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An excerpt from the book Algorithms (4th Edition) by Robert Sedgewick and Kevin Wayne. And I also added a python implementation.

- A
**graph**is a set of vertices and a collection of edges that each connect a pair of vertices. - A
**vertex**is the fundamental unit of which graphs are formed. - An
**edge**is a connection between two vertices. - A
**path**in a graph is a sequence of vertices connected by edges. - A
**simple path**is one with no repeated vertices. - A
**cycle**is a path with at least one edge whose first and last vertices are the same. - A
**simple cycle**is a cycle with no repeated edges or vertices (except the requisite repetition of the first and last vertices). - The
**length of a path or a cycle**is its number of edges. - A graph is
**connected**if there is a path from every vertex to every other vertex in the graph. - A graph that is
**not connected**consists of a set of connected components, which are maximal connected subgraphs. - An
**acyclic**graph is a graph with no cycles. - A
**tree**is an acyclic-connected graph. - A disjoint set of trees is called a
**forest**. - A
**spanning tree**of a connected graph is a subgraph that contains all of that graph’s vertices and is a single tree. - A
**spanning forest**of a graph is the union of spanning trees of its**connected components**. - A
**self-loop**is an edge that connects a vertex to itself. - Two edges that connect the same pair of vertices are
**parallel**. - Mathematicians sometimes refer to graphs with parallel edges as
**multigraphs**and graphs with no parallel edges or self-loops as**simple graphs**. - When there is an edge connecting two vertices, we say that the vertices are
**adjacent**to one another and that the edge is**incident**to both vertices. - The
**degree**of a vertex is the number of edges incident to it. - A
**subgraph**is a subset of a graphâ€™s edges (and associated vertices) that constitutes a graph. - The
**density**of a graph is the proportion of possible pairs of vertices that are connected by edges. - A
**sparse**graph has relatively few possible edges present. - A
**dense**graph has relatively few possible edges missing. - A
**bipartite graph**is a graph whose vertices we can divide into two sets such that all edges connect a vertex in one set with a vertex in the other set. You can colour one set of vertices red and the other set of vertices black.

We have 2 basic requirements to choose the best graph representation (data structure) to use.

- We must have the
*space*to accommodate the types of graphs that we are likely to encounter in applications. - We want to develop
*time-efficient*implementations of Graph instance methods (the basic methods that we need to develop graph-processing clients).

We maintain a V-by-V boolean array, with the entry in row v and column w defined to be true if there is an edge adjacent to both vertex v and vertex w in the graph, and to be false otherwise.

This representation fails on the first requirement. Graphs with millions of vertices are common and the space cost for the V^{2} boolean values needed is prohibitive.

Using an Edge class with two instance variables of type *int*.

This direct representation is simple but it fails on the second requirement, implementing the get_neighbours function of a vertex would involve examining all the edges in the graph.

We maintain a vertex-indexed array of lists of the vertices adjacent to each vertex.

This data structure satisfies both requirements for typical applications and is the one we will use here.

This implementation achieves the following performance characteristics:

- Space usage proportional to V + E
- Constant time to add an edge
- Time proportional to the degree of v to iterate through vertices adjacent to v (constant time per adjacent vertex processed)

```
class Graph:
def __init__(self, V):
self.__V = V
self.__E = 0
self.__adj_list = [[] for i in range(V)]
def V(self): return self.__V
def E(self): return self.__E
def add_edge(self, v1, v2):
self.__adj_list[v1].append(v2)
self.__adj_list[v2].append(v1)
self.__E += 1
def neighbours(self, v):
return self.__adj_list[v]
def __str__(self):
res = ''
for v in range(self.__V):
res += str(v) + ': '
if self.neighbours(v) is not None: res += str(self.neighbours(v))
res += '\n'
return res
```

```
from undirected_graph import Graph
import requests
def create_graph_from(url):
response = requests.get(url)
data = response.text.split('\n')
v = int(data[0])
e = int(data[1])
g = Graph(v)
for idx in range(e):
v1, v2 = data[2 + idx].split(' ')
g.add_edge(int(v1), int(v2))
return g
g = create_graph_from('https://algs4.cs.princeton.edu/41graph/tinyG.txt')
print(g)
```