## Dynamic Algorithms for Graphs with Treewidth 2 (1994)

Venue: | in Proc. 19th Workshop on Graph-Theoretic Concepts in Comp. Sc. (WG'93), LNCS 790 |

Citations: | 10 - 3 self |

### BibTeX

@INPROCEEDINGS{Bodlaender94dynamicalgorithms,

author = {Hans L. Bodlaender},

title = {Dynamic Algorithms for Graphs with Treewidth 2},

booktitle = {in Proc. 19th Workshop on Graph-Theoretic Concepts in Comp. Sc. (WG'93), LNCS 790},

year = {1994},

pages = {112--124},

publisher = {Springer Verlag}

}

### OpenURL

### Abstract

In this paper, we consider algorithms for maintaining treedecompositions with constant bounded treewidth under edge and vertex insertions and deletions for graphs with treewidth at most 2 (also called: partial 2-trees), and for almost trees with parameter k. Each operation can be performed in O(log n) time. For a large number of graph decision, optimization and counting problems, information can be maintained using O(log n) time per update, such that queries can be resolved in O(log n) or O(1) time. Similar results hold for the classes of almost trees with parameter k, for fixed k.

### Citations

366 |
Graph minors. II. Algorithmic aspects of treewidth
- Robertson, Seymour
- 1986
(Show Context)
Citation Context ...found in section 6. 2 Definitions and preliminary results The notion of tree-width and tree-decomposition were introduced by Robertson and Seymour in their series of fundamental papers on graph minors=-=[16]-=-. Definition 1. A tree-decomposition of a graph G = (V; E) is a pair (fX i j i 2 Ig; T = (I ; F )) with fX i j i 2 Ig a collection of subsets of vertices, and T a tree, such that -- S i2I X i = V . --... |

232 | Easy problems for tree-decomposable graphs
- Arnborg, Lagergren, et al.
- 1991
(Show Context)
Citation Context ...s. A powerful language to express many linear time solvable problems on graphs with given tree-decomposition of bounded treewidth is the Monadic Second Order Logic (MSOL) and its extensions. See e.g. =-=[2, 6]-=-. Theorem 3. (i) If a problem P is finite state, then P is td-open. (ii) If a problem P is expressible in extended monadic second order logic, then P is td-open. Proof. (i) The basic idea is: ff(i) co... |

195 | A linear time algorithm for finding treedecompositions of small treewidth
- Bodlaender
(Show Context)
Citation Context ...r graphs. Many problems become linear time solvable for graphs, given together with a tree-decomposition with treewidth bounded by some constant k. (Such a treedecomposition can be found in O(n) time =-=[4]-=- (see also [13, 15]).) These problems include many well known NP-complete problems, like Hamiltonian Circuit, Independent Set, Graph Coloring, etc., counting problems like How many Hamiltonian circuit... |

120 |
Parallel tree contraction and its applications
- Miller, Reif
- 1985
(Show Context)
Citation Context ...G = fdele G , inse G , delv G , insv G , compr G , subdiv G g, and denote OP G = OP \Gamma G [ fcontr G g. We base our approach on applying parallel tree-contraction, as introduced by Miller and Reif =-=[14]-=-. In [3] tree-contraction was used in the first proof of membership in NC of many problems on graphs with constant bounded treewidth. (Later results, especially those of Lagergren [12] gave very large... |

65 |
Automatic Generation of Linear-Time Algorithms from Predicate Calculus
- Borie, Parker, et al.
(Show Context)
Citation Context ...s. A powerful language to express many linear time solvable problems on graphs with given tree-decomposition of bounded treewidth is the Monadic Second Order Logic (MSOL) and its extensions. See e.g. =-=[2, 6]-=-. Theorem 3. (i) If a problem P is finite state, then P is td-open. (ii) If a problem P is expressible in extended monadic second order logic, then P is td-open. Proof. (i) The basic idea is: ff(i) co... |

56 |
Finding approximate separators and computing tree width quickly
- Reed
- 1992
(Show Context)
Citation Context ... problems become linear time solvable for graphs, given together with a tree-decomposition with treewidth bounded by some constant k. (Such a treedecomposition can be found in O(n) time [4] (see also =-=[13, 15]-=-).) These problems include many well known NP-complete problems, like Hamiltonian Circuit, Independent Set, Graph Coloring, etc., counting problems like How many Hamiltonian circuits does G have? and ... |

48 |
A data structure for dynamically maintaining rooted trees
- Frederickson
- 1997
(Show Context)
Citation Context ... of graph problems that can be queried with their approach is much smaller than the class, dealt with in the present paper. The techniques used in [8] and this paper are quite different. Frederickson =-=[10, 11]-=- found independently similar results for trees, forests, and k-terminal trees under several operations, including label changes. The technique in this paper for maintaining trees and forests is very s... |

47 |
Finite automata, bounded treewidth, and well-quasiordering
- Abrahamson, Fellows
- 1993
(Show Context)
Citation Context ... to the problem that is to be solved can be determined in O(1) time. In case each ff(i) contains only a constant bounded number of bits, we call the problem finite state (after Abrahamson and Fellows =-=[1]-=-.) Next, we review some useful results on the structure of graphs with treewidth at most 2. A graph has treewidth at most 1, if and only if it is a forest. Consider a graph G = (V; E) with treewidth a... |

28 | NC-algorithms for graphs with small treewidth
- Bodlaender
- 1989
(Show Context)
Citation Context ...Coloring, etc., counting problems like How many Hamiltonian circuits does G have? and some PSPACE-complete problems. Also, these problems when restricted to graphs with bounded treewidth belong to NC =-=[3, 7, 12]-=-. We consider the problem of solving these problems on graphs with treewidth at most 2 that change dynamically. We allow the following operations: insertion and deletion of isolated vertices, insertio... |

20 | Algorithms finding tree-decompositions of graphs
- MATOU3EK, THOMAS
- 1991
(Show Context)
Citation Context ... problems become linear time solvable for graphs, given together with a tree-decomposition with treewidth bounded by some constant k. (Such a treedecomposition can be found in O(n) time [4] (see also =-=[13, 15]-=-).) These problems include many well known NP-complete problems, like Hamiltonian Circuit, Independent Set, Graph Coloring, etc., counting problems like How many Hamiltonian circuits does G have? and ... |

17 |
Efficient parallel algorithms for tree-decomposition and related problems
- LAGERGREN
- 1990
(Show Context)
Citation Context ...Coloring, etc., counting problems like How many Hamiltonian circuits does G have? and some PSPACE-complete problems. Also, these problems when restricted to graphs with bounded treewidth belong to NC =-=[3, 7, 12]-=-. We consider the problem of solving these problems on graphs with treewidth at most 2 that change dynamically. We allow the following operations: insertion and deletion of isolated vertices, insertio... |

14 |
A simple linear time algorithm for triangulating three-colored graphs
- Bodlaender, Kloks
- 1993
(Show Context)
Citation Context ... a simple cycle, and then iteratively add zero or more cycles, each new cycle sharing exactly one edge with the part of the component made so far. (This characterization is due to Ton Kloks. See e.g. =-=[5]-=-.) We say an edge (v; w) in a completed graph H is a base edge, if, besides the edge (v; w), there are two other vertex disjoint paths from v to w in H . (It is an edge where cycles were `glue-ed toge... |

11 |
Graph Algorithms.” in Handbook of Theoretical Computer Science, Volume A
- Leeuwen
- 1990
(Show Context)
Citation Context ... 2 I , if j lies on the path from i to k in T , then X i " X k ` X j . There are several notions, equivalent to treewidth, e.g. a graph is a `partial k-tree', iff its treewidth is at most k (see =-=e.g. [17]-=-). We say a tree-decomposition is nice, if T is a rooted binary tree, with for root node r: X r = ;. It is easy to transform each tree-decomposition into a nice one with the same treewidth. Ask-bounda... |

7 | G.: Parametric problems on graphs of bounded tree-width
- Fernández-Baca, Slutzki
- 1994
(Show Context)
Citation Context ...al trees under several operations, including label changes. The technique in this paper for maintaining trees and forests is very similar to the technique used in [10, 11]. Fernandez-Baca and Slutzki =-=[9]-=- considered parametrized algorithms on graphs with bounded treewidth. Overview of this paper. In section 2, definitions and some preliminary results are reviewed. In section 3, we discuss the class of... |

2 |
Fast Parallel Algorithms and Enumeration Techniques for Partial k-Trees
- Chandrasekharan
- 1989
(Show Context)
Citation Context ...Coloring, etc., counting problems like How many Hamiltonian circuits does G have? and some PSPACE-complete problems. Also, these problems when restricted to graphs with bounded treewidth belong to NC =-=[3, 7, 12]-=-. We consider the problem of solving these problems on graphs with treewidth at most 2 that change dynamically. We allow the following operations: insertion and deletion of isolated vertices, insertio... |

2 |
Dynamic Algorithms for Bounded Tree-Width Graphs
- Cohen, Sairam, et al.
- 1992
(Show Context)
Citation Context ...eter k, for some constant k. ? This work was partially supported by the ESPRIT Basic Research Actions of the EC under contract 7141 (project ALCOM II). Related results. In a recent paper, Cohen et al =-=[8]-=- designed algorithms for the maintenance of graphs with treewidth at most 2 or 3. For graphs with treewidth at most 2, insertions and deletions can be done in O(log 2 n) time, while queries cost O(log... |

2 |
Draft: Maintaining regular properties dynamically in k-terminal graphs, Unpublished manuscript
- Frederickson
- 1993
(Show Context)
Citation Context ... of graph problems that can be queried with their approach is much smaller than the class, dealt with in the present paper. The techniques used in [8] and this paper are quite different. Frederickson =-=[10, 11]-=- found independently similar results for trees, forests, and k-terminal trees under several operations, including label changes. The technique in this paper for maintaining trees and forests is very s... |