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Questions tagged [treewidth]

Questions regarding the treewidth of graphs. Graphs of low treewidth admit fast divide-and-conquer algorithms for many graph problems that are NP-hard on general graphs.

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16
votes
2answers
600 views

Is there any problem in $\mathsf{\Sigma^P_2}$ which is solvable in bounded tree width graphs?

I'm looking for a problem which belongs to $\mathsf{\Sigma^P_2}$ in general graphs but is in $\mathsf{P}$ in bounded tree width graphs, In fact I think this problems are harder than using normal ...
21
votes
5answers
2k views

Program for computing Tree decomposition of a graph

Does anybody know of an open-source program for computing Tree decomposition of graphs for a fixed "k"(width)? I know that the problem of finding Tree-Decomposition is NP-Hard for variable "k", but my ...
24
votes
1answer
543 views

Reconstruction Conjecture and Partial 2-trees

Reconstruction conjecture says that graphs (with at least three vertices) are determined uniquely by their vertex deleted subgraphs. This conjecture is five decades old. Searching relevant literature,...
61
votes
5answers
5k views

The origin of the notion of treewidth

My question today is (as usual) a bit silly; but I would request you to kindly consider it. I wanted to know about the genesis and/or motivation behind the treewidth concept. I sure understand that ...
12
votes
2answers
728 views

What is the pathwidth of the 3D-grid (mesh or lattice) with sidelength k?

I asked this question some weeks ago at mathoverflow, but I got no reply. Here, by 3D-grid of sidelength $k$ I mean the graph $G=(V,E)$ with $V= \{1,\ldots,k\}^3$ and $E=\{( (a,b,c) ,(x,y,z) ) \mid |...
12
votes
1answer
276 views

Typical hardness of tree decomposition?

Tree decomposition is hard in the worst case but greedy method seems to be near-optimal on small real-life networks. Is anything known about hardness of tree decomposition of a "typical" instance of ...
6
votes
0answers
208 views

Decomposition based on approximate separators in graph

Suppose I want to find vertex subset $S$ of graph $G=(V,E)$ such that any simple closed walk that visits vertices both in $S$ and in $V\backslash S$ has length $\ge g$ The idea is to relax ...
9
votes
3answers
846 views

Tree decomposition for planar graphs

First asked on math.SE with no replies. Suppose I have a planar graph, with a planar embedding, how do I find tree decomposition? What is the optimal tree decomposition of a $d$-by-$d$ square grid? ...
6
votes
1answer
210 views

Ensemble of tree decompositions for all-pairs problem

Suppose we have a bounded tree width graph $G=\{E,V\}$ and want to count the number of self avoiding walks on $G$ passing through nodes $u$ and $v$ for all pairs of nodes $(u,v)$. For a single pair $(...
10
votes
1answer
266 views

CSPs with unbounded fractional hypertree width

At SODA 2006, Martin Grohe and D$\acute{\rm a}$niel Marx's paper "Constraint solving via fractional edge covers" (ACM citation) showed that for the class of hypergraphs $H$ with bounded fractional ...
13
votes
1answer
4k views

What is the correct definition of $k$-tree?

As the title says, what is the correct definition of $k$-tree? There are several papers that talk about $k$-trees and partial $k$-trees as alternative definitions for graphs with bounded treewidth, ...
23
votes
1answer
836 views

Logspace algorithms on graphs with bounded tree width

Tree width measures how close a graph is to a tree. It is NP-hard to compute tree width. The best known approximation algorithm achieves $O(\sqrt{{\log}n})$ factor. Courcelle's theorem states that ...
14
votes
2answers
905 views

How large a treewidth can a tree plus half the edges have?

Let G be a tree on 2n vertices. The treewidth of G, tw(G) = 1. Now suppose we add n edges to G to get a graph H. An easy upper bound on tw(H) is n + 1. Is this essentially the best possible? It ...