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...
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 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 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 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 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 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 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 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 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 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...
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