Strongly connected directed graph

Strongly connected directed graph. Thus, a directed cycle has a single strongly connected component. Now, consider a directed graph that does not have any cycle. It is often used early in a graph analysis process to help us get an idea of how our graph is structured. Directed graphs are widely used to model various objects and processses having nonsymmetric features. Graph basic operations: layout_acyclic_dummy() Compute a (dummy) ranked layout so that all edges point upward. Let v and w be vertices of a directed graph. If G is a directed graph, then two nodes belong to the same strong component only if Show more. The set of all nodes that are connected with each other form a component. Raises: Mar 13, 2022 · Given a directed graph. This is a strongly connected subgraph and the networkx function for that is strongly_connected_component_subgraphs. ) The corresponding result is a theorem of Ghouila-Houri [25]. This generalizes the idea of connectivity from undirected graphs to directed graphs. In a Biconnected Graph, there is a simple cycle through any two vertices. Each tree in the forest computed in step 3 is a strongly A directed graph is weakly connected if there is a path between all pairs of vertices when edge direction is ignored. Cyclic graphs also may contain such trivial components. Strongly connected is usually associated with directed graphs (one way edges): there is a route between every two nodes. , bidirectional path. If Oct 29, 2021 · The transpose graph property is why Kosaraju’s algorithm works successfully to search SCCs. Assume that n <= m. A directed graph is strongly connected if all of its vertices are strongly connected to each other. Component A is strongly connected. The numbers of nonisomorphic simple strongly connected digraphs on n=1, 2 Aug 7, 2020 · What are strongly connected digraphs? That's what we'll be going over in today's graph theory lesson. If BFS or DFS visits all vertices, Nov 8, 2015 · Step 1: Call DFS (G) to compute finishing times f [u] for each vertex u. 1 The nodes found in each explore call will be the connected components; if you wanted to count the number of components, you could count the number of top level explore calls. In a directed graph is said to be strongly connected, when there is a path between each pair of vertices in one component. Call DFS (Transpose (G)), but in the main loop of DFS, consider the vertices in order of decreasing f [u] (as computed in step 1) Okay, so Feb 12, 2024 · A single directed graph may contain multiple strongly connected components. Feb 12, 2024 · A single directed graph may contain multiple strongly connected components. Start at a random vertex v of the graph G, and run a DFS (G, v). It runs in linear time, matching the time bound for alternative methods including Kosaraju's algorithm and the path-based strong component algorithm. Al the 4 graphs in the diagram below are strongly connected directed graph. If BFS or DFS visits all vertices, then the given undirected graph is connected. By convention, two nodes connected by an edge form a biconnected graph, but this does not verify the above properties. Introduction. 5 Strongly connected components Chap 22 Problems Chap 22 Problems Strongly Connected Components Definition A strongly connected component of a directed graph G is a maximal set of vertices C ⊆ V such that for every pair of vertices u and v, there is a directed path from u to v and a directed path from v to u. Note that if a graph is strongly connected (i. Restated, Property Every directed graph is a dag of its strongly connected components. The strongly connected components in a DAG are the singleton vertices. Throughout this paper we assume the graph is strongly connected implying that P is irreducible. easy to find connected components in an undirected graph: run DFS. bins = conncomp(G) returns the connected components of graph G as bins. That is, if G contains an edge (u, v) then the converse/transpose/reverse of G contains an edge (v, u) and vice versa. layout_acyclic() Compute a ranked layout so that all edges point upward. (from page 517) What I can't understand is the second property/definition, the one that says, when you have a directed graph, then if the associated undirected graph is connected, that implies that the directed graph will be Mar 9, 2023 · Directed graphs have several characteristics that make them different from undirected graphs. The vertices in a cycle are strongly connected. In an SCC all nodes are reachable Jun 16, 2020 · Strongly Connected Graphs. Mar 7, 2023 · Finding connected components for an undirected graph is an easier task. The structure of strongly connected components of a directed graph is captured by so called condensation of a directed graph. Prove that the following two conditions for a strongly connected directed graph G are equivalent: (i) G contains a directed cycle of an even length. 173). Compute G T. for the product of two connected undirected graphs to be connected; he further proves that the product graph, if not connected, has exactly two components. Do either BFS or DFS starting from every unvisited vertex, and we get all strongly connected components. 7. The following tables summarized the number of weakly and Test directed graph for strong connectivity. If a particular component in a directed graph is strongly connected then we call that component Strongly Connected Component or SCC. Returns: connectedbool. Consider a bipartite graph G (T,S) between the two sets defined as follows. Oct 21, 2015 · One can also show that if you have a directed cycle, it will be a part of a strongly connected component (though it will not necessarily be the whole component, nor will the entire graph necessarily be strongly connected). Yes, a graph can, according to the provided definitions, definitely be both weakly and strongly connected at the same time. A directed graph GD. But if node ais removed, the resulting graph would be strongly connected. 2 Breadth-first search 22. A connected subgraph of G that is not a proper subgraph of any other connected subgraph of G is a A directed graph is weakly connected if and only if the graph is connected when the direction of the edge between nodes is ignored. 4) identify the vertices in C (ignoring self-loops but possibly creating parallel edges). 1: Connectivity Terminology. Suppose further that the graph is strongly connected. Tarjan's strongly connected components algorithm is an algorithm in graph theory for finding the strongly connected components (SCCs) of a directed graph. Thus, there is no need for the second traversal that identifies the newly found component and the use of a stack is unnecessary. The reason is simple: a cycle containing several strongly connected components would merge them all into a single, strongly connected component. (ii) The vertices of G can be colored by 2 colors (each vertex receives one color) in such a way that for each vertex u there exists a directed edge ( u, v )with v having the color different A directed graph G as a whole is called strongly connected if any pair of nodes in G are strongly connected. A digraph that is not strongly connected consists of a set of strongly connected components, which are maximal strongly connected subgraphs. In this paper we deduce the corresponding results for directed graphs, namely that if G1, G2 * * * , Gr are strongly connected directed graphs then G1 X G2 X X Gr has exactly d(G1) d(G2 Jul 1, 2012 · (Here a directed graph may have two edges between a pair of vertices, but in this case their directions must be opposite. In a directed acyclic graph, there cannot be a v−w path and a w−v path at the same time, because that would form a closed walk. The algorithm is described in a top-down fashion in Figures 2–4. Sep 20, 2022 · Strongly Connected: A graph is said to be strongly connected if every pair of vertices (u, v) in the graph contains a path between each other. Instead, each vertex is its own strongly connected component. Connected is usually associated with undirected graphs (two way edges): there is a path between every two nodes. Obviously strongly connected components are contained in weakly connected components, but in general they provide a ner partition. To sum up, for directed graphs: a directed tree is a connected directed graph without cycles (not to be confused with a connected directed graph without directed cycles—a connected DAG). In an unweighted directed graph G, every pair of vertices u and v should have a path in each direction between them i. Jan 14, 1994 · Improved Algorithm 1 If the input graph G = (V, E) is acyclic then each strongly connected component consists of a single node. Take a transpose of the original graph. A strongly connected component (or just strong component) of a directed graph is a maximal strongly connected subgraph. U is strongly connected if there is a directed path between any two points in U. So there cannot be a strongly connected component containing more than one vertex. In fact, all strongly connected graphs are also weakly connected, since a directed path between two vertices still connect the vertices upon removing the directions. A directed graph is strongly connected if and only if every vertex in the graph is reachable from every other vertex. Jan 27, 2023 · A directed graph is strongly connected if there is a path between any two pair of vertices. Follow. Kosaraju's algorithm. There is no way to traverse from A to B or visa versa from B to A. Graph. In degree is equal to the out degree for every vertex. (The other case is very similar). In this tutorial, you will understand the working of kosaraju's algorithm with working code in C, C++, Java, and Python. Do the following for every vertex v : Explanation: Depth First Search is used in the Generation of topological sorting, Strongly Connected Components of a directed graph and to detect cycles in the graph. All vertices with nonzero degree belong to a single strongly connected component. Weakly Connected: A directed graph G is weakly connected if it lacks a directed path (from u to v or v to u for any pair of vertices u, v). Jan 14, 2020 · A digraph is strongly connected if there is a directed path from every vertex to every other vertex. Nov 8, 2019 · Iteratively do the following: 1) find a cycle C in G, 2) orient the edges of C in an arbitrary but consistent direction w. Before generating all simple circuits in a directed graph, the algorithm first detects whether there is a simple circuit in directed graph, and the detection can divide the directed graph into trivial graphs or strongly connected components. Breadth First Search is used in peer to peer networks to find all neighbourhood nodes. Jun 20, 2020 · For directed graphs there is the notion of strongly connected components, for which multiple algorithms are available, all slightly more complicated than a simple DFS. 1 Representations of graphs 22. Download scientific diagram | A strongly connected and unbalanced directed graph from publication: FROST—Fast row-stochastic optimization with uncoordinated step-sizes | In this paper Feb 20, 2023 · Given a Directed graph of N vertices and M edges, the task is to find the minimum number of edges required to make the given graph Strongly Connected. It is easy for undirected graph, we can just do a BFS and DFS starting from any vertex. Digraph graph data type. In other words, it is a directed graph whose underlying graph is a tree; Oct 19, 2016 · Figure 1: The strongly connected components of a directed graph. Let's see: Oct 1, 2019 · The notion of “connectedness” in undirected graphs has several natural extensions to directed graphs. For a directed graph D = (V,E), aStrongly Connected Component (SCC) is a maximal induced subgraph S = (VS , ES) where, for Weakly Connected A directed graph is weaklyconnected if there is a path between every two vertices in the underlying undirected graph. Such a digraph is called a directed acyclic graph (DAG). Call DFS (Transpose (G)), but in the main loop of DFS, consider the vertices in order of decreasing f [u] (as computed in step 1) Okay, so Oct 27, 2019 · 0. We recall that a directed graph (or subgraph) is strongly connected if and only if for each bipartition (V 1, V 2) of V there is an edge from V 2 to V 1 (and Strongly Connected Components (SCC) Given a directed graph G = (V,E) A graph is strongly connected if all nodes are reachable from every single node in V Strongly connected components of G are maximal strongly connected subgraphs of G The graph below has 3 SCCs: {a,b,e}, {c,d,h}, {f,g} Strongly Connected Components (SCC) 36 Jul 20, 2022 · Consider the following directed graph, and call it G: As a recap, the above graph G is strongly connected because any two nodes can be reached by a directed edge path between them. Jan 31, 2023 · How to check if a directed graph is eulerian? A directed graph has an eulerian cycle if following conditions are true. Nov 8, 2015 · Step 1: Call DFS (G) to compute finishing times f [u] for each vertex u. See [10] for more information. 05C20, 05C21, 05C40, 92B20. For example, the graph in Figure 6. We Feb 22, 2023 · A directed graph is acyclic if and only if it has no strongly connected subgraphs with more than one vertex, because a directed cycle is strongly connected and every non-trivial strongly connected component contains at least one directed cycle. Given a directed graph G = (V;E). This is same as connectivity in an undirected graph, the only difference being strong connectivity applies to directed graphs and there should be directed paths instead of just paths. We can detect singly connected component using Kosaraju’s DFS based simple algorithm. If BFS or DFS visits all vertices, Dec 11, 2021 · An Eulerian trail (or Eulerian path) is a path that visits every edge in a graph exactly once. An undirected graph is called Biconnected if there are two vertex-disjoint paths between any two vertices. 3 Algorithm to find strongly connected components of a directed graph The algorithm we present is essentially two passes of depth-first search, plus some extremely clever additional book-keeping. 6. Rao Kosaraju and Micha Sharir. 1. For example, the below given graph contains 3 strongly Components(highlighted ones) that are: {a,b,e,f} , {f,g} and {c,d,g,h} because in all of these components there is a path from one vertex to every other vertex. princeton. reverse() Return a copy of digraph with edges reversed in This paper presents for the first time an algorithm requiring only3vw bits in the worst case, which has found numerous uses in the literatur e, often as a subcomponent of larger algorithms, such as those for transitive closure, compiler optimisation, and program analysis. r. A directed graph is strongly connected if every vertex is reachable from every other vertex by a directed path. Directed paths and related Mar 3, 2016 · 1. Running DFS starting on vertex A: Please notice RED text formatted as [Pre-Vist, Post-Visit] Step 2: Compute Transpose (G) Step 3. 3) strongly May 14, 2019 · A digraph that has no directed circuit is called acyclic. The idea is to. Let T= {t 1 ,…,t n } be the sinks and {s 1 ,…,s m } be the sources of the DAG. 3. Vertex v is connected to vertex w if there is a path from v to w. A digraph G with vertex set V is strongly connected if for all i and j in V, there is a path i ⇝ j (and thus also a path j ⇝ i). edu: Directed Graphs. What you should do depends on which of the two notions you need. Therefore, if all nodes are connected, we can reach from one vertex to any other vertex (as per the definition of a strongly connected graph in directed graphs). A set is considered a strongly connected component if there is a directed path between each pair of nodes within the set. Suppose G is a directed graph on n vertices where all in-degrees and our-degrees are constant d ≥ 2 (though the adjacency matrix need not be symmetric). a d b c Figure 4: A directed cycle. Robert Israel. Example 1: Input: Output: 3 Explanation: We can clearly see that there are 3 Strongly Connected C We would like to show you a description here but the site won’t allow us. , the vector satisfying πTP = πT and πT1 = 1, and let Π = Diag(π) be the diagonal matrix with the stationary probabilities {πi}n 1 on the diagonal . I read in few communities that you can do this by following the steps below: Take the BFS or DFS of the original graph. De nition. the graph is connected even when we account for directionality), it is by definition weakly connected as well. Let Tr T r be the set of spanning trees of G G with r r as root and all edges pointing to r r. Figure 2 Given a strongly connected directed graph G = (V, E) G = ( V, E), and a node r ∈ V r ∈ V. 452k 27 348 671. directed graph, strongly connected graph, connectivity, connectome AMS subject classifications. The task is to check if the given graph is connected or not. In your case, they all have length 1, so it returns one of them (I believe whichever networkx happened to put into nx. Here are some key characteristics of directed graphs: Directed edges: In a directed graph, edges have a direction associated with them, indicating a one-way relationship between vertices. Next, consider Apr 10, 2011 · The strongly connected components of a directed graph identifies those parts (subsets of vertices) of a graph where everybody can reach everybody, so that it can reasonable to think of each of these subsets as a single 'thing'. A strongly connected clique (or scq for short) is a set C ⊆ V (G) such that G [C] is strongly connected, and u (G) [C] is a clique of u (G). A much weaker form of Strongly Connected Components. It's under Creative Problems (41. 6 days ago · Connected Digraph. number_strongly_connected_components (G) Returns number of strongly connected components in graph. There are 3 types of connectivity when talking about a directed graph G G. Follow the steps mentioned below to implement the idea using DFS: Initialize all vertices as not visited. cs. Call dfs for the graph G T but in the main loop of DFS explore each vertex in decreasing order of finish time. An Eulerian circuit (or Eulerian cycle) is an Eulerian trail that starts and ends on the same vertex. answered Mar 19, 2013 at 5:59. Cite. For example, there is no directed path which starts at vertex \(6\). In particular, as a corollary of this definition, a strongly connected directed graph cannot have any sink vertices or any source vertices. A directed graph. 6 days ago · A strongly connected digraph is a directed graph in which it is possible to reach any node starting from any other node by traversing edges in the direction (s) in which they point. Your example is exactly such a graph. All of its vertices with a non-zero degree belong to a single strongly connected component. There are two distinct notions of connectivity in a directed graph. Approach: Take two bool arrays vis1 and vis2 of size N (number of nodes of a graph) and keep false in all indexes. In the process of detecting the existence of is_strongly_connected (G) Test directed graph for strong connectivity. Strongly-Connected-Components(G) 1 call DFS(G) to compute finishing times f[u] for each vertex u Apr 18, 2023 · Strongly connected component of a directed graph is a subgraph in which there exists a path from every vertex to every other vertex in the subgraph. Take the BFS or DFS of the transpose. Let πbe the unique vector of stationary probabilities over this graph, i. e. A strongly connected component is the portion of a directed graph in which there is a path from each vertex to another vertex. 3 Depth-first search 22. This tells us something important: The connectivity structure of a directed graph is two-tiered. The elements of the path matrix of such a graph will contain Jan 14, 2020 · A digraph is strongly connected if there is a directed path from every vertex to every other vertex. Share. Odd-length directed cycle. If BFS or DFS visits all vertices, This module implements functions and operations involving directed graphs. 52 is not strongly connected. (MWE) Minimal working example: Nov 26, 2019 · Your directed graph has 2 disconnected components. A digraph is strongly connected if for every pair of nodes i and j, there is a directed path from i to j and another from j to i (in other words, every node is reachable from every other node). For example, following is a strongly connected graph. In contrast to Strongly Connected Components (SCC), the direction of relationships on the path Feb 26, 2024 · Transpose of a directed graph G is another directed graph on the same set of vertices with all of the edges reversed compared to the orientation of the corresponding edges in G. Given a graph (represented as adjacency list), we We can now describe the algorithm to compute the strongly connected components for a graph. C, 3) Orient the chords of C in an arbitrary direction. True if the graph is strongly connected, False otherwise. A corollary of this is that for every pair of vertices v and w in the strongly connected sub-graph G'(V',E') where V' ⊂ V and E' ⊂ E there is a directed path from v to w and a directed path Sep 20, 2017 · All your strongly connected components have a single node. Such components are called strong components. In computer science, Kosaraju-Sharir's algorithm (also known as Kosaraju's algorithm) is a linear time algorithm to find the strongly connected components of a directed graph. To solve this algorithm, firstly, DFS algorithm is used to get the finish time of each vertex, now find the finish time of the transposed graph, then the vertices are sorted in descending order by 2 Answers. Examples: Input: N = 3, M = 3, source[] = {1, 2, 1}, destination[] = {2, 3, 3} Output: 1 Explanation: Adding a directed edge joining the pair of vertices {3, 1} makes the graph strongly connected. A directed graph is strongly connected if there is a path between any two pair of vertices. A directed graph is strongly connected if there is a directed path from any vertex to every other vertex. U is a strongly connected component (SCC) if U is strongly connected but no proper superset of U is strongly connected. A directed graph is weakly connected if there is an undirected path between any pair of vertices, and strongly connected if there is a directed path between every pair of vertices (Skiena 1990, p. Oct 2, 2012 · 5. Apr 8, 2019 · An undirected graph's edges can be considered bidirectional edges. Nov 12, 2022 · A directed graph that is not strongly connected consists of a set of strongly connected components, which are maximal strongly connected sub-graphs. So the truth is that DAGs must have no SCCs on more than one vertex, not that they have no SCCs at all. Complete graphs are undirected graphs where there is an edge between every pair of nodes. If there are two or more vertices left in G then there must still be a Key words. The Weakly Connected Components (WCC) algorithm finds sets of connected nodes in directed and undirected graphs. Indegree and Outdegree: Each vertex in a directed graph has two A directed graph G is strongly connected if: For any two vertices u and v: There is a directed path u → v, and There is a directed path v → u: A Aug 17, 2021 · Note 9. In this article I implement Tarjan's algorithm to find strongly connected components in a graph. The subject of study. a b d c Strongly connected a b d c Weakly connected Connected Components The subgraphs of a directed graph Gthat are strongly connected but not contained in larger strongly connected subgraphs, that is, the VI Graph Algorithms VI Graph Algorithms 22 Elementary Graph Algorithms 22 Elementary Graph Algorithms 22. Nov 15, 2021 · Let G = (V, E) be a (directed) graph. Sep 24, 2020 · De nition. Two vertices are strongly connected if they are connected in both directions to one another. 4 Topological sort 22. Two nodes are connected, if there exists a path between them. Mar 27, 2024 · A weakly connected component is a subgraph of a directed graph in which all vertices are connected by some path, irrespective of edge direction. Strongly-Connected-Components(G) 1 call DFS(G) to compute finishing times f[u] for each vertex u 2 compute GT. If you take CS161, you'll see some cool algorithms involving strongly connected components in a graph! Strong Connectivity Apr 16, 2024 · Given a directed graph, find out whether the graph is strongly connected or not. In other words, draw an undirected graph G` of G whose edges Dec 17, 2012 · For directed graphs, I assume a subgraph is a graph such that every node is accessible from every other node. The bin numbers indicate which component each node in the graph belongs to. Feb 16, 2024 · Given a directed graph, find out whether the graph is strongly connected or not. 2) connected - contains a directed path from u u to v v OR a directed path from v v to u u for every pair of vertices u u, v v. A directed acyclic graph (or DAG) is a digraph with no directed cycles. Conversely, if vertices are not connected, it means that a path does not exist between any two pairs of vertices (as Oct 27, 2019 · 0. 1) weakly connected - replacing all of G G 's directed edges with undirected edges produces a connected (undirected) graph. Kosaraju suggested it in 1978 but did not publish it, while Sharir May 18, 2018 · 3. Sep 12, 2023 · Given a directed graph, find out whether the graph is strongly connected or not. In a directed graph, nodes are strongly connected if there exists a path to and from the nodes with May 9, 2020 · This paper proposes a new algorithm for detecting and generating simple circuits within a directed graph. 3 call DFS(GT May 17, 2016 · There actually is the answer to your question on a page you have linked: algs4. (ii) The vertices of G can be colored by 2 colors (each vertex receives one color) in such a way that for each vertex u there exists a directed edge ( u, v )with v having the color different Therefore, our previous directed graph in Fig. Since the graph is strongly connected, for any two vertices v and w, there is a directed path connecting v to w. Here is what they can do. strongly_connected_components(G) first). Theorem 1 Ghouila-Houri [25] Every strongly connected digraph on n vertices with δ + (G) + δ − (G) ≥ n contains a Hamilton cycle. We'll recap connectedness, what it means to be weakly c 22. 4 Condensation of a Graph. strongly_connected_components(G), key=len) it finds the set of nodes which has the longest length and returns it. A is strongly connected because you can traverse to every other vertex in the component from every vertex in the component. Sorted by: 22. For example this directed graph. Aho, Hopcroft and Ullman credit it to S. If G is an undirected graph, then two nodes belong to the same component if there is a path connecting them. 1. Below you will see that if we start DFS on the original graph from any node in SCC1 we will be able to reach all the nodes in all the three components, as all are strongly connected components and there is an outgoing edge from the first node to SCCs. 2 is not strongly connected since there is no directed path from node bto node a. Given a Directed Graph with V vertices (Numbered from 0 to V-1) and E edges, Find the number of strongly connected components in the graph. The Strongly Connected Components (SCC) algorithm finds maximal sets of connected nodes in a directed graph. Make all visited vertices v as vis1 [v] = true. Mar 19, 2013 · If it's possible to get from a a to b b and from b b to a a, and a directed path from a a to b b passes through c c, then you can get from a a to c c (along the path), or from c c to b b and then back to a a. Definition A strongly connected component of a directed graph G is a maximal set of vertices C ⊆ V such that for every pair of vertices u and v, there is a directed path from u to v and a directed path from v to u. So suppose you have a graph with 4 nodes, a1,a2,b1,b2 a 1, a 2, b 1, b 2, with edges a1 → a2 a 1 → a 2, a2 → a1 Oct 21, 2015 · One can also show that if you have a directed cycle, it will be a part of a strongly connected component (though it will not necessarily be the whole component, nor will the entire graph necessarily be strongly connected). Call dfs for the graph G to compute the finish times for each vertex. For strongly connected graphs the condensation is a trivial graph which has one vertex and no edge. Worst-case performance. V;E/is said to be strongly connected if for every pair of nodes u;v2V, there is a directed path from uto v(and vice-versa) in G. Is is it possible that there is an edge e e such that for all t ∈Tr t ∈ T r, we have e ∈ E(Tr) e ∈ E ( T r)? Sep 1, 2022 · I have the following directed graph: I was trying to determine if the above graph is a strongly connected graph or not. This is a natural consequence of the definitions. The nodes in a strongly connected digraph therefore must all have indegree of at least 1. When you do max(nx. t. The condensation of G is a directed graph Oct 11, 2023 · These are a sort of polar opposite of strongly connected digraphs. Your graph has one connected component when viewed as an undirected graph, and two strongly connected components Here is a definition about connected graphs from the book Ralph Grimaldi - Discrete and Combinatorial Mathematics - 5ed. A directed graph has an Eulerian cycle if and only if. ) and the trick is that you can construct a directed odd-length cycle from an undirected odd-length cycle in a strongly connected component. Jun 6, 2023 · Biconnected graph. Parameters: GNetworkX Graph. Strong connectivity augmentation is a computational problem in the mathematical study of graph algorithms, in which the input is a directed graph and the goal of the problem is to add a small number of edges, or a set of edges with small total weight, so that the added edges make the graph into a strongly connected graph . Lets call {4,5} A, and {0,1,2,3} B. Assuming there are no isolated vertices in the graph you only need to add max (|sources|,|sinks|) edges to make it strongly connected. For directed graphs, strong connectivity is the more useful notion. qh bo av td al oa fp qk bh cm