Tree decompositions of optimal width where every vertex is in a bounded...
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...
View ArticleComplexity 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...
View ArticleIs there an online tool to compute the treewidth of a given graph?
I have an explicit graph (not too big) and I want to compute its treewidth. Is there a Web tool where I can provide the graph (given, e.g., by its explicit list of edges) and obtain its treewidth?
View ArticleForbidden 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, infinite...
View ArticleComplexity of the homomorphism problem parameterized by treewidth
The homomorphism problem$\text{Hom}(\mathcal{G}, \mathcal{H})$ for twoclasses $\mathcal{G}$ and $\mathcal{H}$ of graphs is defined as follows:Input: a graph $G$ in $\mathcal{G}$, a graph $H$ in...
View ArticleDirected 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...
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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 it...
View ArticleProblems that are NP-Complete when restricted to graphs of treewidth 2 but...
Do we know any problem that satisfies the following criteria?It admits polynomial-time solvable on trees.It is NP-complete when restricted to the graphs of treewidth 2.The problem can be encoded only...
View ArticleWhat is the smallest graph of treewidth $k$ having less edges than the...
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...
View ArticleTree 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)...
View ArticleMaximum 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}\}$....
View ArticleTractability of computing generalized hypertreewidth on bounded arity...
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...
View ArticleWhat 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...
View ArticleBound 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 \subseteq...
View ArticleWhat'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...
View ArticleNontrivial 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...
View ArticleTreewidth 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 the...
View ArticleTreewidth 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...
View ArticleWhat 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, and...
View ArticlePolynomial time solvable in series parallel graph but NP-hard in graph with...
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?
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