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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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12 votes
2 answers
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Problems that are NP-Complete when restricted to graphs of treewidth 2 but polynomial on trees

Do we know any problem that satisfies the following criteria? It is polynomial-time solvable on trees. It is NP-complete when restricted to graphs of treewidth 2. The problem can be encoded only ...
Prafullkumar Tale's user avatar
6 votes
1 answer
159 views

Tree decompositions with unique witness for each edge

In this question I am concerned with tree decompositions of undirected graphs. Recall that a tree decomposition of a graph $G = (V, E)$ is a tree $T$ whose nodes are subsets of $V$ (called bags) ...
a3nm's user avatar
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5 votes
0 answers
134 views

What is the smallest graph of treewidth $k$ having less edges than the $(k+1)$-clique?

Treewidth is a graph parameter measuring how close a graph is to being a tree. I am interested in what is the minimal number of edges required for a graph to have treewidth $k$. A natural family of ...
a3nm's user avatar
  • 9,419
1 vote
1 answer
119 views

Tractability of computing generalized hypertreewidth on bounded arity hypergraphs

Generalized hypertreewidth is a generalization of treewidth to hypergraphs. Unlike treewidth, it is not tractable, for a fixed width $k \in \mathbb{N}$, given a hypergraph $H$, to determine if $H$ has ...
a3nm's user avatar
  • 9,419
3 votes
1 answer
195 views

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

Here, by 3D-grid of sidelength $n$ I mean the graph $G=(V,E)$ with $V= \{1,\ldots,n\}^3$ and $E=\{( (a,b,c) ,(x,y,z) ) \mid |a-x|+|b-y|+|c-z|=1 \}$. I known how to get the treewidth of $n*n$ grid is ...
Jxb's user avatar
  • 316
1 vote
0 answers
49 views

Bound on the treewidth of a graph from modular contraction

I cannot find a reference for this easy to prove result concerning the treewidth of a graph with respect to the treewidth of a modular contraction of it. Let $G=(V,E)$ be a graph. A module $M \...
holf's user avatar
  • 2,174
0 votes
0 answers
157 views

What's the connection between branchwidth and treewidth

I understand that treewidth and branchwidth are essentially equivalent for a fixed graph, given that $branchwidth(G) = Θ(treewidth(G))$. However, my question pertains to a specific case involving ...
Jxb's user avatar
  • 316
3 votes
1 answer
166 views

Treewidth for hypergraphs that specify connectedness requirements

This question is about an alternative definition of treewidth, called weak treewidth. It is defined on hypergraphs where hyperedges intuitively require that the connected subtrees of occurrences of ...
a3nm's user avatar
  • 9,419
3 votes
1 answer
109 views

Polynomial time solvable in series parallel graph but NP-hard in graph with bounded treewidth

Whether there is a problem to meet the conditions: it is polynomial time solvable in series parallel graphs but NP-hard in graph with bounded treewidth?
Yuhang Bai's user avatar
5 votes
2 answers
245 views

Treewidth relations between Boolean formulas and Tseitin encodings

Suppose you have a propositional formula $\varphi$ in CNF. You want to efficiently obtain an equisatisfiable CNF formula encoding $\neg \varphi$. You use the usual Tseitin encoding with auxiliary ...
Noel Arteche's user avatar
1 vote
1 answer
191 views

Nontrivial Algorithms for Coloring (Parameterized by Pathwidth)

Let $k$ be a positive integer. In the $k$-coloring problem, we are given a graph $G$ on $n$ nodes, and want to determine if there is a way to assign a color to each vertex of $G$ such that no two ...
Naysh's user avatar
  • 686
1 vote
1 answer
121 views

On motivation towards study of width parameters beyond treewidth

Many width parameters are invented to capture the tractability of CSP (and its equivalent problem, conjunctive queries (CQ) evaluation): treewidth, hypertree width, generalized hypertree width, ...
xxks-kkk's user avatar
  • 125
5 votes
1 answer
178 views

Is there a standard axiomatization of graph width parameters?

There are many useful graph properties described as "width parameters" that show up in algorithm analysis (especially for FPT-type algorithms). The most famous example is probably treewidth,...
GMB's user avatar
  • 2,403
3 votes
2 answers
157 views

Parameterized complexity of tree/branch decomposition

I'm looking for an up to date reference for parameterized complexity of tree and branch decompositions. IE, complexity of finding tree/branch decomposition of optimal width in terms of relevant graph ...
Yaroslav Bulatov's user avatar
2 votes
0 answers
79 views

What is the expected treewidth of a large-treewidth graph intersected with Erdos-Renyi graph?

Suppose we have a graph $G$ with treewidth $t$. Let $p \in (0,1)$ be a constant. Then let's independently remove each edge from $G$ with probability $p$. What is the expected treewidth of the ...
Artur Riazanov's user avatar
0 votes
0 answers
78 views

Low-Treewidth Sorting Networks

It was previously asked if there exist Boolean circuits of treewidth $O(\log n)$ that compute the majority function $\text{MAJ}_n$ on $n$ inputs. While a construction using online algorithms and the ...
Cornelius Brand's user avatar
2 votes
0 answers
97 views

How typical are odd-H-minor free graphs?

Can anything be said about how typical are odd-H-minor free graphs? (definition of odd-minor-free is in Section 2.2 of notes, page 20 of slides). For instance, for a random graph with $n$ vertices, $...
Yaroslav Bulatov's user avatar
1 vote
1 answer
66 views

Treewidth of monotone graph classes with bounded cliquewidth

Assume a graph class excludes a certain bicique $K_{n,n}$ and has bounded cliquewidth. Then by a result of Gurski and Wanke, this class also has bounded treewidth. Is there a similar result that ...
Jan's user avatar
  • 61
5 votes
0 answers
182 views

Trading treewidth for depth in Boolean circuits

We know that languages defined by (poly-sized) Boolean formulae equals $\mathbf{NC}^1$: that Boolean formulae can be simulated in $\mathbf{NC}^1$ was shown by Brent/Spira [B,S], and the converse is ...
ckamath's user avatar
  • 223
1 vote
0 answers
96 views

Can someone recommend a reference on graph minors structure theorem and sublinear treewidth?

Can someone recommend a reference on graph minors structure theorem and sublinear treewidth? Doesn't have to be the newest/strongest results as long as it's easier than tracking down all the papers ...
Hao S's user avatar
  • 228
1 vote
0 answers
69 views

Directed tree decompositions on subtrees of DAGs

Given a DAG, is the arboreal decomposition of the DAG with the guarantee that given a node $x$, $v$ such that $x$ is reachable from $v$ are in the subtree of $x$? If not, is there a similar ...
shgr1092's user avatar
4 votes
2 answers
266 views

Does replacing each vertex of $G$ by $H$ increase treewidth of $G$ by at most $\Delta(G)$?

I am asking this question from the context of parameter preserving reductions which has implications for kernelization (See Theorem 18 of [1] for an example). For simplicity, here I am assuming that ...
Cyriac Antony's user avatar
2 votes
0 answers
77 views

Can the theory of Bidimensionality be applied to weighted instances of a problem?

So my understanding of bidimensionality is you are assured the problem solution is about O(k^2) so you can pay O(k) purely to reduce the instance to one of bounded treewidth. As far as I know, this ...
Hao S's user avatar
  • 228
10 votes
1 answer
245 views

Tree decompositions of optimal width where every vertex is in a bounded number of bags?

Let $G$ be a graph on $n$ vertices whose maximum degree is at most $\Delta$ and whose treewidth is at most $k$. Does there exist a function $f(k, \Delta)$, independent of $n$, such that it is possible ...
hdur's user avatar
  • 261
3 votes
1 answer
201 views

When is hypertree width more useful than generalized hypertree width?

In analysis of CSPs, there are three width notions that are analogous to treewidth: hypertree width (hw), generalized hypertree width (ghw) and fractional hypertree width (fhw). Moreover the ...
Laakeri's user avatar
  • 1,786
2 votes
2 answers
115 views

Does distance-2 coloring fit in Telle and Proskurowski 's algorithm for partial-k trees?

This question is on "Vertex Partitioning Problems" framework of Telle and Proskurowski. For solving problems in parital $k$-trees (i.e., graphs of bounded treewidth), the "Vertex ...
Cyriac Antony's user avatar
5 votes
0 answers
148 views

How much does treewidth changes after removal of a path?

Let $G$ be a graph such that $\mathrm{tw}(G)=t$. Let $t' = \min\limits_{u,v \in V(G)} \max\limits_{P \text{ is a path from } u \text{ to } v} \mathrm{tw}(G - P)$. Then how small $t'$ can be? My ...
Artur Riazanov's user avatar
5 votes
1 answer
563 views

Complexity of SAT parameterized by treewidth

Many papers state that Boolean satisfiability is in FPT when parameterized by primal, dual, or incidence treewidth. What are the best known time complexities of these parameterized algorithms? In ...
Laakeri's user avatar
  • 1,786
14 votes
0 answers
232 views

Counting solutions to extended MSO formulas, and sampling -- do these appear in the literature?

I am trying to determine if the literature contains various extensions of Courcelle's theorem. Since I haven't been able to find these in the literature, I guess that these are folklore results, or ...
Elle Najt's user avatar
  • 1,469
5 votes
3 answers
484 views

Is counting simple cycles in $P$ for graphs of bounded tree width?

Motivation: Determining if a graph has a Hamiltonian cycle is $NP$-hard in general. However, determining if there is a Hamiltonian cycle is in polynomial time on graphs of bounded tree width, either ...
Elle Najt's user avatar
  • 1,469
6 votes
1 answer
375 views

Naive definition of treewidth

Treewidth has arguably pretty involved definition. Recently I was thinking about a problem and turns out it easy to solve it for graphs with small ``naive treewidth''. Naive treewidth is defined as ...
Artur Riazanov's user avatar
6 votes
0 answers
351 views

Grid-Minor Theorem of Robertson and Seymour and its Algorithmic Applications

Graph-Minor Theorem of Robertson and Seymour [1] states that if graph G has large treewidth, then it contains a large grid as minor. Most approximation results on general classes of graphs with ...
user avatar
1 vote
0 answers
282 views

Directed NP Hard Problem on DAG

There are problems that are NP-Hard on undirected graphs(maximum weight independent set and graph coloring) but are polynomial time solvable on trees. Tree decomposition is a good tool to talk about ...
am_rf24's user avatar
  • 23
3 votes
1 answer
196 views

Natural (well studied) classes of graphs with treewidth $\Theta(n^\alpha)$ with $1/2 < \alpha < 1$

Which natural (well studied) classes of graphs have treewidth that scales as $\Theta(n^\alpha)$ in the number $n$ of vertices, with $1/2 < \alpha < 1$?
delete000's user avatar
  • 828
5 votes
2 answers
590 views

Maximum Treewidth of a Graph with $m$ Edges

What is the maximum treewidth of a graph with $m$ edges? In other words, what is the correct growth for the following function? $\alpha(m) = max\{\mathrm{treewidth}(G): G \mbox{ has $m$ edges}\}$. ...
Springberg's user avatar
10 votes
2 answers
374 views

Finding subgraphs with high treewidth and constant degree

I am given a graph $G$ with treewidth $k$ and arbitrary degree, and I would like to find a subgraph $H$ of $G$ (not necessarily an induced subgraph) such that $H$ has constant degree and its treewidth ...
a3nm's user avatar
  • 9,419
7 votes
0 answers
120 views

Deterministic approximation algorithms for treewidth

As far as I understand, the factor $O(\sqrt{\log OPT})$ approximation algorithm for treewidth of Feige, Hajiaghayi, and Lee is randomized, and no deterministic approximation algorithm with this factor ...
daniello's user avatar
  • 3,266
2 votes
0 answers
268 views

Paper regarding the complexity of the longest path problem on weighted directed graphs of bounded treewidth

I would like to cite a paper/report/etc that solves the following problem polynomially in $n$: Given a weighted directed graph $G=(V,E)$, $|V|=n$, of bounded treewidth $k \in \mathbb{N}$ and a source-...
cs_student_273's user avatar
10 votes
1 answer
339 views

Is bounded-width SAT decidable in logspace?

Elberfeld, Jakoby, and Tantau 2010 (ECCC TR10-062) proved a space-efficient version of Bodlaender's theorem. They showed that for graphs with treewidth at most $k$, a tree decomposition of width $k$ ...
András Salamon's user avatar
2 votes
1 answer
146 views

Treewidth of two complete binary trees joined at their leaves

Let $T$ be a complete binary trees on $n$ nodes. Let $G'$ be the graph that consists of two disjoint copies of $T$. For a leaf $x \in T$, let $x_1, x_2$ be the two copies of it in $G'$. Then, let $G$ ...
Chris's user avatar
  • 79
14 votes
1 answer
827 views

Are there interesting graph classes where the treewidth is hard (easy) to compute?

Treewith is an important graph parameter that indicates how close a graph is from being a tree (although not in a strict topological sense). It is well known that computing the treewidth is NP-hard. ...
PsySp's user avatar
  • 840
7 votes
0 answers
99 views

Complexity of computing the simplicial width of a graph

Let $G=(V,E)$ be a finite undirected graph. A tree decomposition $(T,\lambda)$ of $G$ is a tree $T$ with labeling function $\lambda : T \to 2^{V}$ such that: For every edge $\{v_1,v_2\} \in E$, there ...
M.Monet's user avatar
  • 1,429
6 votes
1 answer
292 views

Treewidth of deep Sierpiński Sieve Graph

Note $S_n$ the Sierpiński sieve graph of order $n$, which is obtained from the connectivity of the Sierpiński sieve. For $n$ high enough, what is the treewidth of $S_n$? I think that I can show that ...
while False's user avatar
9 votes
2 answers
299 views

Something-Treewidth Property

Let $s$ be a graph parameter (ex. diameter, domination number, etc) A family $\mathcal{F}$ of graphs has the $s$-treewidth property if there is a function $f$ such that for any graph $G\in \mathcal{...
Springberg's user avatar
8 votes
2 answers
302 views

Complexity of testing if a hypergraph has generalized hypertreewidth $2$

A hypergraph $H = (V, E)$ consists of a set of vertices $V$ and a set $E$ of hyperedges, i.e., subsets of $V$. The generalized hypertreewidth (GHW) parameter is a measure of the cyclicity of a ...
a3nm's user avatar
  • 9,419
18 votes
0 answers
421 views

Complexity of the homomorphism problem parameterized by treewidth

The homomorphism problem $\text{Hom}(\mathcal{G}, \mathcal{H})$ for two classes $\mathcal{G}$ and $\mathcal{H}$ of graphs is defined as follows: Input: a graph $G$ in $\mathcal{G}$, a graph $H$ in $...
a3nm's user avatar
  • 9,419
7 votes
1 answer
275 views

Tree-decomposition with clique interfaces

Let $G=(V,E)$ be a finite undirected graph. A tree decomposition $(T,\lambda)$ of $G$ is a tree $T$ with labeling function $\lambda : T \to 2^{V}$ such that: For every edge $\{v_1,v_2\} \in E$, there ...
M.Monet's user avatar
  • 1,429
11 votes
1 answer
398 views

Classes of graphs with superconstant treewidth

There are several interesting classes of graphs with bounded treewidth. For instance, trees (treewidth 1), series parallel graphs (treewidth 2), outerplanar graphs (treewidth 2), $k$-outerplanar ...
Mateus de Oliveira Oliveira's user avatar
5 votes
0 answers
1k views

Tree decomposition for DAGs

Tree decompositions and treewidth are a standard way to measure how close an undirected graph is to a tree. I am studying decompositions of directed acyclic graphs (DAGs), and have come to define them ...
a3nm's user avatar
  • 9,419
25 votes
1 answer
1k views

Is it still open to determine the complexity of computing the treewidth of planar graphs?

For a constant $k \in \mathbb{N}$, one can determine in linear time, given an input graph $G$, whether its treewidth is $\leq k$. However, when both $k$ and $G$ are given as input, the problem is NP-...
a3nm's user avatar
  • 9,419