Hey there, let’s talk about paths inside of a graph. We draw a directed weighted graph and compare node-based paths to edge-based paths. Node lists alone cannot tell which edge you took when parallel edges have different costs. Edge-based paths use tuples of start, destination and weight so every route is clear.

We also cover simple paths with no repeated edges or vertices, cyclic paths that start and end on the same node, and how a cycle proves the graph is cyclic. Then we explain DAGs, directed acyclic graphs, with examples that stay acyclic and ones that gain cycles.

This sets up algorithms for later videos. Leave a comment with topics you want next. Thanks for watching and please subscribe.

00:00 Introduction to Graph Paths
00:28 Drawing a Sample Graph
02:25 What is a Path
02:42 Node-Based Paths
03:27 Adding Weights and Parallel Edges
04:34 Example Paths with Different Costs
07:19 Edge-Based Paths
07:35 Representing Paths with Tuples
11:35 Order Matters in Directed Graphs
12:52 Simple Paths
15:34 Cyclic Paths
18:22 Directed Acyclic Graphs DAGs
19:08 Building a DAG Example
21:41 Adding a Cycle to Break the DAG
23:15 Conclusion and Future Videos

=-=-=-=-=-=-=-=-=

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Hey there, let’s talk about paths inside of a graph.

Okay, so first thing that I’ll say is you probably should have watched my other videos already so that you understand at least the basics of graphs or maybe you already know and you just came here for a little extra.

So let’s just draw a sample graph here.

I’m going to do like random nodes.

Let’s see what am I supposed to do.

I’m supposed to like full screen that.

I’m going to draw some random nodes with some random values inside of them.

So I’m just going to do like a three here.

Maybe like a three like that.

And then I’m going to duplicate this a few times.

So I’m going to go.

This is a valid graph.

It’s just not a connected graph as we talked about in my last video.

So I’m just going to do this.

I guess that’s okay.

I’m going to change all these numbers.

Even though your graph should be able to support duplicate values in the nodes,

it’s going to be really hard to explain.

So I’m just going to change all this stuff.

9 here and then like 22.

I think that’s probably good.

So I’m going to do this.

And this.

Okay.

So still a valid graph, but there’s no chance that we could have a path anywhere in this

entire graph.

anywhere in this entire graph so that sucks we could by the way do a path

with a undirected graph but I kind of feel like I want to do direction on all

the edges so let’s see let’s do direction and also weight I’m gonna like draw an

arrow here and arrow here there I’m gonna draw edges first in a way that it

like I could find a path almost anywhere I want so I’m just going to like draw a bunch of edges

maybe like another edge over here nope that’s wrong how about this maybe another edge going up

here and an edge going up here and an edge going down there I’m just going to draw a whole bunch

of edges in the hopes that we’ll have a decent graph with lots of path possibilities okay I guess

Oh, no, no, this is not a complete graph.

I’ve just drawn a graph where you can kind of get everywhere.

Okay, so, well, I mean, what is a path?

It’s just sort of like, you know, some sort of a representation of how do you get from

one start node to some destination node, like a map.

So the first thing that we could do is we could say, let’s make a path based on nodes.

I’m going to say node-based paths.

Actually, let me get rid of this and put it on the next slide.

We’ll do this, duplicate it, that way I have the original.

Let’s pretend that our path is going to be just a list of nodes that we travel through.

So I’m going to say node-based and we’ll have one path, we’ll say path1 and then path2,

maybe we’ll do like a second path.

well at first we might think this is totally fine but uh later on it’ll get confusing in terms of

which edge did we actually take so let me show you let me show you what i mean i’m first going

to add weights to all the edges so i’m just going to do like you know four and a three over here

and then like a one and a two little one and then like a six and then like a four i’m just adding

random weights or random costs to all of the edges and um then eventually let’s see when i

let’s see when I finish doing all of that eventually we can probably realize that we

could add more than one edge going from the same start to the same destination node but give it a

different weight so that we have a little bit more of a choice so suppose for the sake of argument

that we have this six and the four here the six can go to the four at a cost of four but maybe the

a cost of one. I don’t know why it’s there. I don’t know why the four is there. Also,

we’ll just say this is our graph. So you can have edges like this that go to the same start

and destination nodes, but have different weights. So a path that is based on nodes

doesn’t really have enough information to tell us which edge we took. Let me show you what I mean.

So suppose we want to get from node, I don’t know, node three, I’ll put an S here for start.

for start. And then I guess I’ll put a D for destination or donut. I’ll put a D there on

node number 22. So we’ll say we’re going to go from node three to node 22. So start and destination

a path based, sorry, a node based path, path one, we could say maybe let’s go from the three

to the six to the four to the 12 to the nine to the 22 and that tells you all the notes that

you would hop on another path could be let’s see that first path when we went from three one six

four twelve nine uh 22 when we go from the six to the four we could have a second path here

that goes from three to one to six to four but takes a different edge this is kind of what i’m

this is kind of what i’m trying to show you notice that we have two different paths i’m going to do

one in uh maybe red so it’s going to be three one six four so from three to one and then six

and then we’ll travel along that four edge once we get to the four we’ll go 12 and then once we

Then the other path, path number two, let’s say that we chose the other edge.

We chose the six to the four edge that only cost us one.

So it would kind of look the same for the most part, except when we finally get to that

other node, we’re actually going to be going this way, right?

And then we’re going to go up and then to the right and then down.

So if you look at this carefully for a little while, maybe pause the video.

These are actually two totally different paths with two totally different costs.

different costs. So, um, you know, the cost of the orange, let me write it in orange.

It’s going to be five plus one. So six, and then a seven there, and then a eight, nine there. And

then, uh, then I guess like a 15 there and then a 16 there. So the cost of the orange pass is going

to be 16. If we switch to the red path, the only difference is the choice of edge we took between

we took between the six and the four node.

So we would increase our cost by three.

So the red path would be a cost of 19.

But how can you tell the difference in a path

that is based on a node?

It doesn’t have enough information, right?

So what we’re gonna do instead

is we’re gonna use edge-based paths.

Edge-based paths.

In some places, you know, you can use node-based paths.

That’s totally fine.

fine but for me I’m going to say let’s do edge based paths edge based paths and now we just have

to figure out how to add that extra information into the path so going from the three to the one

we’re actually traveling along an edge right we’re traveling along that five edge so let’s just stick

a tuple in there a tuple meaning just a little collection of values for those of you in c++ if

items we’re going to use three item tuples tuples not always limited to three items it could be two

three four even probably one um or much more so i’m going to say let’s use a tuple that has three

items per item or tuple and we’ll just say that we specify the start node and then the destination

node so we start at three we go to one and then the weight of the edge that we choose to travel

but we’re putting enough information into our hop

that this can be considered an edge-based path.

So when we go to the one node,

we ended on the one node, sorry, we’ve gone to the one node.

We just ended there.

So we could start on the one node for the next tuple

because then we’re going to go to the,

what are we doing first, red?

I think we’re doing red first, yeah.

We go from the one node to the six nodes.

So I’m going to go comma six.

and then we choose which edge to travel along that’s just the one edge and so 161 means start

at the one node and at the six node and take the one weighted edge then do the same thing

again for the next tuple so we’re at the six now where are we going to go we’re going to go to the

four and then we choose the edge that costs four and then for the next tuple we’re on the four

I’m going to say 12 is the destination and which edge did we choose the two even though there’s

only one choice at least we have enough information for all edges in our entire path so then again

we’re at the 12 we’re going to go to the 9 so I’m going to put a common 9 there and then we choose

to take the edge that uh what was that cost six I can’t read anymore um let me actually just look

real fast six I guess I overrode it I’m just going to assume that it’s a six um I’ve got

I guess I could erase it real fast

or move it or something

okay let’s see what is that

yeah it’s a 6 okay

so we’re on the 12 we’re going to go to the 9

and we choose to take the edge that costs 6

then we’re on the 9

and we choose to go to the 22

using the edge that costs 2

and then this last thing doesn’t need to be done

because we have the entirety of the path

in our edge based list

So I’m actually just going to copy paste this entire path and stick it into the second path

because remember the only difference between the first path and the second path that we

chose is when we were going from 6 to 4 we chose the 1 edge instead of the 4 edge.

So now look at this, I can just change that 4 to a 1 and now we definitely have two different

paths and because we’re using edges to describe our path we can tell the difference.

You know in a computer program we would know exactly which edge to travel along or even

which edge to travel along or even just a user moving their finger along a graph or whatever.

So in my opinion, edge-based paths are a little bit better.

They’re also better because if you practice writing down edge lists like this,

then you get a lot closer to a representation of a graph in your machine called an edge list.

So in an edge list, oops, I put ed light.

I don’t know why I did that.

And now the computer’s not responding.

So my mistake is immortalized.

my mistake is immortalized. Edge list representation of a graph is basically a graph is equal to

a tuple where one item in the tuple is a list of vertices and another item in the tuple

is a list of edges. So if you learn how to do edge-based paths, then you’re already a step

closer to being able to represent a graph inside of your computer program in the machine.

So another thing that I want to point out is that since this graph has direction,

the order of the start node and the destination node absolutely matters for example if we have

this three comma one here indicating we start at three and we end at one well that means there must

be an edge that points from three to one otherwise this is an invalid path if for some reason

if for some reason let me get rid of this here there’s too much stuff on the screen

three this is invalid because it’s saying that we have to start at the one node and end at the three

node but the one node has no has no edge going to the three node so this is totally invalid

does that make sense i hope it makes sense to you if this was an undirected graph where every single

edge did not have direction then the order actually wouldn’t even matter you would just

to see if an edge existed.

So let’s see, we talked about different paths.

You know, typically you would want to choose

the lowest cost path,

but there are algorithms that we can use

in future videos to compute paths.

Let’s see, something called a simple path

is where every single edge in the path is distinct.

So for example, I don’t know,

let’s say we had like a really gross path here.

Let me get rid of, let me get rid of some of this.

Let me get rid of some of this.

Get rid of the oranges maybe.

So that this is a little slightly easier to read.

Let’s see.

Did we choose?

Oh, I have to fix this.

Let’s fix the three and then the one.

And then we chose the four.

So this is the red path indeed.

So a simple path is, nope, I accidentally erased like an extra edge or something.

So I got to do one from the four to the 12.

Okay.

to the 12 okay let me just do that real fast from the 4 to the 12. at a cost of 2. okay so a simple

path is something where you don’t repeat edges so i can also write down simple path here simple path

if this was not a simple path then we would just basically have to repeat an edge somewhere

so um if any of the edges or vertices were repeated this would not be a simple path so let me just i

would not be a simple path so let me just I don’t know let me just update this real fast let’s say

we do oh what should we do let’s do I don’t know when we go from three to one to six to four

let’s go back from four to six and then we’ll take the other edge back to four and then we’ll

go up to 12 so this will be like um not a simple path if we say gosh I should have written this

this down before i drew it three to one and then one to six and then six to four and then four back

to six okay so i’m going to add another tuple so starting at four going back to six and taking

that three edge i think is what we’re looking at and then another tuple to get back when we go from

six to four and then take the um one cost edge and then once we’re at four then we can go four

cost of two and then 12 to nine at a cost of six and then nine to 22 at a cost of two okay so

i mean i’m just saying you can make a path where you just repeat a whole bunch of the edges and

nodes and you’re just like going in a circle forever like whatever you really want it’s not

necessarily going to be an efficient path or something that an algorithm should produce

but you can have a path like that maybe your character is just going crazy in the game and

because we repeat stuff on the previous slide.

Since we did not repeat anything, this is a simple path.

Okay, let’s see.

Something else that I wanna talk about is a cyclic path.

A cyclic path is basically a path

where you start and end on the same node.

Being able to find a cyclic path in a graph

is a way that you can prove that the graph is itself cyclic.

is itself cyclic. Let me try to get rid of this here. Yeah, there we go. Okay, so just

pretend for the sake of argument that I’m just kind of looking around and I’m trying to find,

you know, some path that proves I can start at the same node and end on that node without

repeating edges. So let’s see. Right, so I’m going to do take the three and go to the six.

to the six and then so i want to try if i start on the three then i want to try to find a way to get

back to the three again so i go from three to six and then i can go from six to four

and then if i go from four to six over here i’m not repeating that edge going from six to four i’m

using a new edge from four to six and then i can go from six to three again not actually repeating

means I have found a cycle in the graph. Let me just describe this path real fast in terms of

edges. So we started on three. Oops. We started on three and then we went to six and it cost us three.

And then once we were on six, we went to four. So I’m going to go from six to four. And we chose

the edge that cost us four. And then once we were on the four, we just went backwards to the six.

four going to six and we took the edge that cost three once we were on six we took the six to three

edge so six to three whoops six to three edge and that cost us three yeah did I do that right

when we go from the three to the six in the beginning I wrote down a three but I should

have written down a two hopefully you cringed on my behalf right because I was I think that red

on the left side so my brain was looking at the three but we were really taking the arrow that

points down from the three to the six so there’s a two there let me just double check one more time

six to four at cost of four four to six at cost of three six two three at cost of three

so we have started and ended uh on the same node and therefore we have found a cycle in this graph

which means this graph is cyclic what is cyclic just means it’s a graph that has a cycle

It’s a graph that has a cycle anywhere inside of it.

So this is a cyclic graph.

Make sense?

Okay, why is this important?

Well, there’s kind of like a special type of graph called a DAG,

which means a directed…

Here, let me write this down real fast.

On the next slide, so that we don’t get confused.

On the next slide, I’m going to say we have a DAG

is basically a directed…

directed, acyclic graph. What does A mean when it’s in front of something? It just means it’s

not that thing. So acyclic means not cyclic and then graph. So in some algorithms you are required

to, you know, like have a DAG before you can even use the algorithm or if you’re being tested

somewhere. So let’s just start by, you know, drawing like a little bit of a graph and we’ll

have a graph that qualifies. So I have a little graph here. If it was empty,

I don’t know if you could call it a DAG or not. I think I would because it’s empty. It could be

directed. So we have like a graph here. I would call this a DAG because it’s a graph. We don’t

have any edges, but like I could assume that this would be a directed graph. But just to make this

a better example, let’s just say we have like two nodes here. And you could also probably assume

DAG because you could assume that there are no edges in this graph but if we did add an edge

a second from now then it would be directed not the best example but so let’s just add an edge here

oh actually let’s add an edge that’s undirected real fast from from two to six and then maybe

I’ll just kind of duplicate some of these we’ll do a couple more so I can just make this more

so ask yourself is this a DAG directed acyclic well there are no cycles that I can see in the

graph there’s no paths that you can find to kind of like end where you started but this is a it is

a graph and I don’t see any cycles but it’s not directed because these edges don’t have direction

so that means right away we have to upgrade every single edge to have direction so I guess I was

going to draw a bunch more edges and now I’m just going to only draw this one edge

So this is definitely a DAG if we just sort of do it like this, maybe another edge like this,

because I don’t know about you, but I cannot find any path that allows me to start and end

on the same node. Let’s just double check it. We’ll do the two. Can we find our way back to

the two? No, because there are no incoming edges to the two. All right, that’s good. Let me just

do a check here. What about the six? If we go out to the six, from the six to the four to the

6 from the 6 to the 4 to the 11 it would have been dangerous if we could get to the 2 but we

can’t even get to the 2 so that’s fine that means the 6 is okay we cannot find our way back there

the 4 we can go to the 11 but we can’t go any further we’re at a dead end we can’t even go to

the 6 that means the 4 can’t find its way back to itself where you can’t find a path that goes back

the 11 has only incoming edges so that’s automatically you know a dead end so all of

nodes qualify as we could not start and also end at that node with a single path so that’s good

that means this is a DAG a directed acyclic graph if you want to make this into not a DAG just figure

out how to add a path so that you end up with a cycle so what was the first thing we did the 2

goes to the 6 goes to the 4 goes to 11 then it’s a dead end we could either add an edge from 11 to 2

to make a cycle but that’s a little bit too easy right because if you see any pair of nodes

right because if you see any pair of nodes with arrows kind of pointing back and forth between the

two nodes then right away that alone is a cycle so I’m just going to maybe do an arrow or an edge

pointing from four to two then it doesn’t look so much like a cycle although it’s not too hard

to realize that there is a cycle in this graph in this graph so this is not a DAG let me show you

and then from four to two.

And therefore we would have a cycle.

So this whole graph is not considered acyclic.

This graph has a cycle.

And if we want to do an edge-based path,

we don’t have weights on this graph right now.

So we could describe the path as such.

We could say, start at two, go to six.

And if your edge list is of a graph

that doesn’t even have weight at all,

then you don’t really need that third item in the tuple.

item in the tuple you can just do a start node and a destination node this does kind of preclude

you from having parallel edges you know two different edges that point from the same start

node to the same end node so as i go from two to six and then from six to uh four and then from

four to two and then we have a cycle and that’s it okay i think this is all i really wanted to

talk about for this particular video in future videos we’re going to do algorithms and then

some others up already that have graph properties and connectedness and definitions and things like

that. So leave a comment asking about some algorithm or some property of a graph that

you might be interested in. Otherwise, thank you so much for watching this video. I hope you learned

a little bit of stuff. And I hope you had a little bit of fun. I’ll see you in the next video.

Yeah, I’m gonna go eat some cookies. No, I shouldn’t do that. But I’m still going to.

Hey everybody, thanks for watching this video again from the bottom of my heart.

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The post Graph Paths Explained: Node vs Edge Based, Simple Paths, Cycles and DAGs appeared first on NeuralLantern.com.

Hello there. Lets talk about graphs and whether they are connected or not or maybe whether they are strongly connected versus weakly connected. We look at undirected graphs first and show what makes a graph connected meaning every node can reach every other node by a path. We add edges and check the connections step by step while also noting subgraphs. Then we switch to directed graphs and explain strongly connected versus weakly connected. Strongly connected follows the edge directions for every path. Weakly connected ignores the directions and checks if the graph would still be connected. Examples show how adding edges changes the status of the graph.

00:00 Introduction to Graph Connectedness
00:01 Drawing Nodes and a Disconnected Graph
01:56 Defining a Connected Graph
03:21 Introducing Subgraphs
04:48 Adding Edges to Create a Connected Graph
06:32 Adding Directions to Create a Directed Graph
08:03 Strongly Connected Versus Weakly Connected
08:34 Removing an Edge to Show Disconnection
09:29 Checking and Achieving Weak Connectivity
11:18 Adding Edges to Create a Strongly Connected Graph
13:20 Confirming the Strongly Connected Graph
13:32 Conclusion
14:17 Outro and Subscribe Request

=-=-=-=-=-=-=-=-=

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Hello there! Let’s talk about graphs and whether they’re connected or not or maybe whether they’re

strongly connected versus weakly connected versus not connected. You never know.

Okay, so hopefully you’ve seen my previous videos by now where we just talked about the basics of

paper here i’m just going to start drawing a graph remember graphs are nodes and edges where

the nodes are like little t-type data structures that can hold a specific type of value um so i’m

going to say our our nodes hold integers um i think i made that one too small that’s okay

and the edges uh could have weights or maybe not weights at all depending on if you have

a weighted or an unweighted graph so i’m just going to start duplicating my nodes a little bit

going to make a random graph. I need to probably change these values. Technically your graphs

should support duplicate values in the nodes and also the weights, but I don’t want to because

it’s easier to talk about the graph when all the values are distinct. So for starters, we have a

bunch of nodes and they’re not really connected to each other. Is this a valid graph? Yeah it is.

It’s just not a connected graph. So the terminology that we’re going to use for

undirected graphs meaning the edges don’t have direction is going to be connected or not you

know i usually just say a not connected graph or a disconnected graph so just basically like

connected or not connected okay so this is a not connected graph i’m going to say not connected

not cinnamon i’m thinking about food as usual

connected graph why is it not a connected graph well to define what does it mean to be a connected

graph it just means that every single node can find a path to every single other node

in an undirected graph so we can just double check this real fast we can say you know can

node number two find a path to uh you know let me put a question here let’s say we’re going to check

you should be checking all the nodes one by one by one to see if if the node in particular can

first node that can’t do it means the whole graph is not considered connected so you know two it

can’t reach one so that means it’s not a connected graph so this is not a connected graph because one

or more nodes violate the rule of you have to be able to reach every single other node okay so that

means if we want a connected graph we probably just have to start adding edges so I’m going to

go ahead and uh let me let me write the word connected here and then I can like either circle

cross it out. So right now we have no, this is not a connected graph. I’m going to start adding

edges. We’re going to stick with an undirected graph for now. So I’m going to connect two and one

and two and nine, maybe one and 13. Actually, you know what? I’m going to erase this edge

and maybe connect these over here. And the thing is, this is a valid graph that you’re looking at.

it’s just not a connected graph.

Some other terminology I didn’t mention in other videos

is that we can have subgraphs.

We have like a subgraph here and another subgraph here.

Those are basically just two different portions of the same graph

that appear to be connected within themselves.

So those are subgraphs.

You can probably tell right away that this is not a connected graph.

You can double check to be sure.

You know, it’s good to do things the hard way

so you don’t get the wrong answer

when you’re really trying to figure something out.

So let’s look at two.

So let’s look at two. Can two reach one? Yes. So that’s good. Can two reach nine? Yeah,

it just hops to one and then hops to nine. So two can reach one and also nine. That’s good.

Can two reach three? Nope. So this is still not a connected graph.

Okay, I’m going to add one more edge. And then we’ll just check again.

Where do you think I’m going to do it? Maybe if I add, I’ll add a funky edge going from 13 to like

to like nine or something just so it feels a little bit more awkward okay let’s check one by

one by one let me actually get rid of this edge on the previous slide so we can still say that’s

not connected let’s check one by one by one every node must be able to reach every single other

node so I’m gonna look at two and I’m gonna say all right two can it reach one yes can it reach

to nine. So that’s good. Can it reach three? Yeah, it can go from two to one to nine to 13

to three, and then it can reach three. And because we just saw that it went through 13 to reach three,

that means two can reach every single other node. This is going to be more important as a double

check when we have directed graphs in a second. So two is good. So the next thing you want to do is

the next thing you want to do is just move on to some other node and do the exact same process you

know can can nine reach a one yes okay it can reach a one can it reach a two yeah it goes through

the one can it reach the 13 yep can it reach the three yep okay so we just check every single

destination node starting at the node nine but here’s a little bit of a shortcut try to find a

node that is right next to a node that you know can reach everything and if the node in question

that previously connected node, vertex,

then we know that that node in particular is also connected.

So what I’m saying is the one,

instead of checking every single other node in the graph,

which might be really hard

if we have a whole bunch of nodes in our graph,

we’ll just ask, can the one reach the two?

Because if it can, the two can reach everything.

So that means the one actually can reach everything as well.

So the one can reach the two.

So the one is connected to everything.

the three, the three can reach the one and the one can reach everything. I’m doing things the

hard way on purpose just to prove a point. So the three can reach everything. The 13 can reach the

three in one hop. So the 13 can reach everything. The nine, it can reach the one and it can reach

the 13 in one hop. So the nine can also actually reach everything. So that means this is indeed a

the next thing that we’re going to do is add direction to our graph you know we could do like

more examples maybe leave a comment if you want me to do some super extra complicated examples for

just a undirected and connected graph i’m ignoring weight by the way because it doesn’t really matter

for these purposes so let’s upgrade our graph so uh maybe i should put a question mark here

that would have been a good idea we’re just asking you know is it connected or not

And I’m just going to start adding direction to every single edge.

Let’s see if I’m following my notes.

Single node, subgraphs, right?

Well, it wasn’t connected at first and then we started adding.

Okay, so that’s basically okay.

Let’s add direction to every single edge.

So every single edge has to have a direction

or every single edge has to have no direction.

You can’t really mix and match.

So I’m just going to go through every single edge and add a direction like this.

and you know to be fair I think I’ve said this in my other videos an undirected edge is kind of the

same thing as a directed edge going in or two directed edges going in both directions so this

3 and 13 what’s going on with my computer hello this 3 and 13 edge it would have kind of been the

same as just having two edges going in opposite directions so maybe I’ll just do that right now

for this huge weird edge I’m just going to put like an arrow over here instead of totally redrawing

Then I double check that every single edge has a direction so that I have a valid graph.

And now I can ask, is this connected or not?

Actually, when it comes to a directed graph, you don’t really have connected or not anymore.

You have strongly connected versus weakly connected.

Maybe I should just separate these words and then just kind of like move them.

So let’s get rid of this.

to this save this over here move it down so what am i talking about um well first let me remove an

edge real quick i’ll remove this edge right here so now we have two sub graphs in this graph you

can tell for sure if this was an undirected graph it would not be connected but now that we’re

dealing with direction we have to ask is it strongly connected versus weakly so what does

can every node reach every other node if you obey the direction of the edges so that’s clearly not

true there’s no edge at all that even connects this one and nine so it’s definitely not strongly

connected weakly connected on the other hand means if you temporarily ignore the direction

of the edges then could you consider the graph connected so if we pretended that there were no

arrows here would we think that it’s a connected graph if the answer is yes then it’s at least a

So this is not weakly connected because if this was an undirected graph, it’s not even connected at all.

So that sucks.

I’m going to go ahead and add another edge here.

So I’m going to add an edge from 2 to 9.

So the first thing is, oh, sorry, I have to add another edge.

Let me do from 1 to 13 for fun.

So now the first thing is, let’s check to see if this is a strongly connected graph.

node is unreachable by any other node so this is definitely not strongly

connected because all the other nodes would be would have to be able to reach

too alright so first thing you could do just to very quickly check is just look

at every single node and ask is it reachable you know by anything and then

after that you can do things the hard way so it’s not strongly connected what

about weakly connected if we ignore the direction of the graph sorry if we

then yeah, we could reach every single node from every other node. Let’s just double check that

real fast. I’m going to say, you know, the two node, if we’re ignoring direction, can it reach

the one? Yes. Can it reach the 13? Yes. The three? Yes. Can it reach the nine? Yes. So that means the

two node is sort of like weakly connected to every other node in the graph. Do the same thing for the

one node. Previously, we just said, you know, check everything one by one or use a shortcut.

one by one or use a shortcut. So I’m going to use the shortcut. The one node, if we ignore direction

can reach the two nodes. So I’m going to say the one is weakly connected, at least everything else.

I’m not even going to check to see if it’s strongly connected to everything else. I think the answer

is no, but we do the same thing for the 13. The 13 can reach the one pretty fast. The three can

reach the 13. The nine can reach the two. So every single node can weakly reach. That’s not an

a weakly connected graph. Okay, cool. How can we make this a strongly connected graph? Well,

we just have to add a bunch of edges. So that’s what I’m going to do. I’m going to kind of like

look at what is missing here. So we’ll ask the two, can the two reach and let’s obey the direction

of the edges again. Can the two reach the one? Yes. Can the two reach the 13? Yes. Can it reach

Yes, so that means right now the two is strongly connected to everything else.

So we’re on track for a strongly connected graph.

We then look at the one node and we realize that the one node,

it can reach the 13 and the three and it can also reach the nine,

but it can’t reach the two.

That’s one thing that was blocking us last time from saying strongly connected.

So we just have to add another edge if we want this to be a strongly connected graph.

We have to figure out how can we get the one to reach the two.

the 2 we can go through something if we want to that’s fine we can also just

write an edge that just goes back to the 2 but that’s I don’t know that’s kind of

boring I’m gonna make an edge that goes from 1 to 9 and then I’m gonna make an

edge that goes from 9 to 2 so looking at the 1 node it can now reach the 2 if we

go through the 9 so that’s good can the 1 reach the 9 yes can the 1 reach the 13

three yes okay so the one node can now reach every single other node so now we’re a little bit more

on track for a strongly connected graph look at the 13 node can it reach the one node um actually

yes if we go from 13 to 9 and then 2 and then 1 we can reach the one node since we know the one

node is connected to everything else we can now say that the 13 node is connected to everything

13 so i’ll say three is connected to everything the nine node can reach the two node which is

connected to everything so that means the nine node is also connected to everything and at this

point we now have a strongly connected graph it’s not weakly connected it’s stronger than that

any questions um so let’s see we added a bunch of uh edges i think maybe this is actually all we

need to do for this particular video because we’re just going to talk about connectedness

going to talk about connectedness about 15 minutes not too bad all right so i think i’m

going to leave it here drop a comment if you want to do way more complicated examples if you want to

provide me a sample graph that’s awful and i could just i don’t know in the sometime in the future i

could record a video about it um or just let me know your thoughts thank you so much for watching

this video i hope you had a little bit of fun and i hope you learned a little bit of stuff i’m outie

Patowdy.

Alright, let’s see.

I’m going to do this.

Whoa. Can you see that guy in the background?

Oh, shoot.

I’m just kidding.

I set that as the screensaver.

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Thanks for watching this video again from the bottom of my heart.

I really appreciate it.

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Thank you.

so

I’m going to go ahead and get started.

so

Yeah, I think I’m good.

Hello there.

Let’s talk about whether a graph is connected or not,

or maybe whether it’s strongly connected or weakly connected or not connected at all.

You never know.

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