Hey folks,
let [; G ;] be a simple graph (no self-loops) with $n$ vertices and $m$ edges.
We say $G$ is $okp$ iff $G$ admits a drawing where all vertices are on the outer face and any internal edge (chord) is crossed at most $k$ times.
For $k=0$, these graphs are exactly the outerplanar graphs and we have $m_0 \leq 2n-3$.
For $k=1$, Auer et al. showed that $m_1 \leq 2.5n-4$
One can further show that $m_2 \leq 3n-5$ and $m_3 \leq 3.25n-6$.
Can we generalize this to any $k$?:
Chaplick et al. showed that $okp$ graphs are $d$-degenerate with $d = \lfloor \sqrt{4k+1}+1 \rfloor$, hence we have $m_k \leq dn \approx 2\sqrt{k}n$.
Now the best lower bound I can think of (and I strongly suspect to be best possible) only has approximately $\sqrt{k}n$ edges (split the large face into equal parts which all contain the largest possible clique).
Below is the example for $k = 2$:
https://preview.redd.it/leyuj3f66lq41.png?width=263&format=png&auto=webp&s=867044421300c163c03f1cc6805c49dd9e1fb2cd
As it is well defined which two edges cross each other (if both are present), we can also look at the linear algebra formulation of this problem:
Let $A$ be the adjacency matrix for $G$.
$G$ is undirected which implies $A$ is symmetric.
As $G$ has no self-loops we have zeroes in the diagonal and since $G$ is outerplanar we have ones in the off diagonal. Now if there exists an edge $(u_i,u_j)$, we have at most $k$ edges that are incident to $u_p$ with $ i < p < j$ and $u_q$ with $q < i$ or $j < q$, hence we have at most $k$ 1-entries in the marked areas, see Figure below.
https://preview.redd.it/m61xlvh95lq41.png?width=1272&format=png&auto=webp&s=03f40652a48a3a88dbe323bacbd0f4a5965caa34
Thus we want to maximize the Frobeniusnorm of a binary matrix subject to the constraints illustrated above.
Thank your for your time!
I don't want to step on anybody's toes here, but the amount of non-dad jokes here in this subreddit really annoys me. First of all, dad jokes CAN be NSFW, it clearly says so in the sub rules. Secondly, it doesn't automatically make it a dad joke if it's from a conversation between you and your child. Most importantly, the jokes that your CHILDREN tell YOU are not dad jokes. The point of a dad joke is that it's so cheesy only a dad who's trying to be funny would make such a joke. That's it. They are stupid plays on words, lame puns and so on. There has to be a clever pun or wordplay for it to be considered a dad joke.
Again, to all the fellow dads, I apologise if I'm sounding too harsh. But I just needed to get it off my chest.
Do your worst!
I'm surprised it hasn't decade.
For context I'm a Refuse Driver (Garbage man) & today I was on food waste. After I'd tipped I was checking the wagon for any defects when I spotted a lone pea balanced on the lifts.
I said "hey look, an escaPEA"
No one near me but it didn't half make me laugh for a good hour or so!
Edit: I can't believe how much this has blown up. Thank you everyone I've had a blast reading through the replies 😂
It really does, I swear!
Because she wanted to see the task manager.
They’re on standbi
Pilot on me!!
Nothing, he was gladiator.
Dad jokes are supposed to be jokes you can tell a kid and they will understand it and find it funny.
This sub is mostly just NSFW puns now.
If it needs a NSFW tag it's not a dad joke. There should just be a NSFW puns subreddit for that.
Edit* I'm not replying any longer and turning off notifications but to all those that say "no one cares", there sure are a lot of you arguing about it. Maybe I'm wrong but you people don't need to be rude about it. If you really don't care, don't comment.
When I got home, they were still there.
What did 0 say to 8 ?
" Nice Belt "
So What did 3 say to 8 ?
" Hey, you two stop making out "
I won't be doing that today!
Where ever you left it 🤷♀️🤭
[Removed]
This morning, my 4 year old daughter.
Daughter: I'm hungry
Me: nerves building, smile widening
Me: Hi hungry, I'm dad.
She had no idea what was going on but I finally did it.
Thank you all for listening.
There hasn't been a post all year!
You take away their little brooms
It was about a weak back.
It’s pronounced “Noel.”
After all his first name is No-vac
What, then, is Chinese rap?
Edit:
Notable mentions from the comments:
-
Spanish/Swedish/Swiss/Serbian hits
-
French/Finnish art
-
Country/Canadian rap
-
Chinese/Country/Canadian rock
-
Turkish/Tunisian/Taiwanese rap
There hasn't been a single post this year!
(Happy 2022 from New Zealand)
Nothing, it just waved
Him: I can explain everything!
(It's his best joke yet I think)
Bob
So that I could frequently say, "I am going to walk 5 miles now."
Edit: My most popular post on Reddit! 😀 Thank you for the awards.
Just to clarify, 12345678
But nobody ever mentions his sister, Onya, who invented the starter pistol.
Me grabbing a soda from my (what I thought was) half full 12pk...
Notices there's only 2;
Me: "Awe man... This is a damn bird box!" Her: "What the hell does that mean?!" Me: (Pulls both cans out & shows them to her) "It's only got Toucans."
I'm not ashamed to admit the look on her face was glorious.
