 Again, this talk is about connected node search against a lazy rober and I will shortly define what all of the things mean. We start with what is a node search strategy. Basically the idea is as follows. We have a graph, we have some cops and robbers, basically there is a robber which is occupying some of the vertices of the graph and we have some cops which we could place on some vertices and then trying to catch the robber. Formally how it looks like. We have a search strategy which basically says we can either add a searcher somewhere to some vertex of the graph, for instance to vertex A and we can for instance add another searcher to vertex B, like say we could first place to A, place to B and then remove searcher from A. And the idea is as follows. Basically when we do this we know that in A there couldn't be a cop, there couldn't be a robber because we just were there and we are in the next vertex and then we removed a cop from A but still robber couldn't get to A. And formally our search strategy is just a sequence of sets of vertices. Basically for every move we have a new set and initially the set of occupied vertices by researchers is just some vertex. It has size 1 and it doesn't really matter which vertex is this, we can start anywhere. And then we, since we either add one or remove one, a symmetric difference between the previous set and the next set is exactly one. Yeah. And the intuition is that basically the robber is sitting somewhere in a graph we don't really know where but whenever we step, we place a cop on the place of the robber then robber has to move somewhere. If he cannot move somewhere then he loses and we win and that's our goal to actually win this game. But anyhow. So there are different variations of what robber can do. I already said that robber here is invisible so basically we don't really know when he is sitting and our strategy should work for any place when he is actually, when he was sitting initially and where he was going to after our moves. And there are two possible known variations of what robber can do basically. He has eyes are lazy which means that he only can move when we place a searcher at his place exactly. If we don't touch his place then he couldn't move. And add a robber which is basically a robber can move whenever he places but again he always can move only to a neighbouring place. Yeah. So anyhow there are two known versions of this thing and yeah the important notion here is the set of free locations. Basically it's the set of positions where the robber could still be after some steps in our search strategy. So basically first imagine what a lazy robber can do then imagine we start from this third place of the previously described search strategy then what we can do we can go to the top then remove someone from here then go to the top again. See this is a fine strategy because robbers if robber is there and we catch him and if robber is there he still cannot move here because he is lazy so he cannot move until we go to his place. So basically this is fine. So robber can go across only through the edges? Yeah, only through the edges. He cannot just jump. Yeah, yeah. He goes along the edges. So basically in this everything is fine but now if we can also remove this one it's still fine. We can remove this one still fine but now we cannot replace a searcher here because if robber was there he could move back there and it will kind of it would make our job worse because he can be now anywhere on the green parts. So this is really a bad move and the good move is the first place a searcher here and then we can place the searcher there because now robber cannot move there because there is a cop there and after this we can finish the whole thing by occupying these places. So now the robber cannot be anywhere. He must be caught at this place even if he was there and if he was there he will be caught earlier. And anyhow the three locations initially it's just everything which is occupied by a searcher on the first move and then it is a bit complicated formula but generally just we take a set of positions on a step i minus one then definitely robber cannot be at any place which is occupied by a searcher which is si and then he can still be somewhere which is like basically it's the picture a bit before. Basically this part corresponds to robber being possible to that robber can actually go there if you don't know the space. So basically if there is a path to some vertex from the place which is now occupied by a searcher but wasn't occupied previously then robber can go there and this is this part. Location initially it's basically the whole vertex set. Set of location is just a subset of vertices. Initially just whole vertex set without one vertex which is the initial place of the first searcher and then it kind of decreases all the time. But it's always some subset of vertices and we want to have of course that it's empty at the end. So when a strategy finishes there is no possible place for a robber to be. Anyhow continuing to difference to a jail robber. In a jail robber case we really cannot in a case we could have put out this searcher and continue further but now we cannot because a jail robber if he sits here he can still move there even without us stepping on his place. So basically we have to keep this one here and so use another searcher to go there. So basically in this case we have to use three searchers which are in this only two. Yeah. So of course we are interested in complete searcher strategies which are always have empty three locations at the end so robber can be only the empty set of vertices. And there is another possible feature of a searcher strategy is that it's monotone. It is not necessarily generally but it can be restricted to these strategies. Basically it means that for every new set of three locations it's always a subset of previously existing set of like locations. Basically just means that we never recontaminate stuff. So if some vertex was green at some point then it can never became blank. If we definitely know that at some place there is no robber then it cannot be changed afterwards basically. And we can define all these numbers. So basically this agile not search number how many searchers actually need to win a game against an agile robber on this graph. This number is just the maximum number of searches which you use during the strategy. And you of course take the best strategy so you minimize across all strategies this number. You can also define monotone agile not search. You can define lazy not search. You can define monotone lazy not search. So these are all different parameters. Now there is some already well known established relationships among the parameters. And interesting thing about this not search strategies is not just interesting on itself but they're also connected to well known graph parameters once the trivets. So in this work it was proven that trivets of G is exactly the same as monotone lazy not search number of G minus one, minus one just technicality. And the same as lazy not search number of G. So you also can see that in this case monotone doesn't really change anything. Then just the lazy not search number is the same as monotone lazy not search number. So basically it doesn't really make sense to have a strategy which is not monotone which allows Robert to go somewhere which he's not supposed to be. Anyhow this is really nice that trivets is the same as this number. But also there is another characterization of this thing which is called tri-vertex separation. And basically it's defined as follows. We have our graph G. Now we have some layout of vertices. So basically just a permutation of them or an ordering of them. And now in the following number. Basically for every position i in this sequence we look at this number. It's the number of other vertices and where is a red if they can go to i through the place after i. So basically they can have one jump to the set of vertices after i and then somehow get back to i. So for instance yeah. So in this graph these two vertices are red because they can jump there and then go to i. Now you cannot really have two jumps before i. You always must have one jump to i or after and then back to i. No, no, no. These edges are just the edges of the graph. So basically these are vertices of the graph, some are ordered and then we can jump on edges of the graph. We cannot just jump arbitrarily. These are all edges. So this was some graph and it was like drawn on this flat sequence. And for instance for vertex i this size was two. And we're interested in to minimize this number across all the presentations. And the number is maximum over all i. And also nice thing that turns out that this is the same as trivets. Okay let's connect. You could imagine that it's indeed true that the same holds for pathways and it's also well known that... But pathways corresponds to agile strategies while trivets corresponds to lazy strategies. So pathways it's all the same only here it's agile strategy and also monotone doesn't really change anything. And here it's also path vertex separation which is easier it's just the number of edges which go from before e to after e. And it's the same with minimizing over all the sequences. Yeah, so this is nice and well known but what is really the point of this paper? It's how does it change when we also enforce connectedness on the on the thing. And what does connectedness mean? I'll define just now. So basically the idea is we enforce the guarded space so the space that the rubber cannot be to be connected. So for instance this is not allowed. You cannot just teleport somewhere and deploy your new search at some random place of the graph. You always have to have this zone connected so you can only kind of expand the little each time. And this actually makes sense because for instance if you, like on applications of not search games, it's like virus cleaning or like cave searching it does really make sense for for the zone to be connected because you can actually teleport in some cave you can only explore them kind of one by one or in the network. You also kind of always have to preserve the connected space of where we have everything under control basically. Yeah. So when there's now only connected searches and for every step which you must have our guarded space to be connected. So the space which is the complement of the three locations. And we also can define analogously all these parameters. We can define connected a gel node search, monotone connected a gel node search, connected lazy search, monotone connected lazy search. Yeah. And the question is what do we pay for that? Basically like obviously every strategy is connected. Every connected strategy is a strategy but not every strategy is connected strategy. So maybe this is actually much more restricted than the usual strategies. Another known thing from this paper is with pathways the price is not that high. That basically we pay only a constant for this. So basically they connected a gel neighbor search is at most monotone connected gel neighbor search and it's the same as connected pathways. Where connected pathways is imagine now part of the competition when we have some bikes and stuff. So in a usual part of the competition we don't require anything about how bugs are connected apart from the definition of all the pathways. But now we also in a connected path it's required that all every prefix of the bugs if you just take a union on the services it must induce a connected sub graph for every such prefix. So for the first bug for the first two bucks for the first three bucks and so this is how connected pathways is defined. And then in the newspaper it was proven that actually connected pathways is the same as monotone connected a gel neighbor search. And also that's actually multiplying by two just monotone a gel neighbor search we get monotone connected a gel neighbor search. So this is nicely between this and this. But for attributes and in contrast laser node search basically we can see that like really there was a paper from which you can tell that actually connected trivets can be really far from trivets it basically can have a factor of logon in between. Anyhow in this work the other study how can you generalize these results or move these results to trivets and to laser searches. That's the most important thing in this paper. So first what they do is they define this parameter is already defined. They define connected trivets and then they show that this is actually all equal the same as this pathways. So the connected trivets is equal to monotone connected laser neighbor search. And the connected trivets is defined analogously to pathways is basically for every path the connected trivets is as follows. We have some root on a typical decomposition and every path from the root to some vertex must again enforce a connected subgraph. So if we take again the union of all of these bugs and look at an induced subgraph it must be connected. And indeed it somehow generalizes the path because in the pathways it basically if you place the roots at the start then it's the same. All the paths from root to somewhere must be connected must induce connected subgraph. And also they define a notion of connected trivets separation. Again it is defined in the same way but in the same way with the layouts and the parameters but the only difference which is why it's connected is we also require that for every i for every vertex there is some j which is before i and also there is an h between j and i. That's the only difference to the just trivets separation. Yeah. And turns out that all of this is equal. It's just the same definition. Anyhow this is the first part but then another interesting thing you can notice that this parameter is actually closed under edge contraction. So basically if we have some layout with some costs if we contract an edge whenever we increase the cost. So basically you can intuitively of course it requires a proof but intuitively it is like this. Whenever we have something which goes from vertex to somewhere if we contract it then it doesn't really change how paths behave which goes which go from the left to there and back to i. If there was a path from something which goes which went through j to i then it still is after the contraction. Anyhow so we defined this parameter closed under edge contraction. So the interesting thing is what really happens when how can we characterize graphs which have low monotone connected laser neighbor search number. So basically let's define ck as the graphs which have this parameter small at most k and let's define abstraction set to this as the graphs which have large number which is strictly modern k but all the contractions of this graph have at most k. So this is kind of minimal graphs which which do not allow us to have this number at most k and since this all makes sense since this is closed under edge contraction. Yeah and probably the most number result of this paper is the characterization of this set abstraction set of c3 because yeah it is infinite that is true but also data show how exactly does it look like it has kind of small description of this set and this is probably yeah this is like the the complicated part because this is like really technical and long proof and also another thing which they show that's the price of connectivity of place researchers is high basically it's kind of a reformulation of these old results they show that there is a graph gk which has monotone laser neighbor search number as 3 and monotone connected research number as 3 plus k and the size of g is about 2 to the k so it's like some logarithmic difference between there is no response time again just it's not due to response time just due to the most which he allows which the robber is allowed to make the laser robber can move only to places laser robber can only move after his location was occupied by a searcher and a gel robber can move after any smooth with a searcher yeah that's the only difference anyhow just to sketch how this equivalence is shown basically this is not very complicated which is just for instance to show that cts is at most mclns we just show that a search strategy which is connected and monotone provides us a layout which has the same cost and basically just there order of verses in which they are occupied by the cops can see that it's very true and for a connected trivix that connected trivix is at most connected trivix separation is basically we have a connected constructed decomposition and our bugs are just the supporting sets of vertex i supporting set is just this which can reach i through verses after i and the vertex on the i's position itself so if we take this as a bug then you can it can only be shown that these bugs really form a connected to decomposition basically the idea is when we move from i when we have edges from i to some other vertices this will be the edges in a tree of bugs in a decomposition and the last part is that when we have a decomposition we actually can produce the search strategy and just we start from the roots note of that decomposition and then go just along the tree and this basically provides the strategy we just have to place searches in the same place where where the difference is between bugs in this DFS along the tree okay so the interesting part is that the creation of abstraction set is is starting with this it's not the final but just some first examples so just to be on the same page let's verify that K4 is a neat abstraction set of C3 why it is because definitely for all kind of structures you need four searches whenever like does matter monotone connects it lays agile you always if there is at most three of the searchers then the robber can always go to the last part four searches to to search the whole clique but then clearly also again after any contraction we are only remain with K3 and then it is enough to have three searches definitely again for all kinds of startages so basically this shows that K4 is in abstraction set of C3 because it has number of four but every contraction has number of three and obviously you can see that graphs like this are also really in abstraction set of C3 this is a bit more general because in this previous graph it doesn't really matter how many distinct there are there may be two maybe three and so on there must be at least two of these things between these two verses but then it may be arbitrary number of them so to draw this nicely the square on the edge means that there can be arbitrary number of these two paths but at least two of them so basically any graph of this form will be in abstraction set but also there will be an element of abstraction set like this there are these variations of the graph and if you glue any two of them it will still be a graph in abstraction set and this is the theorem that the set of abstraction is precisely this graph this family of graphs and any any of these two graphs connected by this vertex not connected but glued together by this vertex and this is the whole description of this abstraction set which is kind of nice okay so a bit of the like some very slight sketch of how it how it's proved that exactly this is the abstraction set basically first the start is to show that in the graph which is the abstraction set there is always it's either connected or it has exactly one cut vertex and moreover this cut vertex has exactly two vacant components connected to it so basically this is impossible since there are two cut vertices and this is also impossible since there is this is basically to to use the fact that after contraction since it's an abstraction set after contraction the graph has number 3 or less and then construct a layout by seeing that either this or this has at most k either this or this k either this or this k either this or this k and then we can construct a layout which is also at most k so this cannot be in abstraction it must have it must have two connected components that we already saw but this connected components must be exactly in this set so also they prove that another technical step is that whenever we have these two vertices we can actually add arbitrary many of them it doesn't really matter how many of these they are it doesn't change if the graph is an abstraction set or not that's basically why this looks so weird with this abstraction set when we have this arbitrary number of two paths there is some simplicity you cannot have this single edge somewhere you cannot have this kind of separating edge and you cannot have this marginal edge I will sketch briefly how to prove that you cannot have this for instance basically if the graph belongs to city then they cannot be like this and the idea is quite simple actually you just take a layout since you know that after this contraction the graph will not be in abstraction and then if you just take this layout which achieves the number of three and then expand this layout to the layout of the whole graph this is the layout of the graph of the contract in this edge and this is the layout of the whole graph if you just take this vertex which was after the contraction and then expand it to this edge then you can really see that really it has the same it has the same connectivity on the left to the right of this it's the same in this layout since this edge doesn't really change anything so this is how to prove one of these small technical steps okay so yeah so this last part this construction which shows that there is a large price of connectivity is basically this simple example we take this kind of a tree which is a bit which is this is the second this is the kind of T2 tree which is tree which is tree and then it's defined analogously for all the larger ars basically you can see that monotonous laser neighbor search for this is always tree because we will drop a searcher here we will drop a searcher there and we will just with three we will clear everything but with a connected connected laser airbrush here is 3 plus n, when n is the number of layers, because you always have to occupy all these layers before you can start creating up to the top. Show this, and like finally to look again at this abstraction set, the nice thing about it that it's nicely presented, but it also has like two degrees of infinity. Basically you can have in this graphs you can have as many to pass as possible here, but then you can also have here as many at this construction as possible. So like this graph has even two degrees of infinity. And generally I guess this is nice thing, because really it's not that much examples we have about the abstraction set in a contraction, in a only edge contraction families. Like we know a lot about miners, but not too much about edge contractions. Anyhow to start concluding things, there were before also like probably one of the contributions of this paper is also that they define this connected trivets. There were before some notions of connected trivets, for instance this was requiring like the one part in the, if you have some edge in a TD composition, then we can then we can ask that all this edges, all the vertices above the edge are connected and all the vertices below the edge are connected. This was this definition. Or alternatively we can ask if every back in a TD competition is connected. These are all some kind of a notions of connected trivets. But then the one which after present is more consistent with connected trivets, but still has similar parameter references. So like we saw with the connected three vertex separation and the connected search strategies, like the search strategies. Yeah. So an algorithmic aspect here is, yeah, we really saw that it's close under contraction. And basically they say that checking the connected trivets is most true, it should be in polite time, should be because it should follow from the proof of the algorithm, but they didn't do it, do this in the paper. But they kind of, it's kind of, it should be doable. But still it's strictly larger than two, this is wide open. It's, it's not really clear how this looks like at all, even after this work. Okay, so that's all. For combuting connected trivets, are there, there is some approximation? Yeah, yeah, for connected trivets, no, no, it's not really clear. For connected trivets, that is an XP algorithm. So you can, in time, end to the connected trivets. But for connected trivets, it's like, they know nothing at all, apart from this. So what is the motivation we are doing from trivets to connected trivets, like this structural parameter, which is like, which may be larger than the trivets? Yeah, yeah, yeah, but then it's connected. Yeah, I don't, you know that, that well, I know motivation about why it's interesting about connected node searches, because really it's, it is in out of station, more natural to consider connected searches. About trivets is a bit less clear from you why it's important, but yeah, since it's connected, it's the same as connected lazy name, non-search, it's, it's also interesting, I guess, but the connectedness, connectedness is back, so also my, give, give us something, but I don't really know what exactly.