How many vertices should be disconnected to make a graph acyclic?

Given an undirected graph with some cycles:

enter image description here

we can “disconnect” the red vertex by adding a separate vertex to each of the edges adjacent to it:

enter image description here

In this case, disconnecting a single vertex makes the graph acyclic.

My questions:

  • What is a term for the smallest number of vertices that must be “disconnected” like this, to make the graph acyclic?
  • What is an algorithm for finding a smallest such set of vertices?

(I found some related concepts, but they are different:

  • The circuit rank is the number of edges that have to be removed to make the graph acyclic; it always equals the number of edges minus the number of vertices plus the number of components.
  • The vertex connectivity is the number of vertices that have to be removed to make the graph disconnected.)

lie groups – Generating $K$-types of a $(mathfrak g,K)$-module for $K$ disconnected

Let $G$ be a real reductive Lie group, let $K$ be a maximal compact subgroup of $G$, and let $V$ be a $(mathfrak g,K)$-module. For $sigmainwidehat{K}$ we denote the $sigma$-isotypic component of $V$ by $V_sigma$, so that $V=bigoplus_{sigmainwidehat{K}}V_sigma$.

Question: Suppose that $V$ is generated by a $K$-invariant subspace $Wsubseteq V_sigma$ for some $sigma$, i.e., $V=U(mathfrak g)W$. Is it true that $V_sigma=U(mathfrak g)^KW$? Here $U(mathfrak g)^K$ is the $K$-invariant part of the enveloping algebra $U(mathfrak g)$ under the adjoint action of $K$.

This statement is true when $K$ is connected, and basically follows from Proposition 9.1.10 in Dixmier’s book Enveloping Algebras. However, I am interested in the cases where $K$ is disconnected. In the latter cases, I cannot generalize Dixmier’s argument. But it appears to me that in some references/papers it is assumed to be true.

plugins – Create an independent product page disconnected from my store using woocommerce

plugins – Create an independent product page disconnected from my store using woocommerce – WordPress Development Stack Exchange

c# – Entity Framework Core: Connected vs Disconnected Insert/Updates

What are the use cases or scenarios where programmer would use Entity Framework Core Connected vs Disconnected scenario in updating a SQL record? I am trying to find advantages/disadvantages for each (No recommendation), and then given that, our company will find proper time to apply each.

Using Net Core 3.1

Does Windscribe VPN keep logs even when disconnected?

Does Windscribe (or generally any VPN service) keep traffic logs even when they are disconnected? (Disconnected here means when the ‘power-button’ like button in the VPN’s app is switched off). Some do keep logs when the VPN is on (connected to a server) but do they keep it even when not connected to another server?

ubuntu – Stale SMB shares from disconnected ZFS pool

When I connect to my samba server, it lists a bunch of shares that should no longer be available. They were originally from some ZFS volumes that had sharesmb=on, but that zpool is no longer connected to the server.

How can I remove these outdated samba shares?

Disconnected by openvpn server in first two hours

I have setup the openvpn server and client. The client could connect to the server. Then, the server will aways disconnect the client after around two hours(refer to the log of server). And to my surprise, the client shows the connection still on. But I can’t send any data from client to server at that time.

Here is the config of server(part):

keepalive 10 120
nice 3
verb 0
mute 10
reneg-sec 0

Here is the log of server:

Feb  1 07:59:32 OpenVPN server: peer (client) connected - local IP:
Feb  1 09:33:34 OpenVPN server: peer (client) disconnected, sent: 55 KB, received: 56 KB
Feb  2 07:59:08 OpenVPN server: peer (client) connected - local IP:
Feb  2 09:34:56 OpenVPN server: peer (client) disconnected, sent: 56 KB, received: 57 KB
Feb  3 07:58:12 OpenVPN server: peer (client) connected - local IP:
Feb  3 09:58:20 OpenVPN server: peer (client) disconnected, sent: 70 KB, received: 71 KB

Here is the config of client(part):

dev tun
proto udp
remote 2222
resolv-retry infinite
auth SHA1
cipher BF-CBC
comp-lzo adaptive
nice 0
verb 3
mute 10

Here is the log of clinet:

2021-2-3 7:58   VPN profile (client) is connecting to ( (protocol(OpenVPN), IP(, interface(tun0))
2021-2-2 7:59   VPN profile (clinet) is connecting to ( (protocol(OpenVPN), IP(, interface(tun0))
2021-2-1 7:59   VPN profile (clinet) is connecting to ( (protocol(OpenVPN), IP(, interface(tun0))

4g – SMS Only Works When Disconnected from LTE

I have a really weird issue that I can’t find any information on: My SMS only works when I am disconnected from LTE.

Essentially what happens is that if I’m connected to LTE (I can be on WiFi too that doesn’t matter), the little “cell bars” in the top right of the screen turns into a little stop sign and I can’t send or receive SMS. I probably can’t receive calls either but I can make them and when I do it turns the stop sign into cell bars.

If I go to settings -> connections -> mobile networks -> network mode and change it to anything besides “LTE/3G/2G (auto connect)” then the stop sign changes to cell bars and I get all my text messages.

My service provider is mint mobile and my phone is a galaxy note 10. I’ve tried going to network operators and selecting all the options for manual as well as automatic and it doesnt help. I’ve double checked the APN is correct according to mint and I’ve also tried changing it to whatever options are there.

This has happened on multiple sim cards from mint, but on one of them it “fixed itself” somehow in about a week after installing it. I haven’t been able to try another service provider’s sim.

I’ve attached some screenshots of the issue to show whats going on more clearly:

On LTE, note stop sign in top right

On 3G, bars on top right

On LTE, service state: out of service

On 3G, service state: in service

windows – Turn off PC if bluetooth device is disconnected (proximity) or loses connection

Is there such software, I’ve looked into Windows security features for such an option and it doesn’t exist. I have found that you can unlock your PC with your Bluetooth device but can’t shut it down if you leave your PC for instance.

My idea is as follows, I have a smartwatch (or a phone) and I keep it on me the whole time. I would like to just leave my PC and if my Bluetooth (watch) disconnects from my PC I would like it to shutdown.

  • if Bluetooth device disconnects > turn off PC

Why? Many times I forget to turn off my computer either at work or at home and as I wear my watch or have my phone on, I can just connect it over with Bluetooth and if I leave for whatever reason it can just shut down (or hibernate)

  • I have tried basic google search for such software but haven’t found one.

networking – Driver IGB has reset and network disconnected

i use ubuntu 18.04 for my server, and my ethernet card is intel i210 Gigabyte Network, some time my server is disconnected from network and display this error is :

[363794.283022] igb 0000:05:00.0 enp5s0: Reset adapter
[364565.076382] igb 0000:05:00.0 enp5s0: Reset adapter
[366066.216858] igb 0000:05:00.0 enp5s0: Reset adapter
[366088.232214] igb 0000:05:00.0 enp5s0: Reset adapter
[366105.127736] igb 0000:05:00.0 enp5s0: Reset adapter

please help me

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