### Dynamic Graphs: Strongly Connected Components (SCC)

Chapter 4: Strongly Connected Components in Dynamic Graphs

# Chapter 4: Strongly Connected Components (SCC) in Dynamic Graphs

## Introduction

A Strongly Connected Component (SCC) in a directed graph is a subgraph in which there's a directed path from any vertex to any other vertex within the subgraph. In dynamic graphs where edges and vertices can be added or removed, maintaining the SCCs efficiently is a challenging task. This chapter explores how to handle SCCs in dynamic scenarios.

## Standard Algorithms for SCCs

Tarjan's algorithm and Kosaraju's algorithm are commonly used for finding SCCs in static graphs. Below is a brief outline of Tarjan's algorithm.

``````
TarjanSCC(Graph G):
1. Initialize empty stack S, time = 0
2. For each unvisited vertex v:
3. strongconnect(v)

strongconnect(v):
1. Initialize v.index and v.lowlink to time, time++
2. Push v onto S
3. For each neighbor w of v:
4. If w is unvisited, recurse on strongconnect(w)
6. If v is a root node, pop stack and generate SCC
``````

## Dynamic Graphs and SCCs

The main challenges for maintaining SCCs in dynamic graphs are:

• When an edge is added, the SCC structure may need to be updated.
• When an edge is removed, an SCC may split into multiple smaller SCCs.

## SCCs in Dynamic Graphs

Strategies for maintaining SCCs in dynamic graphs include:

1. Recomputing from Scratch: This approach is straightforward but inefficient for large graphs.
2. Incremental Updates: This approach aims to update the SCCs incrementally to save computation.

### Incremental SCC Algorithm

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

## Conclusion

Strongly Connected Components are a critical aspect of directed graphs. Efficient algorithms for maintaining SCCs in dynamic graphs can be valuable for a variety of applications, from network analysis to web crawling.