Graphs and Their Properties

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Questions and Answers

What defines a directed graph?

Edges are not associated with any direction.

Edges form an ordered pair. (correct)

Every node is connected to every other node.

It contains isolated nodes.

Which statement accurately describes a complete graph?

Every node is connected to every other node. (correct)

It contains at least one isolated node.

It has no edges.

It contains only directed edges.

What is a cycle in a graph?

A sequence of nodes with no connection.

A path having no repeated edges or vertices except the first and last. (correct)

A path with repeated edges only.

A closed simple path with distinct nodes.

What does it mean for a graph to be connected?

Some path exists between every two vertices. Signup and view all the answers

Which of the following is NOT a characteristic of a weight graph?

Edges can have multiple directions. Signup and view all the answers

In graph terminology, what defines a closed path?

A path where the initial node and terminal node are the same. Signup and view all the answers

How many edges are in a complete graph with 5 vertices?

$10$ Signup and view all the answers

What is represented by a simple path in graph theory?

A path where all nodes are distinct except for the first and last. Signup and view all the answers

What defines the weight of an edge in a digraph?

The weight of an edge must have positive values. Signup and view all the answers

In a directed graph, when is an entry in an adjacency matrix set to 1?

When there exists an edge directed from Vi to Vj. Signup and view all the answers

Which representation uses an adjacency list to store a graph's structure in memory?

Linked Representation Signup and view all the answers

What can be inferred about a node with a degree of 0?

It has no edges connected to it. Signup and view all the answers

In the adjacency matrix representation of a weighted directed graph, how are the non-zero entries filled?

With the weight of respective edges. Signup and view all the answers

What term describes two nodes that are directly connected by an edge?

Adjacent nodes Signup and view all the answers

Which of the following accurately describes a loop in a graph?

An edge associated with identical endpoints. Signup and view all the answers

Which method of graph representation uses a matrix to show connections between vertices?

Sequential Representation Signup and view all the answers

What is the relationship between the sum of the lengths of adjacency lists and edges in an undirected graph?

It equals the twice of the number of edges present in the graph. Signup and view all the answers

In a directed graph, how does the sum of lengths of all adjacency lists compare to the number of edges?

It equals the number of edges. Signup and view all the answers

Which traversal algorithm involves exploring all neighboring nodes before moving to the next level?

Breadth First Search (BFS) Signup and view all the answers

What is the initial action performed by the BFS algorithm?

Putting a vertex at the back of a queue. Signup and view all the answers

What do you need to do in order to avoid cycles when implementing BFS?

Mark each vertex as visited. Signup and view all the answers

In the context of DFS, what characterizes the search as it progresses?

It goes deeper until a goal node or a node with no children is found. Signup and view all the answers

What additional information does a node in a weighted directed graph contain?

The weight of the node. Signup and view all the answers

What is the main purpose of the Graph Traversal Algorithm?

To examine all the nodes and vertices of the graph. Signup and view all the answers

What method does BFS use to keep track of the next location to visit?

Queue Signup and view all the answers

Which of the following applications is NOT typically associated with BFS?

Finding a path between two nodes Signup and view all the answers

Which algorithm ensures a shallow path solution when searching through a graph?

BFS Signup and view all the answers

Which of these statements about DFS backtracking is correct?

DFS follows a backtrack according to the path depth. Signup and view all the answers

What is a common application of topological sorting?

Scheduling jobs based on dependencies Signup and view all the answers

In topological sorting, what must be true for a vertex to be placed before another?

It has a lower in-degree. Signup and view all the answers

What occurs if two vertices have the same in-degree during topological sorting?

There are multiple valid topological sorts possible. Signup and view all the answers

What does DFS directly aim to find during its traversal?

A path from the start to an end node. Signup and view all the answers

What data structure is primarily used in Depth First Search (DFS)?

Stack Signup and view all the answers

In the context of DFS, what are discovery edges?

Edges leading to an unvisited node Signup and view all the answers

Which of the following statements about DFS is true?

DFS utilizes backtracking when encountering dead ends Signup and view all the answers

What does the DFS algorithm do after reaching a node with no unvisited adjacent vertices?

It backtracks to the previous node in the stack Signup and view all the answers

What type of list structure does BFS use compared to DFS?

DFS uses LIFO while BFS uses FIFO Signup and view all the answers

Which algorithm is generally faster for traversing deep paths in graphs?

DFS is faster due to its depth-focused approach Signup and view all the answers

What is one major limitation of BFS regarding decision trees?

It can only traverse the first level of the tree Signup and view all the answers

In which scenario does DFS perform better than BFS?

When the target node is farther from the source Signup and view all the answers

Study Notes

Graphs

Graphs are a collection of vertices and edges that connect them.
They can be represented as cyclic trees where vertices have complex relationships.
A graph G is defined as an ordered set G(V, E), where V(G) is the set of vertices and E(G) is the set of edges connecting them.

Directed and Undirected Graphs

Undirected graphs: Edges have no direction; traversing an edge is possible in both directions between connected vertices.
Directed graphs: Edges represent a specific path from one vertex to another. They form an ordered pair (initial node, terminal node).

Graph Terminology

Path: A sequence of nodes followed to reach a terminal node from an initial node.
Closed Path: A path where the initial and terminal nodes are the same.
Simple Path: A closed path with all nodes distinct except the first and last.
Cycle: A path with no repeated edges or vertices except the first and last.
Connected Graph: A graph where a path exists between every pair of vertices. There are no isolated nodes.
Complete Graph: A graph where every node is connected to all other nodes. It contains n(n-1)/2 edges, where n is the number of nodes.
Weighted Graph: Each edge has an assigned weight value, representing the cost of traversing the edge.
Digraph: A directed graph where each edge has a specified direction.
Loop: An edge connecting a vertex back to itself.
Adjacent Nodes: Two nodes connected by an edge.
Degree of a Node: The number of edges connected to a node. A node with degree 0 is an isolated node.

Graph Representation

Sequential Representation: Uses an adjacency matrix to store the mapping of vertices and edges.
- The rows and columns of the matrix are represented by graph vertices.
- An entry Mij in the matrix is 1 if there's an edge between Vi and Vj (for undirected graphs).
Linked Representation: Uses an adjacency list to store the graph.
- Each node has an adjacency list containing its connected nodes and pointers to the next adjacent nodes.
- For undirected graphs, the sum of list lengths is twice the number of edges.
- For directed graphs, the sum of list lengths equals the number of edges.
- Weighted graphs include the weight of each edge in the adjacency list.

Graph Traversal Algorithms

Traversal: Exploring all nodes and edges of a graph.
Breadth First Search (BFS): Starts from a root node and explores all neighbors before moving to deeper nodes.
- Utilizes a queue data structure to manage node exploration.
- Visited and Not Visited categories are used to prevent cycles.
Depth First Search (DFS): Starts from an initial node and explores as deep as possible before backtracking.
- Employs a stack data structure.
- Uses discovery edges (unvisited node) and back edges (already visited node).
- Also uses Visited and Not Visited categories.

BFS vs DFS

Parameter	BFS	DFS
Data Structure	Queue	Stack
Speed	Relatively slower compared to DFS	Relatively faster compared to BFS
Distance from Source	Better suited for targets close to the source	Better suited for targets far from the source
Suitability for Decision Tree	Not well-suited for decision trees	More suitable for decision trees; allows traversal within decisions
Type of List	FIFO (First-In, First-Out)	LIFO (Last-In, First-Out)
Tracking Method	Uses the queue to keep track of the next location to visit	Uses the stack to keep track of the next location to visit
Type of Solution	Ensures a shallow path solution; finds the shortest path first	Doesn't ensure a shallow path solution; backtracks to find a solution
Tree Path	Traverses a path according to tree level	Traverses a path according to tree depth
Backtracking	Backtracking is not needed in BFS	Backtracking is needed in DFS

Applications of BFS and DFS

BFS:
- Checking graph connectivity
- Generating a spanning tree
- Finding the shortest path
- GPS navigation
DFS:
- Testing connectivity
- Finding a path between two nodes
- Solving puzzles

Topological Sort

A linear ordering of vertices in a directed acyclic graph (DAG).
For every edge U-V in the DAG, vertex U appears before vertex V in the ordering.
It determines a sequence in which tasks can be executed without violating dependencies.

Applications of Topological Sort

Scheduling jobs based on dependencies
Instruction scheduling in compilers
Determining compilation task order
Data serialization

Example of Topological Sort

The example illustrates how to find different topological orderings for a given graph.
It demonstrates the process of removing vertices with the least in-degree, updating the in-degree of other vertices, and finding possible orderings.

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Description

Explore the fundamental concepts of graphs, including directed and undirected graphs, and important terminology like paths and cycles. This quiz will challenge your understanding of how vertices and edges interact within graph theory.