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Graduate
By
Shiva Kintali
on June 10, 2012 | Updated Dec. 6, 2017
Basics of Treewidth
Treewidth of an undirected graph \(G\) measures how close \(G\) is to a tree. A tree decomposition of a graph \(G(V, E)\) is a pair \(\mathcal{D} = ({X_i\ |\ i \in I}, T(I, F))\) where …
Mathematics
Graph Theory
basics
treewidth
0
Graduate
By
Shiva Kintali
on June 7, 2012 | Updated Dec. 6, 2017
Recognizing series-parallel graphs in linear time
Let \(G(V,E)\) be a simple undirected graph with \(|V| = n\) and \(|E| = m\). Let the treewidth of \(G\) be at most \(k\). Derive a tight upper bound of \(m\) in terms of \(n\) and \(k\). Show that …
Computer Science
Mathematics
Algorithms
Graph Theory
chordal graph
linear time algorithms
series parallel
treewidth
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Graduate
By
Shiva Kintali
on May 31, 2013 | Updated Dec. 6, 2017
Treewidth and circumference
Circumference of a graph is the length of its longest cycle. Prove that if a graph has circumference \(k\) then its treewidth is at most \(k-1\). Give a class of graphs showing that this bound is tigh…
Mathematics
Graph Theory
circumference
treewidth
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Graduate
By
Shiva Kintali
on June 6, 2012 | Updated Dec. 6, 2017
Treewidth vs Pathwidth
Let \(G(V,E)\) be an undirected graph with \(|V|=n\) vertices. Let \(tw(G)\) and \(pw(G)\) represent the treewidth and pathwidth of \(G\) respectively. Let \(H\) be a tree. Prove that …
Mathematics
Graph Theory
pathwidth
treewidth
0
Graduate
By
diego
on Aug. 13, 2012 | Updated Dec. 6, 2017
Treewidth, the natural way
Given a graph \(G\), we define \(\mathrm{la}(G)\) as the smallest \(r\) such that \(G\) is a minor of some \(T\times K_r\), where \(T\) is a tree (and \(\times\) in this context is the cartesian produ…
Mathematics
Graph Theory
treewidth
0
Graduate
By
Shiva Kintali
on Feb. 20, 2013 | Updated Dec. 6, 2017
Treewidth and Cliques
A tree decomposition of a graph \(G(V, E)\) is a pair \(\mathcal{D} = ({X_i\ |\ i \in I}, T(I, F))\) where \({X_i\ |\ i \in I}\) is a collection of subsets of \(V\) (called bags) and \(T(I, F)\) is a …
Mathematics
Graph Theory
treewidth
0
Graduate
By
Shiva Kintali
on June 4, 2013 | Updated Dec. 6, 2017
Treewidth and feedback vertex set
A feedback vertex set is a set of vertices whose removal results in an acyclic graph. Prove that if a graph has a feedback vertex set of size \(k\), then it has treewidth at most \(k+1\).
Mathematics
Graph Theory
treewidth
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