### Dynamic Graphs: Connected Components in Dynamic Graphs

Chapter 3: 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.