Graphs and Graph Theory

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Questions and Answers

What is the time complexity of Depth-First Search (DFS) when using adjacency lists to represent a graph?

O(n)

O(n log n)

O(n^2)

O(e) (correct)

In the BFS algorithm, what data structure is primarily used to keep track of vertices to be visited?

Array

Queue (correct)

Linked List

Stack

If a graph is represented as an adjacency matrix, what is the time complexity for determining all vertices adjacent to a given vertex v?

O(e)

O(n) (correct)

O(1)

O(n^2)

Which of the following statements about Breadth-First Search (BFS) is true?

BFS produces a spanning tree as a final result. Signup and view all the answers

When BFS completes and the queue is empty, what is the final step regarding the graph?

Produce a final spanning tree by removing unused edges. Signup and view all the answers

What is an Eulerian walk?

A path that traverses every edge exactly once and returns to the starting vertex. Signup and view all the answers

How is a graph defined mathematically?

As a collection of vertices and edges, represented as G = (V, E). Signup and view all the answers

What guarantees the existence of an Eulerian walk for a graph?

All vertices having even degree. Signup and view all the answers

What characterizes an undirected edge in a graph?

A bidirectional link between two vertices. Signup and view all the answers

In the example graph G = (V, E), if V = {A, B, C, D, E}, what is the degree of vertex A?

3 Signup and view all the answers

What is the difference between directed and undirected graphs?

Undirected graphs allow movement in any direction, while directed graphs restrict movement. Signup and view all the answers

What is the representation of the edge connecting vertices B and D?

(B, D) Signup and view all the answers

Which statement is true about weighted edges?

They represent a variable cost for traversing between vertices. Signup and view all the answers

What characteristics define a complete graph?

All nodes are connected to each other. Signup and view all the answers

Which statement about a regular graph is true?

Each node is connected to another by edges equally. Signup and view all the answers

What is a key property of a cycle graph?

The first and last nodes are the same. Signup and view all the answers

Which statement correctly describes an acyclic graph?

No cycles are present in the graph. Signup and view all the answers

What does a weighted graph represent?

Values are assigned to edges based on distance. Signup and view all the answers

Which of the following correctly defines the indegree of a vertex?

The total number of incoming edges to the vertex. Signup and view all the answers

In the context of graph theory, what distinguishes a simple graph?

It has no parallel edges or self-loops. Signup and view all the answers

What is the significance of adjacent nodes in a graph?

They are connected by an edge. Signup and view all the answers

What data structure is primarily used to implement Depth-First Search (DFS)?

Stack Signup and view all the answers

What is the time complexity of DFS when using an adjacency matrix?

O(|V|^2) Signup and view all the answers

What is a spanning tree in the context of DFS traversal?

A graph that connects all vertices without any loops Signup and view all the answers

Which condition must be satisfied for the DFS traversal to terminate?

There are no unvisited vertices reachable from visited ones Signup and view all the answers

During DFS traversal, how are adjacent vertices explored?

Recursively, one by one Signup and view all the answers

Which step is involved in backtracking during DFS?

Pop a vertex from the stack to revisit it later Signup and view all the answers

What should you remove from the graph to finalize the spanning tree after DFS?

Unused edges Signup and view all the answers

What is the primary characteristic of the graphs discussed in the DFS context?

Undirected graphs only Signup and view all the answers

What characterizes the adjacency matrix of an undirected graph?

It is symmetric. Signup and view all the answers

What defines a circuit in graph theory?

A closed walk with all vertices appearing only once, except the initial and final vertex. Signup and view all the answers

What is a distinguishing feature of the adjacency list representation in a graph?

It provides a list of adjacent vertices for each vertex. Signup and view all the answers

In an adjacency multilists representation, how are edges represented?

By two entries, one for each vertex incident to the edge. Signup and view all the answers

Which statement accurately describes a connected graph?

A graph where all vertices are connected to at least one other vertex. Signup and view all the answers

What is one purpose of using adjacency multilists in graph representation?

To easily mark edges as examined. Signup and view all the answers

What is the degree of a vertex in an undirected graph?

The total number of edges incident to that vertex. Signup and view all the answers

How is weight information generally stored in a graph using adjacency lists?

By including an additional field in the list nodes. Signup and view all the answers

How is the indegree of a vertex defined?

The total number of edges leading to the vertex. Signup and view all the answers

Which representation of a graph uses a matrix format to denote edges?

Adjacency Matrix Signup and view all the answers

What are weighted edges useful for in graph applications?

Capturing distances or costs between vertices. Signup and view all the answers

In graph representation, what does a value of 1 in the adjacency matrix signify?

There is an edge from the row vertex to the column vertex. Signup and view all the answers

What is a characteristic of a directed graph's adjacency matrix?

It can be asymmetric. Signup and view all the answers

In the context of graph theory, what is typically true about the adjacency list in relation to memory usage?

It uses less memory than an adjacency matrix when graphs are sparse. Signup and view all the answers

Which of the following accurately defines a subgraph?

A graph whose vertices and edges are entirely contained within another graph. Signup and view all the answers

What is characterized as the outdegree of a vertex?

The total number of edges directed from that vertex. Signup and view all the answers

Study Notes

Graphs

A graph is a non-linear data structure
Graphs are comprised of vertices (nodes) and edges (arcs) connecting the vertices.
A map is a common example. Cities are vertices, and roads connect them via edges.
Graph theory was used to solve the Seven Bridges of Königsberg problem.

Graph Terminology

Vertex (Node): An individual data element in a graph. In the example, A, B, C, D, and E are vertices.
Edge (Arc): Connects two vertices. Represented as (starting vertex, ending vertex). Example: (A,B).
Undirected Edge: Bidirectional. (A, B) is equal to (B, A).
Directed Edge: Unidirectional. (A, B) is not equal to (B, A).

Types of Graphs

Undirected Graph: Contains only undirected edges.
Directed Graph: Contains only directed edges.
Complete Graph: Each vertex is adjacent to all other vertices. Edges = n(n-1)/2, where n is the number of vertices.
Regular Graph: All vertices have the same degree. All vertices are adjacent to each other.
Cycle Graph: A graph containing a cycle; the first and last vertices are the same.
Acyclic Graph: A graph without cycles.
Weighted Graph: Edges have assigned costs or values, often representing distances. Also known as a network.
Outgoing Edge: Directed edge originating from a vertex.
Incoming Edge: Directed edge ending at a vertex.
Degree: Total number of (undirected) edges attached to a vertex.
Indegree: Number of incoming edges to a vertex in a directed graph.
Outdegree: Number of outgoing edges from a vertex in a directed graph.
Parallel Edges/Multiple Edges: Two or more edges connect the same vertices.
Self-Loop: An edge where the start and end vertices are the same.
Simple Graph: A graph with no parallel edges or self-loops.
Adjacent Nodes: Vertices connected by an edge.
Incidence: In an undirected graph, an edge is incident on each of its vertices.
Closed Walk: A walk that begins and ends at the same vertex.
Open Walk: A walk that begins and ends at different vertices.
Path: An open walk in which no vertex repeats.
Circuit: A closed walk in which no vertex repeats (except the first/last vertex).
Subgraph: All vertices and edges of a smaller graph are also part of the larger graph.
Connected Graph: There's a path between any two vertices.
Disconnected Graph: Not connected.

Graph Representations

Adjacency Matrix: A 2D matrix representing a graph; 1 indicates an edge, 0 otherwise.
Adjacency List: A list-based representation; each vertex stores a list of its adjacent vertices.
Adjacency Multilists: Each edge is represented in the lists of both vertices it connects.

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Description

Explore the fundamentals of graphs, including their definition, terminology, and different types. This quiz will also cover examples like the Seven Bridges of Königsberg problem and the distinctions between directed and undirected graphs.