How does treewidth behave under graph minor operations?












3












$begingroup$


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 tree-width graph-minors







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    $begingroup$
    Might I suggest adding the tags 'treewidth' or 'graph-minors'? I'm unable to do so, as they are unused tags. Searching for them as normal keywords brings up quite a few results.
    $endgroup$
    – SmeltQuake
    6 hours ago










  • $begingroup$
    @Raphael Can you replace the tag tree-width with treewidth? "Treewidth" is much more popular than "tree-width". I typed "tree-width" accidentally. Thanks!
    $endgroup$
    – Apass.Jack
    4 hours ago
















3












$begingroup$


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 tree-width graph-minors







share|cite|improve this question









New contributor




SmeltQuake is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$








  • 1




    $begingroup$
    Might I suggest adding the tags 'treewidth' or 'graph-minors'? I'm unable to do so, as they are unused tags. Searching for them as normal keywords brings up quite a few results.
    $endgroup$
    – SmeltQuake
    6 hours ago










  • $begingroup$
    @Raphael Can you replace the tag tree-width with treewidth? "Treewidth" is much more popular than "tree-width". I typed "tree-width" accidentally. Thanks!
    $endgroup$
    – Apass.Jack
    4 hours ago














3












3








3


1



$begingroup$


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 tree-width graph-minors







share|cite|improve this question









New contributor




SmeltQuake is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$




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 tree-width graph-minors




graph-theory discrete-mathematics tree-width graph-minors






share|cite|improve this question









New contributor




SmeltQuake is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question









New contributor




SmeltQuake is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|cite|improve this question




share|cite|improve this question








edited 4 hours ago









Apass.Jack

10.8k1939




10.8k1939






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asked 6 hours ago









SmeltQuakeSmeltQuake

183




183




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SmeltQuake is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






SmeltQuake is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.








  • 1




    $begingroup$
    Might I suggest adding the tags 'treewidth' or 'graph-minors'? I'm unable to do so, as they are unused tags. Searching for them as normal keywords brings up quite a few results.
    $endgroup$
    – SmeltQuake
    6 hours ago










  • $begingroup$
    @Raphael Can you replace the tag tree-width with treewidth? "Treewidth" is much more popular than "tree-width". I typed "tree-width" accidentally. Thanks!
    $endgroup$
    – Apass.Jack
    4 hours ago














  • 1




    $begingroup$
    Might I suggest adding the tags 'treewidth' or 'graph-minors'? I'm unable to do so, as they are unused tags. Searching for them as normal keywords brings up quite a few results.
    $endgroup$
    – SmeltQuake
    6 hours ago










  • $begingroup$
    @Raphael Can you replace the tag tree-width with treewidth? "Treewidth" is much more popular than "tree-width". I typed "tree-width" accidentally. Thanks!
    $endgroup$
    – Apass.Jack
    4 hours ago








1




1




$begingroup$
Might I suggest adding the tags 'treewidth' or 'graph-minors'? I'm unable to do so, as they are unused tags. Searching for them as normal keywords brings up quite a few results.
$endgroup$
– SmeltQuake
6 hours ago




$begingroup$
Might I suggest adding the tags 'treewidth' or 'graph-minors'? I'm unable to do so, as they are unused tags. Searching for them as normal keywords brings up quite a few results.
$endgroup$
– SmeltQuake
6 hours ago












$begingroup$
@Raphael Can you replace the tag tree-width with treewidth? "Treewidth" is much more popular than "tree-width". I typed "tree-width" accidentally. Thanks!
$endgroup$
– Apass.Jack
4 hours ago




$begingroup$
@Raphael Can you replace the tag tree-width with treewidth? "Treewidth" is much more popular than "tree-width". I typed "tree-width" accidentally. Thanks!
$endgroup$
– Apass.Jack
4 hours ago










1 Answer
1






active

oldest

votes


















2












$begingroup$

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 $text{tw}(G) leq text{tw}(T) = 1 + text{tw}(G') = 1 + w'$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Indeed, it seems trivial now. Thank you.
    $endgroup$
    – SmeltQuake
    4 hours ago











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1 Answer
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1 Answer
1






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2












$begingroup$

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 $text{tw}(G) leq text{tw}(T) = 1 + text{tw}(G') = 1 + w'$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Indeed, it seems trivial now. Thank you.
    $endgroup$
    – SmeltQuake
    4 hours ago
















2












$begingroup$

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 $text{tw}(G) leq text{tw}(T) = 1 + text{tw}(G') = 1 + w'$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Indeed, it seems trivial now. Thank you.
    $endgroup$
    – SmeltQuake
    4 hours ago














2












2








2





$begingroup$

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 $text{tw}(G) leq text{tw}(T) = 1 + text{tw}(G') = 1 + w'$.






share|cite|improve this answer











$endgroup$



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 $text{tw}(G) leq text{tw}(T) = 1 + text{tw}(G') = 1 + w'$.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited 5 hours ago

























answered 5 hours ago









Pål GDPål GD

6,8802342




6,8802342












  • $begingroup$
    Indeed, it seems trivial now. Thank you.
    $endgroup$
    – SmeltQuake
    4 hours ago


















  • $begingroup$
    Indeed, it seems trivial now. Thank you.
    $endgroup$
    – SmeltQuake
    4 hours ago
















$begingroup$
Indeed, it seems trivial now. Thank you.
$endgroup$
– SmeltQuake
4 hours ago




$begingroup$
Indeed, it seems trivial now. Thank you.
$endgroup$
– SmeltQuake
4 hours ago










SmeltQuake is a new contributor. Be nice, and check out our Code of Conduct.










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