly chordal graphs (see Section 6.3). 4.3 Weighted independent domination in interval graphs There are many algorithms for variants of domination in interval graphs, see =-=[15, 17, 40, 42, 44, 155, 156]-=-. Among them, Ramalingam and Pandu Rangan [156] gave a unified approach to the weighted independent domination, the weighted domination, the weighted total domination and the weighted connected domina... |

8 | A linear-time algorithm for connected r-domination and Steiner tree on distance-hereditary graphs
- Brandstädt, Dragan
- 1998
(Show Context)
Citation Context ...N ′ (v). 9.2 Weighted connected domination D’Atri and Moscarini [67] gave O(|V ||E|) algorithms for connected domination and Steiner tree problems in distance-hereditary graphs. Brandstädt and Dragan =-=[26]-=- presented a linear-time algorithm for the connected r-domination and Steiner tree problems in distance-hereditary graphs. In this section, we present a linear-time algorithm given by Yeh and Chang [1... |

6 | Dominating sets and domatic number of circular arc graphs - Bonuccelli - 1985 |

5 |
Rangan, Connected domination and Steiner set on asteroidal triple-free graphs
- Balakrishnan, A
- 1993
(Show Context)
Citation Context ...n O(n+m) algorithm computing a dominating path for connected AT-free graphs (see also [62]). An O(n 3 ) algorithm computing a minimum connected dominating set for connected AT-free graphs is given in =-=[7]-=-. An O(n + m) algorithm computing a minimum connected dominating set in connected AT-free graphs with diameter greater than three is given in [59]. 9 Distance-hereditary graphs 9.1 Hangings of distanc... |

5 | Finding efficient dominating sets in oriented graphs - Barkauskas, Host - 1993 |

5 | Clique r-domination and clique r-packing problems on dually chordal graphs - Chepoi, Dragan |

5 |
On the restriction of some NP-complete graph problems to permutation graphs
- Brandstädt, Kratsch
- 1985
(Show Context)
Citation Context ... and Even in 1971 [152, 77]. Since that time quite a few polynomial time algorithms have been constructed on permutation graphs. For example, Atallah, Manacher and Urrutia [6], Brandstädt and Kratsch =-=[28]-=-, Farber and Keil [82] and Tsai and Hsu [171] have constructed polynomial domination and independent domination algorithms, Brandstädt and Kratsch [28] and Colbourn and Stewart [54] have constructed p... |

4 | Rangan. Connected domination and Steiner set on weighted permutation graphs - Arvind, Pandu - 1992 |

4 | Dynamic Programming on Intervals - Asano |

4 |
Finding a minimum independent dominating set in a permutation graph
- Atallah, Manacher, et al.
- 1988
(Show Context)
Citation Context ...introduced by Pnueli, Lempel and Even in 1971 [152, 77]. Since that time quite a few polynomial time algorithms have been constructed on permutation graphs. For example, Atallah, Manacher and Urrutia =-=[6]-=-, Brandstädt and Kratsch [28], Farber and Keil [82] and Tsai and Hsu [171] have constructed polynomial domination and independent domination algorithms, Brandstädt and Kratsch [28] and Colbourn and St... |

3 |
E cient algorithms in cocomparability and trapezoid graphs, manuscript
- Arvind, Breu, et al.
- 1996
(Show Context)
Citation Context ...nation and the weighted independent domination problems in cocomparability graphs. These algorithms are designed by dynamic programming using cocomparability orderings. Breu and Kirkpatrick [30] (see =-=[2]-=-) improved this by giving O(nm 2 )-time algorithms for the domination and the total domination problems and an O(n 2.376 )-time algorithm for the weighted independent domination problem in cocomparabi... |

3 |
Dominating sets for split and bipartite
- Bertossi
- 1984
(Show Context)
Citation Context ...on to a bipartite graph. Corollary 5.5 The total and the connected domination problems are NP-complete for bipartite graphs. There are many other NP-complete results for variations of domination, see =-=[9, 14, 21, 56, 63, 66, 84, 102, 110, 145, 180, 181]-=- and [86, 115, 116]. Most of the proofs are more or less similar to the above two. There are only very few are proved by different ways. As an example, Booth and Johnson [21] proved that the dominatio... |

3 |
Independent domination in trees
- Beyer, Proskurowski, et al.
- 1977
(Show Context)
Citation Context ...(vn), γ 0 (vn)}. The advantage of the dynamic programming method is that it also works for problems whose solutions have no local property. As an example, Beyer, Proskurowski, Hedetniemi and Mitchell =-=[18]-=- solved the independent domination problem by this method. Moreover, the method can be used to solve the vertex-edge-weighted cases. The following derivation for the vertex-edgeweighted domination in ... |

3 | The computational complexity of feedback vertex set, Hamiltonian circuit, dominating set, Steiner tree, and bandwidth on special perfect graphs - Brandstädt - 1987 |

2 | Efficient sets in graphs - Bernhard, Hedetniemi, et al. - 1993 |

2 |
On the domatic number of interval graphs
- Bertossi
- 1988
(Show Context)
Citation Context ...≤ n} 4.2 Domatic numbers of interval graphs Another interesting usage of the interval ordering is the following rewriting for the result on the domatic numbers of interval graphs obtained by Bertossi =-=[16]-=-. He transformed the domatic number problem on interval graphs to a network flow problem as follows. This is a typical example of the transformation method. We first describe Bertossi’s method. Suppos... |

2 | adt, Domination and the Use of Maximum Neighbourhoods - Behrendt, Brandst - 1992 |

1 | A class of polynomially solvable set covering problems - Bertolazzi, Sassono - 1988 |