Dynamic Graphs: Connected Components in Dynamic Graphs

Chapter 3: Connected Components (CC) in Dynamic Graphs

Introduction

Connected Components are a fundamental concept in graph theory, partitioning a graph into disjoint subgraphs within which any two vertices are connected by a path. In dynamic graphs, vertices and edges can be added or removed over time, which poses challenges for efficiently identifying connected components. This chapter explores these challenges and solutions.

Standard Connected Components Algorithm

The most common algorithm for finding connected components in a static, undirected graph is Depth-First Search (DFS). Here's a quick review of the algorithm:

  ConnectedComponents(Graph G):
    1. Initialize an empty list to store connected components
    2. Mark all vertices as unvisited
    3. For each unvisited vertex u:
        4. Initialize an empty list for the current component
        5. DFS(u, current component)
        6. Add the current component to the list of connected components

  DFS(u, current component):
    1. Mark u as visited
    2. Add u to current component
    3. For each neighbor v of u:
        4. If v is not visited:
            5. DFS(v, current component)

Dynamic Graphs and Connected Components

The primary challenges in identifying connected components in dynamic graphs are:

• When vertices or edges are added, existing components may merge.
• When vertices or edges are removed, a single component may split into multiple components.

Connected Components in Dynamic Graphs

Strategies to adapt Connected Components algorithms for dynamic graphs include:

1. Recomputing from Scratch: Simple but computationally expensive.
2. Incremental Updates: Efficiently update the connected components as changes occur.

Incremental Connected Components Algorithm

  IncrementalConnectedComponents(Graph G_t, CC_List):
    1. Detect changes (added/removed vertices and edges)
    2. Update CC_List according to the changes
    3. Correct any inconsistencies in CC_List

Conclusion

Connected components are fundamental to many graph algorithms and applications. Like BFS and SSSP, connected components can be efficiently computed in dynamic graphs using incremental methods.