How does treewidth behave under graph minor operations?
$begingroup$
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 operations of (1) vertex deletion, (2) edge deletion and (3) edge contraction. I am curious as to whether one can bound the decrease in treewidth when applying any single operation from (1)-(3).
Of particular interest for me is the question whether said decrease can always be bounded by some constant.
graph-theory discrete-mathematics graph-minors treewidth
edited Feb 17 at 20:51
D.W.♦
99.5k12121286
asked Feb 17 at 13:41
SmeltQuakeSmeltQuake
283
$endgroup$
add a comment |
$begingroup$
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 operations of (1) vertex deletion, (2) edge deletion and (3) edge contraction. I am curious as to whether one can bound the decrease in treewidth when applying any single operation from (1)-(3).
Of particular interest for me is the question whether said decrease can always be bounded by some constant.
graph-theory discrete-mathematics graph-minors treewidth
edited Feb 17 at 20:51
D.W.♦
99.5k12121286
asked Feb 17 at 13:41
SmeltQuakeSmeltQuake
283
$endgroup$
add a comment |
$begingroup$
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 operations of (1) vertex deletion, (2) edge deletion and (3) edge contraction. I am curious as to whether one can bound the decrease in treewidth when applying any single operation from (1)-(3).
Of particular interest for me is the question whether said decrease can always be bounded by some constant.
graph-theory discrete-mathematics graph-minors treewidth
edited Feb 17 at 20:51
D.W.♦
99.5k12121286
asked Feb 17 at 13:41
SmeltQuakeSmeltQuake
283
$endgroup$
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 operations of (1) vertex deletion, (2) edge deletion and (3) edge contraction. I am curious as to whether one can bound the decrease in treewidth when applying any single operation from (1)-(3).
Of particular interest for me is the question whether said decrease can always be bounded by some constant.
graph-theory discrete-mathematics graph-minors treewidth
graph-theory discrete-mathematics graph-minors treewidth
edited Feb 17 at 20:51
D.W.♦
99.5k12121286
asked Feb 17 at 13:41
SmeltQuakeSmeltQuake
283
edited Feb 17 at 20:51
D.W.♦
99.5k12121286
asked Feb 17 at 13:41
SmeltQuakeSmeltQuake
283
edited Feb 17 at 20:51
D.W.♦
99.5k12121286
edited Feb 17 at 20:51
D.W.♦
99.5k12121286
edited Feb 17 at 20:51
D.W.♦
99.5k12121286
99.5k12121286
asked Feb 17 at 13:41
SmeltQuakeSmeltQuake
283
asked Feb 17 at 13:41
SmeltQuakeSmeltQuake
283
asked Feb 17 at 13:41
SmeltQuakeSmeltQuake
283
283
add a comment |
add a comment |
1 Answer
1
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oldest
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$begingroup$
The decrease in tree width of any of the three operations will be 0 or 1. The proof is simple (I will take the case of the vertex deletion).
Let $G$ be the original graph and $G' = G - v$ be the graph obtained by deleting $v$.
Let $T'$ be an optimal tree decomposition of $G'$, with $text{tw}(G') = w'$.
Now, construct $T$ from $T'$ by adding $v$ to every bag of $T'$. Clearly, $T$ is a tree decomposition for $G$ with width $text{tw}(G) leq text{tw}(T) = 1 + text{tw}(G') = 1 + w'$.
answered Feb 17 at 14:14
Pål GDPål GD
6,9652342
$endgroup$
1
$begingroup$
Indeed, it seems trivial now. Thank you.
$endgroup$
– SmeltQuake
Feb 17 at 15:06
add a comment |
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1 Answer
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active
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1 Answer
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active
oldest
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$begingroup$
The decrease in tree width of any of the three operations will be 0 or 1. The proof is simple (I will take the case of the vertex deletion).
Let $G$ be the original graph and $G' = G - v$ be the graph obtained by deleting $v$.
Let $T'$ be an optimal tree decomposition of $G'$, with $text{tw}(G') = w'$.
Now, construct $T$ from $T'$ by adding $v$ to every bag of $T'$. Clearly, $T$ is a tree decomposition for $G$ with width $text{tw}(G) leq text{tw}(T) = 1 + text{tw}(G') = 1 + w'$.
answered Feb 17 at 14:14
Pål GDPål GD
6,9652342
$endgroup$
1
$begingroup$
Indeed, it seems trivial now. Thank you.
$endgroup$
– SmeltQuake
Feb 17 at 15:06
add a comment |
$begingroup$
The decrease in tree width of any of the three operations will be 0 or 1. The proof is simple (I will take the case of the vertex deletion).
Let $G$ be the original graph and $G' = G - v$ be the graph obtained by deleting $v$.
Let $T'$ be an optimal tree decomposition of $G'$, with $text{tw}(G') = w'$.
Now, construct $T$ from $T'$ by adding $v$ to every bag of $T'$. Clearly, $T$ is a tree decomposition for $G$ with width $text{tw}(G) leq text{tw}(T) = 1 + text{tw}(G') = 1 + w'$.
answered Feb 17 at 14:14
Pål GDPål GD
6,9652342
$endgroup$
1
$begingroup$
Indeed, it seems trivial now. Thank you.
$endgroup$
– SmeltQuake
Feb 17 at 15:06
add a comment |
$begingroup$
The decrease in tree width of any of the three operations will be 0 or 1. The proof is simple (I will take the case of the vertex deletion).
Let $G$ be the original graph and $G' = G - v$ be the graph obtained by deleting $v$.
Let $T'$ be an optimal tree decomposition of $G'$, with $text{tw}(G') = w'$.
Now, construct $T$ from $T'$ by adding $v$ to every bag of $T'$. Clearly, $T$ is a tree decomposition for $G$ with width $text{tw}(G) leq text{tw}(T) = 1 + text{tw}(G') = 1 + w'$.
answered Feb 17 at 14:14
Pål GDPål GD
6,9652342
$endgroup$
The decrease in tree width of any of the three operations will be 0 or 1. The proof is simple (I will take the case of the vertex deletion).
Let $G$ be the original graph and $G' = G - v$ be the graph obtained by deleting $v$.
Let $T'$ be an optimal tree decomposition of $G'$, with $text{tw}(G') = w'$.
Now, construct $T$ from $T'$ by adding $v$ to every bag of $T'$. Clearly, $T$ is a tree decomposition for $G$ with width $text{tw}(G) leq text{tw}(T) = 1 + text{tw}(G') = 1 + w'$.
answered Feb 17 at 14:14
Pål GDPål GD
6,9652342
edited Feb 17 at 14:39
answered Feb 17 at 14:14
Pål GDPål GD
6,9652342
answered Feb 17 at 14:14
Pål GDPål GD
6,9652342
answered Feb 17 at 14:14
Pål GDPål GD
6,9652342
6,9652342
1
$begingroup$
Indeed, it seems trivial now. Thank you.
$endgroup$
– SmeltQuake
Feb 17 at 15:06
add a comment |
1
$begingroup$
Indeed, it seems trivial now. Thank you.
$endgroup$
– SmeltQuake
Feb 17 at 15:06
1
1
$begingroup$
Indeed, it seems trivial now. Thank you.
$endgroup$
– SmeltQuake
Feb 17 at 15:06
$begingroup$
Indeed, it seems trivial now. Thank you.
$endgroup$
– SmeltQuake
Feb 17 at 15:06
add a comment |
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