Questions tagged [treewidth]
Treewidth is a graph parameter which measures how close the graph is to a tree (smaller is better). Many problems can be solved more efficiently on graphs with bounded treewidth, using dynamic programming.
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MSOL and Courcelle's theorem for maximum clique
The Clique Problem is known to be NP-complete but is known to be fixed-parameter-tractable (FPT) if the treewidth of the underlying graph is fixed.
The traditional proof is by a dynamic programming ...
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MSOL for a vertex-cover enlargement problem
Consider the following problem.
Given a graph $G=(V,E)$, and two positive integers $k$ and $\gamma$, decide if there is a set of new edges to be added such that $|E'|\le k$, and any subset $V'\...
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Tree width given path decomposition
I have a family of graphs whose path decompositions I know. Is it possible to compute the tree-width of these graphs in polynomial time?
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Use of the degree variable in an MSOL formula
I am working on giving an MSOL formula for an NP-hard problem; this proves that the problem is linear-time solvable on bounded treewidth graphs. Given a graph $G = (V, E)$, the problem would be to ...
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On definitions of graph width
Wikipedia shows graph width $k$ as the degeneracy, an ordering of the vertices $v_1,\ldots , v_k$ for which, if we orient each edge $(v_i, v_j)$ towards $i$ where $i<j$, the maximal degree is at ...
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Clarifications about tree-width definition
I have read the definition of treewidth/tree-decomposition both in Wikipedia and in here:
https://medium.com/@karlrombauts/treewidth-how-all-graphs-are-trees-in-disguise-ec699b69e2fb
I'm finding ...
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Tree Width of Directed Graph
I'm Nestor Mermoz Thea. I have two definition over the Directed strong pseudoforest and Directed weak pseudoforest that I don't really well understand.
Directed weak pseudoforest: A directed weak ...
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Relationships between path width and clique size of interval graphs
I faced the following claim on wikiepdia about interval graphs (https://en.wikipedia.org/wiki/Interval_graph):
The pathwidth of an interval graph is one less than the size of its maximum clique.
I ...
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Treewidth of a subdivision of a graph
Let $G$ be a graph and $G_s$ its $s$-subdivision, i.e. the graph obtained from $G$ by subdividing each edge $s$ times.
Why is the treewidth of $G$ the same as the treewidth of $G_s$?
(Subdividing an ...
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Does FPT allow for doubling the parameter?
I have recently come across a result that showed that a given problem is in FPT when parameterized by the treewidth of a graph. However, they did this by showing that the problem is in FPT when ...
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The length of the formula in Monadic second-order logic
The Courcelle's theorem as following:
Obviously, we need the length of the $\varphi$~(that is $||\varphi||$)~in Theorem. However, how to caculate the length of $\varphi$?
For example, $$\begin{...
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Dynamic Programming for Feedback Vertex Set - bounded treewidth
Saw it on another post that there is a way of solving FVS in polynomial time if the treewidth is constant, using dynamic programming?...
If I'm given the treewidth of a graph, how do I solve it in ...
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If G has no simple path on x vertices ,then the treewidth of G is upper bounded by x
Statement:
If G has no simple path on x vertices ,then the treewidth of G is upper bounded by x.
Hint: Begin by computing a DFS tree, and prove an upper bound on its height.
I am supposed to prove the ...
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Is there a graph theory textbook that covers treewidth thoroughly?
Can someone recommend a graph theory textbook that covers treewidth thoroughly?
Something that focuses on the graph-theoretic structure of bounded treewidth graphs rather than solving problems on them....
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Bounded treewidth implies bounded clique-width
We have a graph G of treewidth $\operatorname{tw}(G)\leq k$, for some $k\in\mathbb{N}$. I've seen a claim that that implies that the clique-width of the same graph is at most $k \cdot 2^k$. This ...
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Size of tree decomposition
Given a graph $G$ with $n$ vertices, let $(X, T)$ be a tree decomposition of $G$ with the smallest width. Is the number of nodes in $T$ upper bounded by $n$? I have googled it but all materials I ...
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treewidth of a given graph
Is the treewidth of this graph equal to 2?
I have tried to prove it through the definition of a tree decomposition.
If its not correct can someone give any hints?
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How does treewidth behave under graph minor operations?
It is a well-known fact that for any minor H of a graph G (commonly written as $H \leq_m G$), the treewidth of H is smaller than or equal to that of G.
Minors of a graph are created through the ...
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