Graph Theory - Weakly Connected Graphs

Weakly Connected Graphs

A weakly connected graph is a directed graph in which there is a path between any two vertices if we ignore the direction of edges. In other words, a graph is weakly connected if replacing all its directed edges with undirected edges results in a connected graph.

Properties of Weakly Connected Graphs

Weakly connected graphs have several important properties, such as −

• Undirected Connectivity: If we ignore the direction of edges, there is a path between any pair of vertices in the graph. However, the directed edges may not allow full bidirectional reachability.
• Multiple Components: A weakly connected graph may contain multiple weakly connected components, as some vertices may not be reachable from others when considering the direction of edges.
• Not Fully Reachable: Unlike strongly connected graphs, weakly connected graphs may have vertices that are unreachable due to the directionality of edges, but there will be a path if directionality is ignored.
• Not Cyclic: A weakly connected graph can contain cycles in the undirected version of the graph, but it might not have cycles when considering edge direction.

Weakly Connected Components (WCCs)

In a weakly connected graph, it can be helpful to break the graph into smaller subgraphs known as weakly connected components (WCCs). An WCC is a maximal subgraph where any two vertices are connected by a path when ignoring the edge directions.

The following image shows a graph broken into weakly connected components −

• Maximal: A WCC is the largest subgraph where all vertices are connected when directionality of edges is ignored.
• Independence: Each WCC is weakly connected, and no vertex in one WCC is reachable from a vertex in another WCC when ignoring edge direction.
• Partition: The entire graph can be partitioned into WCCs, where each WCC is a connected component in the undirected version of the graph.

Conditions for Weak Connectivity

A directed graph is weakly connected if and only if it satisfies the following condition −

• When we ignore the direction of edges, there is a path between any pair of vertices u and v. That is, for every pair of vertices u and v, there exists a path from u to v and a path from v to u when the directions are ignored.

Example: Weakly Connected Graph

In the below graph, there is a path between all pairs of vertices when ignoring edge directions, but the directed edges do not guarantee full reachability −

Applications of Weakly Connected Graphs

Weakly connected graphs have practical applications in scenarios where the direction of edges is not as important, but overall connectivity matters −

• Routing Algorithms: In some routing algorithms, especially in communication networks, weak connectivity ensures that all nodes are reachable regardless of edge direction.
• Social Networks: In social networks, weak connectivity can represent groups where any two users can interact indirectly, even though the interactions are not necessarily bidirectional.
• Infrastructure Systems: In infrastructure networks (like water or electricity grids), weak connectivity ensures that all points in the network are indirectly connected, ensuring strongness even in the case of edge directionality being ignored.
• Graph Partitioning: Weakly connected graphs are used in graph partitioning to ensure that the network can be split into smaller subgraphs while maintaining connectivity in the undirected version.

Testing Weak Connectivity

There are several methods for testing whether a directed graph is weakly connected, such as −

• DFS-Based Algorithm: A depth-first search (DFS) can be used to check if there is a path between all vertices when we ignore the direction of edges. If all vertices are reachable from a starting vertex in an undirected version of the graph, the graph is weakly connected.
• Breadth-First Search (BFS): Similar to DFS, BFS can be used to check if the graph is weakly connected by treating all edges as undirected and confirming reachability between all pairs of vertices.
• Connected Components Algorithm: One way to check for weak connectivity is by finding all the connected components in the undirected version of the graph. If there is exactly one connected component, the graph is weakly connected.

Weak Connectivity in Various Graphs

Following are the lists of various types of directed graphs and their weak connectivity status −