Efficient algorithm for enumerating $k$-vertex cuts

Let $ G = (V, E)$ be a connected undirected graph. We call $ G$ $ k$ -connected if the removal of any $ k – 1$ vertices leaves $ G$ connected. The 1-connected graphs are the connected graphs, the 2-connected graphs are also known as biconnected graphs, etc. The largest $ k$ so that $ G$ is $ k$ -connected is called the connectivity of $ G$ .

For 1-connected graphs, there is an efficient algorithm that splits a graph into so-called biconnected components, which are connected via articulation points, the points which upon removal would disconnect the graph.

For 2-connected graphs, there similarly is an efficient algorithm that gives triconnected components and so-called 2-vertex cuts: those pairs of vertices whose removal would disconnect the graph.

I am interested in the more general situation for $ k$ -connected graphs. I know that we can determine the connectivity of $ G$ in polynomial time (see for example here), although much slower than linear time like the algorithms above. However, at least at first glance, these algorithms only give me the connectivity as a number. For practical(-ish) reasons, I am interested in enumerating the $ k$ -vertex cuts. So my question is:

What is a (relatively) efficient algorithm for enumerating those sets of $ k$ vertices in a $ k$ -connected (but not $ k+1$ -connected) graph that will disconnect the graph?

We may assume for simplicity that not too many $ k$ -vertex cuts exist, if that makes the question more interesting.

(Because I want to use the algorithm for practical(-ish) reasons, I am potentially interested in fairly fine-grained distinctions: an exponential algorithm that is fast “most of the time” is still useful to me, as is the distinction between polynomial time algorithms.)

Posting Short Cuts to make my Posts look more professional? [migrated]

I’m new here, but i am already addicted to reading and answering questions. I am predominantly on my phone.

What i am wondering, is how can i beef up my posts? I recently learned about using > before a line to create the yellow box effect. Are there any other simple tips and tricks to help make my posts look that much more professional??

find the union of all min cuts of a flow network

I’m trying to solve the following question :

Given a flow network $ N = (G=(V,E),c,s,t)$ . Let $ \mathcal F$ be the set of all minimum cuts. Prove that $ \mathcal F$ is closed under intersections and unions, i.e. for every $ S_1,S_2\in\mathcal F , S_1 \cup S_2 \in \mathcal F$ and $ S_1 \cap S_2 \in \mathcal F $ .

that part I took care of just fine using the min-cut-max-flow theorem.

the other part is the one that I had trouble with :

Given a max flow $ f $ ,find $ S_{min} = \bigcap_{S\in\mathcal F} S \text{ and } S_{max} = \bigcup_{S\in\mathcal F } S $ .

I realized that when considering a min cut $ (S,T)$ , and given the residual graph (which can be built from the given maximum flow) , every vertex that’s reachable from $ s$ ( source node) , must be in $ S$ , so I was able to use that to come up with an algorithm to find $ S_{min}$ .

But as for finding an algorithm for $ S_{max}$ , I’m kinda having trouble putting my finger on a property of a min cut edge, i.e. what does it take to be a cut edge( or for a vertex to be in S) of some min cut?

I’m not looking for the full answer but rather a hint.. any help is appreciated.

Thanks in advance.

Minimum number of tree cuts so that each pair of trees alternates between strictly decreasing and strictly increasing

I want to find the minimum number of tree cuts so that each pair of trees in a sequence of tree alternates between strictly decreasing and strictly increasing. Example: In (2, 3, 5, 7) , the minimum number of tree cuts is 2 – a possible final solution is (2, 1, 5, 4).

My search model is a graph where each node is a possible configuration of all tree heights and each edge is a tree cut (= a decrease of the height of a tree). In this model, a possible path from the initial node to the goal node in the above example would be (2,3,5,7) – (2,1,5,7) – (2,1,5,4). I have used a breadth-first search in it to find the goal node. As BFS don’t traverse already traversed nodes, the part of the graph that I traverse during the search is in fact a tree data structure.

The only improvement to this algorithm that I was able to think was using a priority queue that orders the possible nodes to be explored in increasing order 1st by number of cuts (as traditional BFS already does) and 2nd by the number of strictly increasing/decreasing triplets. This increases the probability that a goal node with the minimum number N of cuts will be within the first nodes of all nodes with N cuts to be evaluated and the search can finish a little faster.

The time required to execute this algorithm grows exponentially with the number of the trees and the height of the trees. Is there any other algorithm/idea which could be used to speed it up ?

Audio levels rapidly increase and then cuts out

So I’m getting this strange error. At first I thought the audio wasn’t working but then I realized that if I quickly got to the volume control at the top right corner when I start the computer the audio works briefly. When I change the volume to a certain level I hear a beep sound as I should. But when I keep on clicking the same level the beep sound gets louder and louder and then suddenly becomes mute.

I’m running Ubuntu 19.

How can an archer beat a foe that cuts all his arrows out of the air?

My PC is an archer—a human no-archetype fighter 4/hawkguard warder 5—who has taken only feats that let him make more ranged attacks and deal more damage with ranged attacks. My PC doesn’t even have a melee weapon.

The enemy is also level 9 but he’s taken the feats Combat Reflexes and Cut from the Air. The enemy’s feats, attack bonus, and really high Dexterity bonus mean that his attack rolls with arrow cutting are on average about 7 points higher than my PC’s attack rolls with his arrows and that the number of times he can cut arrows launched at him exceeds my PC’s rate of fire. The enemy has multiple special senses, including darkvision, tremorsense and see invisibility.

What can my PC do to beat this guy?

Is there an app to lock the screen or something when the wi-fi cuts off?

I think this would be useful if you’re entering data into a form, but you don’t want to accidentally press “submit” when there’s no Internet, causing you to have to re-enter everything.

Also, it would be nice for macros. Say you have a macro doing something on an app that requires Internet. When the Internet fails, the macro goes haywire by clicking on random stuff it’s not supposed to click. It would be best to lock the screen in that case.

I have Android 4.4.2

Perfect matchings and edge cuts in cubic graphs – part 1

Let $ G$ be a bridgeless cubic (simple) graph, and let $ M$ be a perfect matching in $ G$ . $ G-M$ will necessarily be a set of circuits. For example, if we delete a perfect matching from $ K_{3,3}$ we end up with one circuit. If we delete a perfect matching from the Petersen graph we end up with two circuits. And in general, one could end up with many. The following question comes to mind.

Question Let $ G$ be a cubic bridgeless (simple) graph which is cyclically edge-5-connected, and let $ M$ be a perfect matching in $ G$ . Is there a subset $ K$ of edges in $ M$ such that $ G-K$ has exactly two components, such that either none of the two components are circuits, or both are circuits?

For example, any perfect matching in the Petersen graph is an example of such an edge cut.

A graph $ G$ is cyclically edge-5-connected if no set of fewer than 5 edges is cycle separating. A set $ K$ of edges is cycle separating if $ G-K$ is disconnected and at least two of its components contain circuits.

Smooth real line and Dedekind cuts

I am reading Bell’s A primer of infinitesimal analysis, and the real numbers he considers have certain properties for doing synthetic differential geometry. He calls this object the smooth real line. I am not an expert in topos theory, but know some category theory. I know that we can construct the real numbers object in a topos by Dedekind cuts or by Cauchy sequences, and that these do not always coincide in a topos. My question is Is the smooth real line the real numbers object constructed by Dedekind cuts, by Cauchy sequences in a model for synthetic differential geometry? If not, how is related the smooth real line to the other two real numbers object?

After accessing certain data, internet cuts out for all devices on network switch [on hold]

This is hard to describe, so I’ll try to be thorough. Not sure if this is the correct site to ask this.

My setup is that I have a small network switch (1GBPS) in order to be able to wire in my home PC and my work PC for a better connection.

When I access a certain page on the web app I’m working on with my work pc, both my home PC and my work PC completely lose all internet connection for 30 seconds to a minute. The only error I get in my code is a timeout because the connection drops. The page it tends to happen on is a page that loads a customer’s order. It tends to happen most consistently when I do a cache refresh. On a hard refresh, chrome dev tools reports 2.3 MB of data transferred in the network tab. While this is a decent size, it’s not some huge payload that my switch can’t handle.

I’m a ASP.NET developer, so I’m not doing anything crazy with the internet connection (i.e. i’m not writing my own network code that could be tanking it), just using built-in ASP.NET libraries. It does seem to happen more consistently when doing a hard-refresh (CTRL+F5 in chrome) than on a normal refresh.

I have two work PCs, as we’re in transition from a win7 machine to a win10 machine. I have never experienced this with the win7 machine, only the win10. My machine is on 1803 windows 10 build.

What is something that could potentially cause this problem? I don’t know where to look first. I know our win10 PCs have new kinds of group policies on them that could be the cause, but I’d like to try to steer our IT team towards finding this as they wouldn’t be able to reproduce the issue easily.