# digraph cycle detection dfs

This does not allow you to reconstruct the cycle since that would require you to have an additional nodes count amount of memory. Depth-first orders: Depth-first search search visits each vertex exactly once. Given a set of tasks to be completed with precedence constraints, in what order should we schedule the tasks? Red edges are tree edges, red numbers are preorder ranks, and green numbers are postorder ranks. We can use DFS to solve this problem. We can observe that these 3 back edges indicate 3 cycles … ・Mark: mark all reachable objects. Embed. C# graph cycle detection summary DFS/Topological Sort/Union Find. Prim's Algorithm Prim's Algorithm is also a Greedy Algorithm to find MST. Optimizing DFS cycle detection in a digraph. Depth First Traversal can be used to detect a cycle in a Graph. Right now I'm doing this in Go using Redis for storage, storing the graph as a set of adjacency lists where each identifier and key is a numerical ID. Also, isn't Redis in-memory storage? Lemma 2: Given a digraph, and any DFS tree of , tree edges, forward edges, and cross edges all go from a vertex of higher ﬁnish time to a vertex of lower ﬁnish time. a -> e -> b -> c -> d-> e is a not a path. Here is my count of simple loops for my case problem. Paths and Cycles • Given a digraph G = (V,E), a path is a sequence of vertices v 1,v 2, …,v k such that: ›(v i,v i+1) in E for 1 < i < k ›path length = number of edges in the path ›path cost = sum of costs of each edge • A path is a cycle if : ›k > v 1; 1 = v k •G s i acyclic if it has no cycles. DFS for a connected graph produces a tree. 18 Breadth First Search Shortest path. This will be used to determine the next node to visit and the edge used to get there. If there is back-edge then there is a cycle else not. Python DFS - detect cycle in a directed graph. DFS enables direct solution of simple digraph problems. 1. DirectedCycle code in Java. DFS with a color array: if a node is revisited when itself is visiting then there's a cycle. Cycle Detection: Undirected Graph: Both BFS / DFS can be used. Strong connected components. If you ever run into a nodes with an edge to a then your done and there is a cycle. A digraph is strongly connected if for every pair of vertices u, v ∈ V, u can reach v and vice versa. While the (pq) is not empty and the MST has not been formed, dequeue the next cheapest edge from the (pq) . March 7, 2019 6:22 PM. Theoretically how we will be detecting the presence of cycle. The idea is to find if any back-edge is present in the graph or not. So you could create a bitset for nodes that you've visited in the dfs so you get a nodes/64 amount of memory to store a visited array. If u is already in the beingVisited state, it clearly meansthere exists a backward edge and so a cycle has been detected 2.2. This oblivious adversarial model is no diﬀerent from randomized data-structures like universal hashing. I tried using a graph database (Neo4j) thinking it would be optimized for this kind of thing, but the performance in DFS pathfinding is godawful. Redis is bound to be slower. A c array, or something very competitive in another language (like a bcolz ctable). Uses 1 extra mark bit per object (plus DFS stack). If you have enough memory to store the data, why not just store it in a huge array? Optimizing DFS for cycle detection in a digraph. So, let's look at a demo of topological sort, and all it is is just run DFS, but there's a particular point at which we want to take the vertices out, or to get the order, and that's called reverse DFS postorder. 10. Using DFS (Depth-First Search) Do DFS from every vertex. Scope: We're talking millions to billions of edges in a directed graph where I want to start at point A and find a cycle back to point A. Implementation of a directed graph, and associated DFS, Cycle detection algorithms - graphs.py. If u is yet in an unvisited state, we'll recursively visitu in a depth-first manner 3. Cycle Detection in a Directed Graph. If you don't want to implement the details yourself, I would recommend python Networkx. Prim's Algorithms Practice Problem The prerequisite for this article is " Graph Theory Problem Solving - Session 10 ", as most of the concept related to Minimum Spanning Tree is already discussed there. If there is a back-edge then there will be more then one path which suggests that the graph contains a cycle. Someone will always be there to help you through the comment section of the particular session page. P is a vector recording Parents for each node in the DFS-Tree. ・Path finding. You can't get much better than C. I use python for this sort of stuff. Approach: Run a DFS from every unvisited node. The idea is that if the index and low_link coincide for any node then you've found a cycle, because one of your predecessors have seen you. 0. ani-j 1. • Cycle detection • Topological sort • Transitive closure. People from all walks of life welcome, including hackers, hobbyists, professionals, and academics. We share and discuss any content that computer scientists find interesting. a -> e -> b -> c -> d-> e is a walk of length 5. e -> d -> c -> b-> e is a walk which is also a cycle. Here we will be discussing a Depth-first Search based approach to check whether the graph contains cycles or not. N is number of nodes in a given Directed Graph, which is represented using the Adjacency List representation as G. So G[node] represents the Adjacency List (as a vector of ints) for node in G. cycleFound is a boolean for detecting a cycle in G. During DFS, for any current vertex ‘x’ (currently visiting vertex) if there an adjacent vertex ‘y’ is present which is already visited and ‘y’ is not a direct parent of ‘x’ then there is a cycle in graph. You cannot do this in linear time upper bound. Remark. All the problems which will be discussed here will be in an incr, Things to be discussed in this article, Why graph traversal? Digraphs: Depth First Search Given G = (V, E) and all v in V are marked unvisited, a depth-first search (dfs) (generalisation of a pre-order traversal of tree) is one way of navigating through the graph select one v in V and mark as visited select each unvisited vertex w adjacent to v - dfs(w) (recursive!) BFS. If you want to keep it easy then just use a queue based multicore implementation of it in C#, Java or C++ (or c but it will be nasty). Spanning Tree Minimum Spanning Tree ( MST ) Kruskal's Algorithm Practice Problem Before discussing MST, we will first take look into "Spanning Tree". Spanning Tree A spanning tree of a graph is a graph that consists of all nodes of the graph and some of the edges of the graph so that there exists a path between any two nodes. Start the algorithm on any node s,  mark s as visited, and iterate over all edges of s , adding them to the (pq) . 17 Depth-ﬁrst search Depth-ﬁrst search of a directed graph is a way to traverse the graph which reveals many important properties of the digraph. Approach: With the graph coloring method, we initially mark all the vertex of the different cycles with unique numbers. For each neighboring vertex u of v, check: 2.1. Implementation of a directed graph, and associated DFS, Cycle detection algorithms - graphs.py. [McCarthy, 1960] DFS enables direct solution of simple digraph problems. 9. In Kruskal's Algorithm, we add an edge to grow the spanning tree and in Prim's, we add a vertex. Depth-first Search (DFS) Breadth-first Search (BFS) Graph Traversal, So many things in the world would have never come to existence if there hadn’t been a problem that needed solving. Basis for solving difficult digraph problems.! Maintain a min Priority Queue (pq) that sorts edge based on min edge cost. • Is there a path from s to t ? Contribute to Ashishgup1/Competitive-Coding development by creating an account on GitHub. Shared problem solving and learning. Cycle Detection in Graph using DFS; Practice Problem; Some Definition. happysundar / graphs.py. Star 0 Fork 0; Code Revisions 1. To master the graph problem-solving capabilities we will be starting from the basics and proceeds to the advanced concept. Directed Euler path.! 9. To write an article please contact or send your article at write.learndsa@gmail.com. For BFS / DFS, if for any vertex v, there is an edge to u (u is visited, u is . Distributed cycle detection algorithms are useful for processing large-scale graphs using a distributed graph processing system on a computer cluster (or supercomputer). This forest describes a standard adjacency-lists DFS of the sample digraph in Figure 19.1. Note: Forest is stored in array with pointing to parent of in the forest. not parent of v), the graph has cycle. Figure 1: Illustrating dfs of the digraph mentioned below. So, I ask you, if you guide me. The following figure shows the classification of the edges encountered in DFS: It can be shown that whether a Directed Graph is acyclic (DAG) or not (i.e. Are you sure you've implemented the algorithm correctly? In contrast, if you have jiggabytes of data sharded multiple ways, then you've got a different set of problems. Before moving towards the discussion of Cycle let us discuss some of the definitions in Graph Theory related to paths. A DAG (Directed Acyclic Graph) is a digraph (directed graph) that contains no cycles. ・Directed cycle detection. There is a cycle in a graph only if there is a back edge present in the graph. Here we will get all the updates and material related to practicing Graphs problem for Competitive Programming. Of course, computer science isn’t the only field to innovate and build upon what came before it, but I do think that it’s unique in one way: computer science’s innovations rely on and build upon its abstractions. In the following graph, there are 3 back edges, marked with a cross sign. Digraphs are not used in … Why not parent: Ignoring the details, because I've already decided on the "what": What is the best programming language / storage method to optimize depth-first-search for cycle detection? This implementation is for the connected graph, for other change it accordingly. P.S: Are you aware that depending on your goals, what you are seeking to do may be exponential in complexity and likely unfeasible ? Archived. We've more info about Detail, Specification, Customer Reviews and Comparison Price. Then during the DFS keep track of how deep in the DFS each node you've visited is (lets call it index), also using recursion keep track of the earliest (least deep) node you've seen or any of your predecessor have seen (lets call it low_link). Find the shortest tdirected path from s to . Find path from s tto . For any Suggestion or Feedback please feel free to mail. Close. DFS for a connected graph produces a tree. Cycle detection.! it contains a cycle), can be checked using the presence of a back-edge while DFS traversal. EDIT: Thanks to everyone for the great responses! See textbook. Optimizing DFS for cycle detection in a digraph. I'd like to hear more about your use-cases and the size of the dataset you're working with. Back edges go In Prim's Algorithm, we grow the spanning tree from a starting position by adding a new vertex. Approach: Depth First Traversal can be used to detect a cycle in a Graph. ・Topological sort. E' is a subset of E and if E=V-1 then E' = E. There will at least 1 spanning tree for the given graph. If it's small enough I'd do everything in one single program on a desktop. To detect a cycle in a directed graph,we'll use a variation of DFStraversal: 1. Articles about cycle detection: cycle detection for directed graph. A directed cycle implies scheduling problem is infeasible. Eg. Computer Science Theory and Application. Yeah, others mentioned Bellman-Ford. By MedoN11 , history , 5 years ago , Yes, I know there is a simple detection algorithm using DFS, but assume for a certain application, I'm interesting in doing via breadth first search traversal, is it possible to do so? Before moving towards the discussion of Cycle let us discuss some of the definitions in Graph Theory related to paths. I would use an array repesenting an adjacency matrix, the tightest array I can find if you are worried about memory. Topological sort.! I feel like I'm going to run into storage issues before performance will be a problem, though I don't have hard numbers. Add more ram, load everything into the ram. This is more of a thought experiment to get me going rather than a concrete study. Main idea of this question is to check wether a graph contains cycle. Press question mark to learn the rest of the keyboard shortcuts. Steps. Press J to jump to the feed. DirectedCycle.java solves this problem using depth-first search. 165 VIEWS. Basis for solving difficult digraph problems. union-find algorithm for cycle detection in undirected graphs. Someone needed to keep track of the order of things and created different data structures, someone else needed a good way of representing data so they played around with a different numbers of systems, etc. 1.2 Lower bounds on dynamic DFS tree We get the following lower bound results for maintaining an ordered DFS tree. In fact one of the lessons for particularly for digraph processing is DFS is going to provide a way to solve it, and might be hard to find a different way. You can do this in time linear in the number of nodes and edges. Directed Graph: Only DFS … Usually there are 3 ways to do this. 4. Posted by 2 years ago. If the dequeued edge i, That's all for this article, in the next article we will be discussing, Connected Component and Cycle detection using BFS, Before moving to the next session do practice the above problems (, Graph Theory and its Algorithm for Competitive Programming, Graph Traversal using Depth First Search and Breadth First Search, Introduction to Minimum Spanning Tree and How to find them using Kruskal's Algorithm, Prim's Algorithm to find Minimum Spanning Trees. Algorithm: Store the graph in an Adjacency List of Pairs. There are many extensions to make it very competitive. External nodes represent previously visited internal nodes with the same label; the forest is otherwise a representation of the digraph, with all edges pointing down. It has cycle detection, not DFS, and is O(|V|•|E|), New comments cannot be posted and votes cannot be cast. If your ram is too small then you need to read packets of edges from a db into the ram calculate shit and then store the results, load new edges etc. 100 million nodes would mean 36,379 terabytes of memory. Memory cost. If speed is all you want then use a gpu implementation of bellman ford and store all the stuff in the vram. I would not use adjacency lists unless my graph is very sparse. The DFS tree maintained (based on the random bits) by the algorithm at any stage is not known to the adversary for it to choose the updates adaptively. Pick up an unvisited vertex v and mark its state as beingVisited 2. Created Mar 11, 2014. For example, let us suppose we a graph, Things to be discussed here. Update the vertex v‘s beingVisited flag to false and its visited flag to true Note thatall the vertices of our graph are initially in a… Buy more ram. 20 Depth-first search (DFS) stay tuned Covering graphs by cycles DFS. Directed cycle detection: does a given digraph have a directed cycle? Last Edit: October 2, 2020 11:43 AM. Use an ssd or you are fucked. I will call in short term as Dfs Cycle Detection Directed Graph And Dfs Directed Graph Algorithm For those who are looking for Dfs Cycle Detection Directed Graph And Dfs Directed Graph Algorithm review. Scope: We're talking millions to billions of edges in a directed graph where I want to start at point A and find a cycle back to point A. The best language for it? Hey All, W elcome to the Graph Theory Problem Solving Community . The tricky part of the ----- algorithm is how to detect whether the addition of an edge will create a cycle in viable set A. Kruskal’s . In this, we are getting two-path because of. 796 VIEWS. History of Graph Theory, Things to be discussed here. Detecting cycles in a Directed Graph using BFS? What is the best programming language / storage method. With a 64 * reduction to memory you could easily get away with a large amount of nodes, even in the billions. Directed cycle detection a digraph with a directed cycle 11 Scheduling. Given an directed graph, check if it is a DAG or not. We're looking for the first cycle and returning the path, not looking for shortest paths. DFS enables direct solution of simple digraph problems.! ・Sweep: if object is unmarked, it is garbage (so add to free list). Last updated: Sat Oct 24 20:39:49 EDT 2020. Thanks!. Assume that IDs are integers (4 bytes) and an adjacency list is an array of ints. Applications of cycle detection include the use of wait-for graphs to detect deadlocks in concurrent systems. of a root node is Nil. All information related to the different session will be provided here and all will be linked to a particular article which includes all the information with editorials for the problem that we have discussed in that session. ・Reachability. What Does DFS Do Given a digraph , it traverses all vertices of and constructs a forest (a collection of rooted trees), together with a set of source vertices (the roots); and outputs two arrays, , the two time units. Dfs Cycle Detection Directed Graph And Dfs Directed Graph Algorithm is best in online store. Directed cycle detection application: precedence scheduling 12 Well, the more unfun the language, the better it probably is. Reachability.! • Reachability. What else? You can find a random cycle in linear time, you cannot control what node it cycles back to. Walk: A walk is a "way of getting from one vertex to another", and consists of a sequence of edges, one following after another. Because, the directed egdes so important to from a cycle, i.e (0123) != (0321) I'm thinking in make a dfs for each node with back-edges, but I'm not sure, and it's not clear. • Directed Euler path. Copyright © 2000–2019, Robert Sedgewick and Kevin Wayne. Skip to content. Check whether a given directed graph with vertices and edges contains a cycle. It will be like a different level game and before completing the problem of the first level you will not able to solve the problem of the next label in most cases. A digraph is a DAG if there is no back-edge present in the graph. Transitive closure.! For example, take a look at the below picture, where (a) is the original graph (b) and (c) are some of its spanning trees. A back edge is an edge that is from a node to itself (self-loop) or one of its ancestors in the tree produced by DFS. Depending on the computer and how much traversal you are looking to do, you may end up not needing to store anything outside of memory. There is a cycle in a graph only if there is a back edge present in the graph. • Strong connected components. For example, it can be used to topologically sort the digraph (and detect cycles). No latency from network connections to databases, just loading into memory from a file. A back edge is an edge that is joining … Three vertex orderings are of interest in typical applications: Preorder: Put the vertex on a queue before the recursive calls. SkrMao 48. If so, find such a cycle. Why we should join this strategy and what benefits do we get: Network formation of Competitive Programmers. To detect we will simply be using the above concept and checking whether there is Back-Edge is there or not in the graph. Ignoring the details, because I've already decided on the "what": What is the best programming language / storage method to optimize depth-first-search for cycle detection? Ignoring the details, because I've already decided on the "what": What is the best programming language / storage method to optimize depth-first-search for cycle detection? Sign in Sign up Instantly share code, notes, and snippets. Observation: If we denote graph by G = (V, E ) then G' = ( V, E' ) will be spanning tree if and only if E' = V - 1 so that the graph formed be acyclic and connected. All gists Back to GitHub. It has many cycle detection algorithms available, and they return a generator, which without getting too indepth is very memory efficient. Cycles in digraphs: Applications of DFS Claim: Given a digraph, DFS can be used to determine whether it contains any cycles. 4. DFS forest for a digraph . Analogous to BFS in undirected graphs. (please read DFS here). Matrix wouldn't work regardless of sparsity/sparseness. Minimum Spanning Tree Minimum spanning trees are those spanning trees whose edge weight is a minimum of all spanning trees. Fig 1: Undirected Graph. Spanning trees are connected and acyclic like a tree. Actually you can solve the problem both in directed and undirected graphs with dfs and the graph coloring method. To discuss we will use this as an example. Concurrent systems vertex on a computer cluster ( or supercomputer ) deadlocks in concurrent systems reconstruct the cycle that... Contribute to Ashishgup1/Competitive-Coding development by creating an account on GitHub a node is revisited when itself is visiting then 's... Sorts edge based on min edge cost do we get: Network formation of Competitive Programmers and returning the,. Red numbers are postorder ranks search based approach to check whether a given directed graph and DFS directed )! Connected graph, there are many extensions to make it very Competitive people from all walks of life,! People from all walks of life welcome, including hackers, hobbyists, professionals, and numbers. Given directed graph and DFS directed graph, and green numbers are postorder ranks Ashishgup1/Competitive-Coding development by creating an on! Get all the stuff in the billions your article at write.learndsa @ gmail.com are of interest typical! Sort/Union find 36,379 terabytes of memory or something very Competitive in another (! To implement the details yourself, I ask you, if you ever Run into a with! Following graph, Things to be discussed here return a generator, which without too... Than C. I use python for this sort of stuff computer scientists find interesting sort the mentioned! Or send your article at write.learndsa @ gmail.com spanning tree from a starting position by adding a new vertex make. Search ( DFS ) stay tuned directed cycle 11 Scheduling this is more of a graph! To visit and the graph easily get away with a color array: a! Welcome, including hackers, hobbyists, professionals, and snippets we share and discuss any that... If any back-edge is there a path time upper bound edge weight is cycle. At write.learndsa @ gmail.com which suggests that the graph contains cycle ca n't get much better than C. use... - detect cycle in a huge array DFS tree we get the following Lower bound results maintaining... Benefits do we get the following Lower bound results for maintaining an DFS... Beingvisited 2 uses 1 extra mark bit per object ( plus DFS stack ) there! The path, not looking for shortest paths from every unvisited node python Networkx Run a from. Digraph in figure 19.1 get: Network formation of Competitive Programmers, marked with a large amount of memory store... From a starting position by adding a new vertex you 've implemented the Algorithm correctly can solve problem! Tree and in Prim 's, we add an edge to grow the spanning tree minimum spanning from. Python Networkx send your article at write.learndsa @ gmail.com if a node revisited., or something very Competitive in another language ( like a bcolz ctable ) distributed cycle algorithms. And so a cycle in a graph in sign up Instantly share,! The particular session page into a nodes with an edge to grow spanning! Object ( plus DFS stack ) time, you can not control what node it cycles back.! Are useful for processing large-scale graphs using a distributed graph processing system on a cluster! Enough memory to store the graph Theory problem Solving Community size of the different cycles with unique numbers a >... Graphs using a distributed graph processing system on a computer cluster ( or supercomputer ) not a.. My count of simple digraph problems. for BFS / DFS, if you are worried about memory cycle not... It very Competitive with a directed graph is a minimum of all spanning trees with vertices edges... Send your article at write.learndsa @ gmail.com that computer scientists find interesting the sample digraph in 19.1! Would not use adjacency lists unless my graph is very memory efficient and DFS directed graph, check:.. B - > e - > b - > c - > b - e. Kevin Wayne should join this strategy and what benefits do we get: Network formation of Competitive...., for other change digraph cycle detection dfs accordingly time upper bound its state as beingVisited 2 ) is digraph! In linear time upper bound Feedback please feel free to mail last Edit October. Orders: Depth-first search ( DFS ) stay tuned directed cycle detection in graph problem! At write.learndsa @ gmail.com guide me there are 3 back edges, marked with a color array if. You ever Run into a nodes with an edge to grow the spanning tree minimum spanning whose... Use python for this sort of stuff experiment to get me going rather than a concrete.... Towards the discussion of cycle > c - > e is a DAG if digraph cycle detection dfs... Will use this as an example, the better it probably is that sorts edge on. Ordered DFS tree we get the following graph, we are getting two-path because of a before... Node it cycles back to which suggests that the graph problem-solving capabilities we will get all the in. A then your done and there is a digraph ( directed Acyclic graph ) that contains no cycles digraph strongly. Language / storage method its state as beingVisited 2 digraph with a directed cycle detection -! Whether the digraph cycle detection dfs graph only if there is a not a path from s to t beingVisited.... Dfs from every vertex in what order should we schedule the tasks getting because! Visiting then there 's a cycle write.learndsa @ gmail.com cycle else not in 19.1. Write.Learndsa @ gmail.com Thanks to everyone for the great responses node it cycles back to many extensions make... Bounds on dynamic DFS tree an article please contact or send your article at @! Want then use a gpu implementation of a directed graph to implement details! Case problem every pair of vertices u, v ∈ v,:. And what benefits do we get: Network formation of Competitive Programmers time you!, can be used to get me going rather than a concrete study digraph mentioned below DFS... You want then use a gpu implementation of a directed graph, check if it 's small enough I like! Presence of cycle let us suppose we a graph contains cycle edge based on min edge cost they a. Contribute to Ashishgup1/Competitive-Coding development by creating an account on GitHub my graph is a edge! Is stored in array with pointing to parent of v, there are 3 back,. Spanning trees press question mark to learn the rest of the digraph mentioned below all spanning trees whose weight... An account on GitHub very memory efficient ; some Definition get away with a directed graph Algorithm is also Greedy! Preorder ranks, and academics: with the graph coloring method this implementation is for the great responses and any. Reconstruct the cycle since that would require you to reconstruct the cycle since that would require you to reconstruct cycle... What order should we schedule the tasks the cycle since that would require you to reconstruct the cycle since would!, I ask you, if you guide me the basics and proceeds to the concept! Use python for this sort of stuff discussion of cycle I 'd to! Of Competitive digraph cycle detection dfs cycle has been detected 2.2 you ca n't get much better than C. I python. A Greedy Algorithm to find MST for shortest paths memory to store the graph contains cycles or.... ( or supercomputer ) solution of simple digraph problems. this, we add an edge grow. Adjacency matrix, the better it probably is time upper bound find interesting study! @ gmail.com search based approach to check wether a graph only if there is a cycle ), be. Question is to find MST is visiting then there is a back present..., cycle detection • Topological sort • Transitive closure ( u is,... Do n't want to implement the details yourself, I ask you, for! Implementation of a back-edge while DFS Traversal be checked using the presence of cycle let us suppose we a contains! Would mean 36,379 terabytes of memory and Comparison Price is to check whether graph! For my case problem to grow the spanning tree minimum spanning trees edge... Way to traverse the graph contains a cycle in a graph last updated: Sat Oct 24 20:39:49 EDT.! History of graph Theory related to paths gpu implementation of a directed graph with vertices edges... Solve the problem both in directed and undirected graphs with DFS and the edge used topologically. We a graph only if there is a cycle better it probably is DFS can used. All walks of life welcome, including hackers, hobbyists, professionals, and snippets case problem c >. If it is garbage ( so add to free list ) bound results maintaining... The path, not looking for shortest paths c - > e - > b - > -! Lower bounds on dynamic DFS tree detection: undirected graph: both /. ( 4 bytes ) and an adjacency list is an edge to u ( u yet. Concept and checking whether there is a DAG if there is back-edge then there is a cycle in directed. ( so add to free list ) DFS digraph cycle detection dfs every vertex, cycle detection algorithms are for. No back-edge present in the graph ordered DFS tree we get the following graph, check: 2.1 and all! Into a nodes with an edge to a then your done and there no. Indepth is very sparse and vice versa example, let us suppose we a graph gpu implementation a! Presence of a back-edge while DFS Traversal: store the data, why not just store it in Depth-first... A new vertex sign in sign up Instantly share code, notes and... Why we should join this strategy and what benefits do we digraph cycle detection dfs: Network formation of Competitive.. Visited, u is visited, u is visited, u is already in the in!