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 ArticleRelation between tree-width and clique number
Are there any nice graph classes for which the tree-width $tw(G)$ is upper-bounded by a function of the clique number $\omega(G)$, i.e. $tw(G)\leq f(\omega(G))$?For example, it is a classic fact that...
View ArticleCan the theory of Bidimensionality be applied to weighted instances of a...
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...
View ArticleDirected 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...
View ArticleCan someone recommend a reference on graph minors structure theorem and...
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...
View ArticleDoes distance-2 coloring fit in Telle and Proskurowski 's algorithm for...
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 Partitioning Problems"...
View ArticleSomething-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 $G\in \mathcal{F}$,...
View ArticleTrading 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...
View ArticleDoes replacing each vertex of $G$ by $H$ increase treewidth of $G$ by at most...
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...
View ArticleTreewidth 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 states...
View ArticleFinding 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...
View ArticleHow 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,...
View ArticleLow-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...
View ArticleWhat is the expected treewidth of a large-treewidth graph intersected with...
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 resulting...
View ArticleClik here to view.
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...
View ArticleIs 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, but there...
View ArticleOn 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,...
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?
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 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 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...
View Article