2685 count the number of complete components dfs graph

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count the number of complete components in a graph using depth-first search (dfs)

this tutorial provides a comprehensive guide to identifying and counting complete components in a graph using depth-first search (dfs). we'll cover the problem definition, necessary background, a detailed dfs-based algorithm, code implementation with explanations, and potential optimizations.

**1. problem definition: complete components**

a **complete graph** (also called a clique) is a graph where every pair of distinct vertices is connected by an edge. in other words, every vertex is directly connected to every other vertex in the graph.

a **component** of a graph is a maximal connected subgraph. that means you can reach any vertex in the component from any other vertex in the component by following a path of edges. the component is *maximal* because if you add any other vertex from the graph that wasn't already in the component, it's no longer connected.

a **complete component** is a component that is also a complete graph.

the problem asks us to count the number of complete components in a given undirected graph.

**example:**

consider a graph with the following edges:

* 0-1
* 0-2
* 1-2
* 3-4
* 3-5
* 4-5
* 6-7

this graph has three components:

* component 1: {0, 1, 2} - this is a complete component because vertices 0, 1, and 2 are all connected to each other.
* component 2: {3, 4, 5} - this is also a complete component because vertices 3, 4, and 5 are all connected to each other.
* component 3: {6, 7} - this is a complete component because vertices 6 and 7 are connected to each other.

therefore, the answer would be 3.

**2. prerequisites: graph representation and dfs**

before diving into the algorithm, let's review the essentials:

* **graph representation:** we'll use an adjacency list to represent the graph. an adjacency list is a list (or array) where each index `i` represents a vertex in the graph, and the value at index `i` is a list of the vertices to which ...

#DFS #GraphTraversal #numpy
count complete components
DFS graph
graph traversal
connected components
depth-first search
graph algorithms
component counting
undirected graph
algorithm efficiency
graph theory
data structures
recursive traversal
adjacency list
cycle detection
graph representation

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