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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5
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1answer
477 views

Long Cycle in Bounded Tree-Width Graphs using DFS and Dynamic Programming

For fixed parameter $k$, I would like to find a long cycle of length $\geq k$ in an undirected graph $G(V,E)$. This can be done in $O(k!2^k|V|)$-time [2] using 1) depth-first search (DFS) and 2) ...
11
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1answer
389 views

MSO properties, planar graphs and minor-free graphs

Courcelle's theorem states that every graph property definable in monadic second-order logic can be decided in linear time on graphs of bounded treewidth. This is one of the most well-known ...
14
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2answers
448 views

A graph parameter possibly related to treewidth

I am interested in graphs on $n$ vertices which can be produced via the following process. Start with an arbitrary graph $G$ on $k\le n$ vertices. Label all the vertices in $G$ as unused. Produce a ...
7
votes
2answers
274 views

Graph addition chains

An addition chain for $n \in \mathbb{N}$ is a sequence of natural numbers $$1 = a_0,\ldots,a_l =n$$ such that each $a_t$ is the sum of two previous elements in the sequence. The length of minimal ...
4
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0answers
111 views

Approximating Front Size of Asymmetric Matrices

The front size of a matrix $A$ is the largest number of non-zeros below the diagonal in any column of its Cholesky factor. If $A$ is symmetric then the minimum front size of $A$ is equal to the ...
9
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3answers
601 views

Special cases of Graphic TSP

In Graphic TSP, you are given an unweighted undirected graph $G$ and the goal is to find a shortest tour in $G$ that visits every vertex at least once. Note that this is NOT same as finding a ...
23
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1answer
512 views

How big is the variance of the treewidth of a random graph in G(n,p)?

I am trying to find how close $tw(G)$ and $E[tw(G)]$ really are, when $G \in G(n,p=c/n)$ and $c>1$ is a constant not depending on n (so $E[tw(G)] = \Theta(n)$). My estimate is that $tw(G) \leq E[tw(...
31
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1answer
885 views

Treewidth and the NL vs L Problem

ST-Connectivity is the problem of determining whether there exists a directed path between two distinguished vertices $s$ and $t$ in a directed graph $G(V,E)$. Whether this problem can be solved in ...
6
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1answer
844 views

Tree width of a particular graph

What is the tree-width of the graph $G = (V_1 \cup V_2 \cup \dotsb V_n, E)$ where the connected components of an induced subgraph of any neighboring set of vertices (i.e. $G[V_i \cup V_j], i = j - 1$)...
2
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1answer
942 views

TSP in bounded tree-width (or bounded branch width) graphs

I see there are some papers/thesis which says TSP is solvable in $O(n)$ in bounded tree-width graphs and some of them implicitly refer to Cook and Seymour (not exact paper just said that they solve it)...
16
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1answer
724 views

Making a minimum-width tree decomposition lean in polynomial time

As is well known, a tree decomposition of a graph $G$ consists of a tree $T$ with an associated bag $T_v \subseteq V(G)$ for each vertex $v \in V(T)$, which satisfies the following conditions: Every ...
17
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2answers
1k views

Forbidden minors for bounded treewidth graphs

This question is similar to one of my previous questions. It is known that $K_{t+2}$ is a forbidden minor for graphs of treewidth at most $t$. Is there a nicely-constructed, parameterized, ...
17
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2answers
632 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 ...
22
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5answers
3k 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
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1answer
578 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,...
64
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5answers
6k 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
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2answers
794 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
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1answer
303 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
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0answers
211 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
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3answers
939 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
213 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
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1answer
306 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
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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
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1answer
867 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
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2answers
1k 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 ...

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