still looking at networks here we're going to look at trails and paths now in a previous video we talked about something called a walk and a walk is simple you just go from A to B to C to D to C to be any sort of trip through a network is a walk so everything I show you from now until the end of this video is going to be a walk but there are specific types of walks called trails and paths and that's what we're looking at here now when you're talking about a trail the way that I like to think about drones is to think about a trick like a trekker someone walking through looking around on the road now a trekker does not like to walk on the same Road twice and that's what the trail is now this is an example of a trail someone started at Point E and then they moved to point a and into B and then to C and then to e and then to D now at no point did they walk on the same Road twice now if they were to say from D walk back to e then this would no longer be a trail because they've walked on the same road twice so ensure trails are for trekkers it's for people that don't like to repeat edges no Road gets walked down the same twice now trails are walks and trails can also be opened or closed now this is an open walk because it starts and finishes on different vertices now of course if we shorten the trail slightly if we just get rid of this arrow we now have the trail e a to B to C and back to E and that is a closed trail no repeated edges and it starts and finishes at the same vertex so that's trails now paths paths for posties now imagine a B C D and E are houses and a post E is delivering mail to each house now the post he doesn't care if he goes down the same road twice that's not an issue what really matters is that the post he gets to every house once a post he doesn't want to go past the house multiple times that doesn't make sense so no repeated vertices so here's an example of a path we start at C here we go from house C to house B to house to house E and to house D now this is a path no repeated vertices you can see he's only gone to each vertex once now we can have open and closed paths as well that seems a little bit odd so we need a small change to our rules about what a path is now this first of all is an open path because it starts and finishes at different vertices now we can have a closed path if the this was like home base the post office post office to house one to house two to have three to ask for and back to the post office we call that a closed path but you might already be spotting a problem with that so a closed path has the start and the finish of the same vertex C around back to see the wait a minute paths have no repeated vertices that's my definition right there so we just need to tweak my definition slightly no repeated if vertices except first and last now it doesn't always have to be first and last because you can have an open path but you can still consider something a path and have a repeated vertex as long as the repeated vertex is the first one at the last one so now that we know all of this we can create a walk and then define it as either a trail or a path or neither whatever so let's go for it let's go from see real the loop to E to C to B that's it okay let's do that C D E C B now is it closed or open first of all well started at seed and ended B so it's not starting and finishing at the same vertex so it must be an open something an open walk these are a trailer path well no repeated vertices right so we went from C to D to e back to C and then to B C's are repeated vertex and it's not the first in the last one it's the first and the second last one so this is not a path I wonder if it's a trail right trail let's say you this road this edge this edge this edge this edge this edge okay no edge has been repeated there we can say that this is a trail because there are no repeating edges this is an open trail and you should be able to it doesn't matter what I drop there you should be able to categorize it as a trail or path or just a walk if it doesn't fit into either of those categories those are trails and paths