17
votes
Accepted
Is the 3-coloring problem NP-hard on graphs of maximal degree 3?
The answer is no: the 3-coloring problem can be solved in linear time on graphs of maximal degree 3 or less, by application of Brooks' theorem. I wasted some time figuring this out, so I thought I'd ...
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14
votes
Accepted
Finding subgraphs with high treewidth and constant degree
See the paper by Julia Chuzhoy and myself on Treewidth sparsifiers.
We show that one can obtain a subgraph of degree at most 3 with treewidth $\Omega(k/polylog(k))$ where $k$ is the treewidth of $G$. ...
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1
vote
NP-hardness: (planar) directed feedback vertex set problem with bounded degree
Thanks to Prof. @Yixin Cao's construction of the reduction, I think I have found a way to show that Planar-DFVS remains $\textsf{NP}$-hard when $(\Delta_{in}, \Delta_{out}) = (2, 2)$.
We use the ...
- 421
1
vote
NP-hardness: (planar) directed feedback vertex set problem with bounded degree
Warning: this does not completely solve your question because the gadget is not planar. But since I could not find a proof of this, I think it might be worth posting.
Speckenmeyer proved that the ...
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1
vote
Finding subgraphs with high treewidth and constant degree
In the case of pathwidth, reposting here a comment made to me by email by Benjamin Rossman back in 2020 (see also the comments to the answer https://cstheory.stackexchange.com/a/38943):
Every graph G ...
- 8,896
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