Cross-System Mapping Principle: Different body systems interact in structured ways. Objective: To trace anatomical and functional connections across systems (e.g., dental-bone-lymphatic). Method: Graph Theory

introduction

Some structures miss the

Graph Theory - 8.1 Introduction, Data Structures - 8.2 Graphs and Multigraphs - 8.3 Subgraphs, Isomorphic and Homeomorphic Graphs - 8.4 Paths, Connectivity - 8.5 Traversable and Eulerian Graphs,Bridges of Königsberg - 8.6 Labeled and Weighted Graphs - 8.7 Complete, Regular, and Bipartite Graphs - 8.8 Tree Graphs - 8.9 Planar Graphs - 8.10 Graph Colorings - 8.11 Representing Graphs in Computer Memory - 8.12 Graph Algorithms - 8.13 Traveling-Salesman Problem SolvedProblems

Directed Graphs - 9.1 Introduction - 9.2 Directed Graphs - 9.3 Basic Definitions - 9.4 Rooted Trees - 9.5 Sequential Representation of Directed Graphs - 9.6 Warshall’s Algorithm, Shortest Paths - 9.7 Linked Representation of Directed Graphs - 9.8 Graph Algorithms: Depth-First and Breadth-First Searches - 9.9 Directed Cycle-Free Graphs, Topological Sort - 9.10 Pruning Algorithm for Shortest Path

• Binary Trees
• 10.1 Introduction
• 10.2 Binary Trees
• 10.3 Complete and Extended Binary Trees
• 10.4 Representing Binary Trees in Memory
• 10.5 Traversing Binary Trees
• 10.6 Binary Search Trees
• 10.7 Priority Queues, Heaps
• 10.8 Path Lengths, Huffman’s Algorithm
• 10.9 General (Ordered Rooted) Trees Revisited

Properties of the Integers - 11.1 Introduction - 11.2 Order and Inequalities, Absolute Value - 11.3 Mathematical Induction - 11.4 Division Algorithm - 11.5 Divisibility, Primes - 11.6 Greatest Common Divisor, Euclidean Algorithm - 11.7 Fundamental Theorem of Arithmetic - 11.8 Congruence Relation - 11.9 Congruence Equations

Ordered Sets and Lattices - 14.1 Introduction - 14.2 Ordered Sets - 14.3 Hasse Diagrams of Partially Ordered Sets - 14.4 Consistent Enumeration - 14.5 Supremum and Infimum - 14.6 Isomorphic (Similar) Ordered Sets - 14.7 Well-Ordered Sets - 14.8 Lattices - 14.9 Bounded Lattices - 14.10 Distributive Lattices - 14.11 Complements, Complemented Lattices - CHAPTER 15 - Boolean Algebra - 15.1 Introduction - 15.2 Basic Definitions - 15.3 Duality - 15.4 Basic Theorems - 15.5 Boolean Algebras as Lattices 15.6 Representation Theorem - 15.7 Sum-of-Products Form for Sets - 15.8 Sum-of-Products Form for Boolean Algebras 15.9 Minimal Boolean Expressions, Prime Implicants 15.10 Logic Gates and Circuits - 15.11 Truth Tables, Boolean Functions - 15.12 Karnaugh Maps

Vectors and Matrices - A.1 Introduction - A.2 Vectors A.3 Matrices - A.4 Matrix Addition and Scalar Multiplication A.5 Matrix Multiplication A.6 Transpose - A.7 Square Matrices - A.8 Invertible (Nonsingular) Matrices, Inverses A.9 Determinants - A.10 Elementary Row Operations, Gaussian Elimination (Optional) - A.11 Boolean (Zero-One) Matrices

Graph theory is a part of mathematics that studies graphs, which are structures made of nodes (points) and edges (lines) connecting them

Learning graph theory helps you understand how pathology flows in anatomy

Formally, a graph is a pair of sets (V, E), where V is the set of vertices and E is the set of edges, connecting the pairs of vertices. Take a look at the following graph −

[image]

V = {a, b, c, d, e} E = {ab, ac, bd, cd, de}

Basic Definitions and Concepts Here are some basic terms in graph theory −

Vertex (Node): A fundamental unit representing a point or an object in a graph. Edge (Link): A line connecting two vertices, representing a relationship or pathway. Adjacent Vertices: Vertices connected by an edge. Degree of a Vertex: The number of edges connected to a vertex. Path: A sequence of vertices connected by edges, with each vertex visited only once. Cycle: A path that starts and ends at the same vertex, forming a loop. Connected Graph: A graph where there is a path between every pair of vertices. Subgraph: A graph made from a subset of vertices and edges of another graph

Types of Graphs Graphs can be classified into different types −

Undirected Graph: A graph where edges have no direction, meaning connections are bidirectional. Directed Graph (Digraph): A graph where edges have a direction, indicating one-way relationships between vertices. Weighted Graph: A graph where edges have weights or costs, representing the strength or capacity of connections. Unweighted Graph: A graph where all edges are equal, with no weights assigned. Simple Graph: A graph with no loops (edges connecting a vertex to itself) and no multiple edges between the same vertices. Multigraph: A graph that allows multiple edges between the same vertices. Bipartite Graph: A graph whose vertices can be divided into two sets where no two vertices within the same set are adjacent. Complete Graph: A graph where there is an edge between every pair of vertices.

Graph Representation Graphs can be represented in various ways −

Adjacency Matrix: A square matrix where the element at row i and column j indicates the presence of an edge between vertices i and j. Adjacency List: A collection of lists where each list contains the vertices adjacent to a particular vertex. Incidence Matrix: A matrix where rows represent vertices and columns represent edges, showing which vertices are connected by which edges.

Adjacency List: 1: [2] 2: [1, 3] 3: [2, 4] 4: [3, 5] 5: [4]

Adjacency Matrix: [[0. 1. 0. 0. 0.] [1. 0. 1. 0. 0.] [0. 1. 0. 1. 0.] [0. 0. 1. 0. 1.] [0. 0. 0. 1. 0.]]

Incidence Matrix: [[1. 0. 0. 0.] [1. 1. 0. 0.] [0. 1. 1. 0.] [0. 0. 1. 1.] [0. 0. 0. 1.]]

Directed vs. Undirected Graphs The directed and undirected graphs are defined below −

In an undirected graph, the edges have no direction, meaning the relationship is mutual. In a directed graph (or digraph), each edge has a direction, meaning the relationship is one-way.

Weighted Graph A weighted graph is a graph in which each edge has a weight or cost associated with it. In a simple graph, the weights could represent distances, time, or other quantities depending on the application.

Multigraph A multigraph is a type of graph that allows multiple edges between the same pair of vertices. These multiple edges are also called parallel edges.

Directed graph

Example Consider a directed graph with vertices V = {A, B, C, D} and edges E = {(A B), (B C), (C D), (D A)}. This is a directed graph because −

The edges have a direction (for example, A B means the edge starts at vertex A and ends at vertex B). The direction of each edge is important, and it determines the flow of information, control, or dependencies.

Adjacency Matrix Representation The adjacency matrix of a directed graph is a square matrix where each entry a[i][j] indicates whether there is a directed edge from vertex i to vertex j. In the matrix, 1 indicates the presence of a directed edge, and 0 indicates the absence of one.

Example

For the directed graph with vertices V = {A, B, C} and edges E = {(A B), (B C), (A C)}, the adjacency matrix would look like this −

A B C A 0 1 1 B 0 0 1 C 0 0 0 This matrix indicates −

There is a directed edge from A B. There is a directed edge from B C. There is a directed edge from A C. There are no edges starting or ending at C.

Adjacency List Representation The adjacency list representation is a more space-efficient way to represent a directed graph. Each vertex has a list of its adjacent vertices, representing the direction of the edges.

Example

For the directed graph with vertices V = {A, B, C} and edges E = {(A B), (B C), (A C)}, the adjacency list representation would be

A: [B, C] B: [C] C: [] Here, vertex A points to B and C, while vertex B points to C, and vertex C has no outgoing edges.

Edge List Representation In the edge list representation, each directed edge is simply represented as a pair of vertices, indicating the direction of the edge.

Example

For the directed graph with edges E = {(A B), (B C), (A C)}, the edge list representation would be −

(A B) (B C) (A C)

Degree of a Vertex in Directed Graphs The degree of a vertex in a directed graph is broken down into two parts: the in-degree and out-degree.

In-degree The in-degree of a vertex is the number of edges that point to that vertex. It represents the number of incoming edges to the vertex.

Example

For the directed graph with edges E = {(A B), (B C), (A C)}, the in-degree of each vertex is

degin(A) = 0 (no edges point to A) degin(B) = 1 (one edge points to B: A B) degin(C) = 2 (two edges point to C: B C and A C) Out-degree The out-degree of a vertex is the number of edges that originate from that vertex. It represents the number of outgoing edges from the vertex.

Example

For the directed graph with edges E = {(A B), (B C), (A C)}, the out-degree of each vertex is −

degout(A) = 2 (edges A B and A C) degout(B) = 1 (edge B C) degout(C) = 0 (no outgoing edges)

Properties of Directed Graphs

Directed graphs have unique properties that distinguish them from undirected graphs. These properties are important in understanding the structure and behavior of directed graphs in different applications.

Transitivity In a directed graph, if there is a directed edge from vertex A to vertex B and from vertex B to vertex C, the graph is said to be transitive if there is a direct edge from vertex A to vertex C.

Strongly Connected Graph A directed graph is said to be strongly connected if there is a directed path from every vertex to every other vertex. In other words, there is a way to travel from any vertex to any other vertex by following the directed edges

Weakly Connected Graph A directed graph is weakly connected if replacing all directed edges with undirected edges results in a connected graph. In a weakly connected graph, there may not be a directed path between every pair of vertices, but there is a way to connect them by disregarding the direction of edges.

Acyclic Graph A directed graph is acyclic if it does not contain any directed cycles, i.e., there is no path that starts and ends at the same vertex, following the directions of the edges. Directed acyclic graphs (DAGs) are important in applications like task scheduling, dependency resolution, and data processing.

Rooted Directed Graph A directed graph is said to be rooted if there is a special vertex, called the root, from which all other vertices can be reached by following the directed edges.

Applications of Directed Graphs

Directed graphs are used in various fields, including computer science, biology, social networks, and more. Their ability to represent relationships with direction makes them versatile for various applications.

Computer Networks In computer networks, directed graphs are used to model network topology, where vertices represent routers or computers, and directed edges represent communication links between them. Routing algorithms such as Dijkstras algorithm operate on directed graphs to find the shortest paths between nodes.

Task Scheduling In task scheduling, directed acyclic graphs (DAGs) are used to represent dependencies between tasks. The vertices represent tasks, and the directed edges represent dependency relationships, indicating which tasks must be completed before others can start.

Social Networks In social network analysis, directed graphs are used to model relationships between users. For example, in Twitter, a directed graph can represent followers, where the directed edge from A to B indicates that A is following B.

Weighted Graphs

A weighted graph is a graph in which each edge is assigned a numerical value, known as the weight. The weight represents the cost, distance, or any other metric that quantifies the relationship between two vertices.

Weighted graphs are extensively used in applications such as network routing, pathfinding, and resource allocation where the weight of edges influences the decision-making process.

Example Consider a weighted graph with vertices V = {A, B, C, D} and edges with weights E = {(A - B, 2), (B - C, 3), (C - D, 1), (A - D, 4)}. Here, the weights represent the cost associated with traveling from one vertex to another

Types of Weighted Graphs

Weighted graphs can be classified based on the nature of the graph and the weights assigned to the edges.

Undirected Weighted Graph

In an undirected weighted graph, the edges have no direction, and the weights represent the cost or distance between the two connected vertices, irrespective of direction.

Directed Weighted Graph

In a directed weighted graph, the edges have a specific direction, and the weights represent the cost or distance from the starting vertex to the ending vertex.

Positive and Negative Weights Weighted graphs can have positive, negative, or zero weights. Positive weights typically represent costs or distances, while negative weights can be used to model benefits or credits. Zero weights indicate no cost or distance between vertices.

Representation of Weighted Graphs Weighted graphs can be represented using adjacency matrices, adjacency lists, and edge lists, similar to unweighted graphs, but with additional information about the weights.

Adjacency Matrix Representation An adjacency matrix for a weighted graph is a square matrix where each entry a[i][j] represents the weight of the edge from vertex i to vertex j. If there is no edge, the entry is typically set to infinity or a designated value indicating no connection.

For a weighted graph with vertices V = {A, B, C, D} and edges with weights E = {(A - B, 3), (B - C, 1), (C - D, 4), (A - D, 2)}, the adjacency matrix would be −

A B C D A 0 3 2 B 0 1
C 0 4 D 0 Here, (infinity) indicates no direct edge between the vertices.

Adjacency List Representation An adjacency list for a weighted graph consists of lists where each vertex points to its adjacent vertices along with the corresponding edge weights.

Example

For the weighted graph with vertices V = {A, B, C, D} and edges with weights E = {(A - B, 3), (B - C, 1), (C - D, 4), (A - D, 2)}, the adjacency list representation would be −

A: [(B, 3), (D, 2)] B: [(C, 1)] C: [(D, 4)] D: [] Edge List Representation An edge list for a weighted graph is a list of edges where each edge is represented as a tuple containing the start vertex, end vertex, and the weight.

Example

For the weighted graph with edges E = {(A - B, 3), (B - C, 1), (C - D, 4), (A - D, 2)}, the edge list representation would be −

(A, B, 3) (B, C, 1) (C, D, 4) (A, D, 2)

Bipartite Graphs

A bipartite graph is a type of graph where the vertices can be divided into two disjoint and independent sets, U and V, such that every edge connects a vertex in U to a vertex in V.

Bipartite graphs are also known as bigraphs. These graphs are used in various applications like modeling relationships between two different classes of objects, such as jobs and workers, or students and courses.

Example Consider a bipartite graph where U = {A, B, C} and V = {1, 2, 3}, and the edges are E = {(A, 1), (A, 2), (B, 2), (C, 3)}. This graph shows the connections between two distinct sets of vertices

Properties of Bipartite Graphs Bipartite graphs have several important properties that distinguish them from other types of graphs.

No Odd-Length Cycles A graph is bipartite if and only if it does not contain any odd-length cycles. This means that any cycle in a bipartite graph must have an even number of edges.

Two-Colorability A bipartite graph can be colored using two colors such that no two adjacent vertices share the same color. This property is known as two-colorability and is a key characteristic of bipartite graphs.

Matching and Perfect Matching In a bipartite graph, a matching is a set of edges without common vertices. A perfect matching is a matching that covers all vertices of the graph. Bipartite graphs are often analyzed for maximum matching problems.

Algorithms for Bipartite Graphs Several algorithms are specifically designed to handle problems in bipartite graphs, such as testing bipartiteness, finding matchings, and solving network flow problems.

Bipartiteness Test To test whether a given graph is bipartite, one can use a graph traversal algorithm like BFS or DFS to check if the graph can be colored using two colors.

Breadth-First Search (BFS) Approach Using BFS, we can start from any vertex and attempt to color the graph using two colors. If we find a vertex that has the same color as one of its neighbors, the graph is not bipartite.

Example

Consider the graph with vertices V = {A, B, C, D, E} and edges E = {(A, B), (A, C), (B, D), (C, D), (D, E)}. Starting from vertex A and using BFS, we can attempt to color the graph.

Depth-First Search (DFS) Approach DFS can also be used for the bipartiteness test by coloring vertices during the traversal. The process involves recursively visiting adjacent vertices and coloring them alternately. If a conflict arises, the graph is not bipartite.

Complete Graphs A complete graph is a type of graph in which every pair of distinct vertices is connected by a unique edge. In other words, in a complete graph, every vertex is adjacent to every other vertex

Subgraphs

In graph theory, a subgraph is a graph formed from a subset of the vertices and edges of another graph. Subgraphs plays an important role in understanding the structure and properties of larger graphs by examining their smaller, constituent parts.

Induced Subgraph vs. Spanning Subgraph The main difference between an induced subgraph and a spanning subgraph is that an induced subgraph can omit vertices but must include all edges between its vertices, whereas a spanning subgraph includes all vertices but can omit edges.

tree

A tree is a special type of graph that is connected and acyclic, meaning it has no cycles or loops. It consists of nodes (vertices) and edges (connections between nodes), where there is exactly one path between any two nodes.

In other words, a tree is a graph where, for any two nodes, you can traverse from one to the other without retracing any edge, and there are no circular paths.

Properties of Trees

A tree has several defining properties, which are important in understanding its behavior and characteristics. These include the following −

Connectivity A tree is a connected graph, meaning there is a path between any two vertices. Unlike general graphs, trees have no isolated vertices.

Acyclic Nature A tree is acyclic, meaning it does not contain any cycles. This is one of the defining characteristics of a tree, distinguishing it from general graphs.

Number of Edges For a tree with n vertices, there are always n-1 edges. This is a fundamental property of trees and is used in algorithms involving trees.

Leaf Nodes A leaf in a tree is a vertex that has only one edge connected to it (also known as a terminal vertex). A tree can have one or more leaf nodes.

Rooted Trees A tree can be considered as a rooted tree if one vertex is designated as the "root," and all edges have a direction pointing away from this root. In a rooted tree, the hierarchy is clearly defined, and each vertex has a parent and potentially multiple children.

Subtrees Any tree can be divided into smaller trees by removing edges. A subtree is a part of a tree that is itself a tree, consisting of a vertex and all its descendants.

Depth of a Tree The depth of a tree is the length of the longest path from the root to any leaf. The depth of a tree gives insight into its hierarchy and is important for various tree-based algorithms.

Height of a Tree The height of a tree is the number of edges on the longest path from the root to any leaf. The height is a measure of the longest possible chain of nodes in the tree.

Level of a Node The level of a node is defined by its distance from the root. The root is at level 0, its direct children are at level 1, and so on.

Balanced Trees A tree is said to be balanced if the height difference between the left and right subtrees of any node is minimal (usually no greater than 1). Balanced trees are important for performance optimization in search operations.

DAT

Types of Trees

Trees come in various types, each with a different purpose or solving a specific problem. These include −

Binary Tree A binary tree is a tree where each node has at most two children. Binary trees are widely used in computer science for tasks like searching, sorting, and hierarchical data storage.

Binary Search Tree (BST) A binary search tree is a binary tree in which each node follows the property that all the values in the left subtree are less than the node's value, and all the values in the right subtree are greater than the node's value. This structure allows for search, insertion, and deletion operations.

AVL Tree An AVL tree is a self-balancing binary search tree. In an AVL tree, the difference in height between the left and right subtrees of any node is at most 1. This balancing property ensures that operations like search, insertion, and deletion are efficient (O(log n) time complexity).

real world applications

Network routing is the process of selecting a path in a network along which data or traffic is transferred from one node (source) to another node (destination).

In the context of graph theory, the nodes represent entities (such as routers, switches, or intersections), and the edges represent links between them (such as network cables or highways).

The goal of routing is to ensure that data is transmitted from the source to the destination with the least cost, time, or resources. Routing algorithms decide how to traverse the network by finding an optimal or near-optimal path based on different criteria like network traffic, available bandwidth, and delays.

Routing Algorithms Several algorithms are used in network routing to determine the best path for data transmission. These algorithms can be classified into two major categories −

Shortest Path Algorithms Flow-Based Algorithms

The following are the steps of Dijkstra's algorithm −

Start with the source node having a distance of 0, and set all other nodes to infinity. Mark the source node as visited, then choose the unvisited node with the smallest distance. For each unvisited neighbor of the current node, calculate its distance and update the shortest distance if it's smaller. Repeat this process until all nodes are visited or the destination node is reached. Time Complexity: O(V2) for basic implementations, and O(E + V log V) for optimized versions using priority queues.

Example

The following code creates an undirected graph with weighted edges and uses Dijkstra's algorithm to find the shortest path from node 1 to node 4 using Python and NetworkX library −

import networkx as nx

G = nx.Graph()
G.add_weighted_edges_from([(1, 2, 1), (1, 3, 2), (2, 3, 2), (2, 4, 1), (3, 4, 3)])

# Find the shortest path from node 1 to node 4
shortest_path = nx.dijkstra_path(G, source=1, target=4)
print(shortest_path)

The result, shortest_path, is a list of nodes representing the path with the minimum total weight −

[1, 2, 4]

Bellman-Ford Algorithm Bellman-Ford is another algorithm for finding the shortest path from a single source node to all other nodes. Unlike Dijkstra's algorithm, Bellman-Ford can handle graphs with negative edge weights, although it is slower in comparison.

The following are the steps of Bellman-Ford algorithm −

Start by setting the distance to the source node as 0 and all other nodes as infinity. Go through all edges and update the shortest distance for each node for V - 1 iterations, where V is the total number of nodes. If a shorter path is found, update the shortest distance for that node. After completing the V - 1 iterations, check for any negative-weight cycles in the graph. Time Complexity: O(V * E), where V is the number of vertices and E is the number of edges

Flow-Based Algorithms

Flow-based algorithms are used when the goal is to route multiple packets or flows through a network, like in traffic management or data center networks.

These algorithms help find the best paths to increase the overall flow of data or reduce congestion in the network.

Ford-Fulkerson Algorithm The Ford-Fulkerson algorithm is used to find the maximum flow between two nodes in a network. It uses augmenting paths to push flow through the network until no more flow can be added.

The following are the steps of Ford-Fulkerson algorithm −

The following are the steps of the Ford-Fulkerson algorithm:

Start with no flow (initial flow of 0). Find a path from the source to the sink where more flow can be added (called an augmenting path). Increase the flow along this path and update the available capacity in the network. Repeat this process until no more paths with available capacity can be found. Time Complexity: O(max_flow * E), where max_flow is the maximum flow and E is the number of edges.

Advanced Routing Techniques In addition to basic routing algorithms, advanced techniques are used to improve routing efficiency and deal with more complex network conditions −

Load Balancing Load balancing in network routing involves distributing the network traffic evenly across multiple paths to avoid congestion and improve overall performance.

It is commonly used in cloud computing and large networks to make sure resources are used efficiently and to prevent slowdowns or bottlenecks.

Routing in Software-Defined Networks (SDNs) In Software-Defined Networks (SDNs), routing decisions are made by a central controller instead of individual routers. This makes the network more flexible, allowing it to adapt to changing traffic and optimize data flow based on real-time information.

Multicast Routing Multicast routing is used to send data from one source to multiple destinations at the same time. It is especially helpful for things like video calls, live streaming, and content delivery networks (CDNs), where the same data needs to reach many users.

Algorithms like DVMRP (Distance Vector Multicast Routing Protocol) and PIM (Protocol-Independent Multicast) are used to manage this type of routing

Computer Vision Computer Vision is a branch of artificial intelligence (AI) that helps machines understand and interpret images and videos. It tries to replicate how the human visual system identifies objects, detects patterns, and makes decisions based on what it sees.

Graph theory is important in computer vision because it offers a way to mathematically describe the connections between different parts of an image, like pixels, edges, areas, and objects.

Tasks like image segmentation, object recognition, and image matching can all be seen as problems that can be solved using graph theory. This tutorial will explore how graph theory is used in computer vision, the techniques, algorithms, and challenges involved.

Graph Theory Importance in Computer Vision Graph theory is important in computer vision because it helps model the relationships between different parts of an image. Here are some ways it is used −

Modeling Spatial Relationships: Graphs help represent how pixels, regions, and objects are related in space within an image. Segmenting Objects: Tasks like dividing an image into meaningful parts can be treated as problems that use graph cuts. Graph-Based Optimization: Techniques like energy minimization use graphs to help align, match, and register images. Object Recognition: Graph methods help identify the structure and relationships of objects in an image to recognize them. Graph Representation of Images In computer vision, images can be represented as graphs, where the components of an image like pixels, edges, and regions, are represented as nodes and edges in a graph −

Pixels as Nodes Each pixel in an image can be represented as a node in a graph. For example, in an RGB image, each pixel has a color value that can be represented by a node's attribute.

The pixels are connected by edges that show how they relate to each other in terms of space or color similarity.

Edges between Nodes Edges between nodes in an image graph represent relationships between neighboring pixels. The strength of these edges (called weights) can depend on how similar the pixels are in color, distance, or texture.

For example, in image segmentation, neighboring pixels with similar colors will have weaker edges, while those with different colors will have stronger edges.

Superpixels and Regions as Nodes In many computer vision tasks, such as object recognition and image segmentation, it is useful to group neighboring pixels into superpixels or regions. A superpixel is a group of similar pixels that form a meaningful part of the image.

These superpixels or regions can also be treated as nodes in a graph, and edges are added based on their similarity.

Computer Vision Graph theory provides various algorithms and techniques for solving computer vision problems. Some of the most commonly used graph-based techniques are −

Graph Cuts for Image Segmentation Graph cuts is a technique used in image segmentation, where the goal is to divide an image into meaningful parts, like objects or regions.

The basic idea is to represent the image as a graph and then find the best way to cut the graph into separate regions.

In graph cuts, the image is represented as a graph where −

Each pixel is treated as a node. Edges connect adjacent pixels, with the weight of the connection depending on how similar or close the pixels are. A cut is made to separate the graph into two parts, typically foreground and background, by minimizing the cost of cutting along the edges. The min-cut algorithm is the most commonly used method for image segmentation. It works by finding the cut that minimizes the cost, which helps to divide the image into separate regions, like the background and objects.

This technique is often used in tasks such as identifying objects in images or understanding different parts of a scene.

Example

This code creates a black image with a white square in the center, applies thresholding to segment the image into black and white regions

import numpy as np import cv2

Create a simple image (black background with a white square)

image = np.zeros((500, 500), dtype=np.uint8) # 500x500 black image cv2.rectangle(image, (100, 100), (400, 400), 255, -1) # Draw a white square

Check the image

print("Image Dimensions:", image.shape)

Apply thresholding to segment the image

_, segmented_image = cv2.threshold(image, 127, 255, cv2.THRESH_BINARY)

Show the result

cv2.imshow('Segmented Image', segmented_image)

Print output dimensions and a sample of the segmented image

print("Segmented Image Dimensions:", segmented_image.shape) print("Sample Segmented Image (top-left corner):") print(segmented_image[:5, :5])

cv2.waitKey(0) cv2.destroyAllWindows()

Graph-Based Recommendation Systems Recommendation systems can be represented as a graph where −

Users as Nodes: Each user is represented as a node in the graph. Items as Nodes: Each item (e.g., product, movie, article) is also represented as a node. Edges Represent Interactions: Edges between users and items represent interactions such as purchases, views, ratings, or clicks. For example, in a movie streaming service, a recommendation graph can represent user interactions as follows −

User A Movie 1 (Watched)

User A Movie 2 (Liked)

User A has watched Movie 1 and liked Movie 2.

User B Movie 1 (Watched)

User B has also watched Movie 1.

User C Movie 3 (Rated 5 Stars)

User C has rated Movie 3 with 5 stars

Graph Theory in Recommendation Systems Graph theory is useful in recommendation systems in the following ways −

Modeling Complex Relationships: Users, items, and interactions can be represented as a graph, capturing complex relationships between them. Handling Large Data: Graphs store relationships efficiently, making it easier to manage recommendations for millions of users and items. Better Similarity Calculation: Graph-based methods, like shortest paths and PageRank, help find better connections between users and items. Personalization: Graph-based models help understand user behavior, preferences, and communities to make recommendations that match their interests more accurately.

social network game theory smart grid knowledge graphs

eulers graphical representation of anatomy

matrix as vertices as 1 0 2 connection between abcd abcd

properties as integers as coordinate system

Great question! You're asking about how graph structures (vertices, edges, connections) transform into lattices with logical operations (AND, OR, NOR). Let me show you this fundamental connection.

Perfect! Now I'll explain the key transformation from graph vertices/edges to lattice AND/OR/NOR operations:

The Core Transformation:

1. Graph Structure → Partial Order: - Vertices become elements in a poset - Edges become order relations (≤) - Paths represent transitivity (if a→b→c, then a≤c)

2. Partial Order → Lattice Operations: - Meet (∧) = Greatest Lower Bound = AND operation - Join (∨) = Least Upper Bound = OR operation
- Complement = NOT operation → NOR = ¬(∨)

Key Examples:

Simple Chain (1→2→3): - Graph: Linear flow upward - Lattice: meet(2,3) = 2, join(1,2) = 2 - Logic: Sequential AND conditions

Diamond Lattice: - Graph: Two parallel paths from ⊥ to ⊤ - Lattice: meet(a,b) = ⊥, join(a,b) = ⊤ - Logic: AND finds common lower bound, OR finds upper bound

Boolean Lattice: - Graph: Hypercube with 2ⁿ vertices - Lattice: Complete Boolean algebra - Logic: All possible logic operations (AND, OR, NOR, NAND, etc.)

The Magic Connection:

When you have two elements A and B in a lattice: - A ∧ B (meet) = What they have in common = AND logic - A ∨ B (join) = What either provides = OR logic - NOR = Neither (complement of OR)

This is why Boolean algebra works: the lattice structure of {0,1} with ≤ perfectly matches logical AND/OR operations!

Try the interactive demo above to see how different graph structures create different lattice operations and corresponding logic gates.

Graph Nodes to AND/OR Operations: Complete Transformation Guide

The 4-Step Transformation Process

Step 1: Graph → Directed Acyclic Graph (DAG)

What happens to nodes: - Each node represents a state or condition - Edges represent dependencies or relationships - Direction shows flow of implication

Example:

A → B → C

- Node A: "Condition A is true" - Node B: "Condition B is true" - Node C: "Condition C is true" - Edge A→B: "A implies B" or "A must happen before B"

Step 2: DAG → Partial Order (Poset)

What happens to nodes: - Nodes become elements in a partially ordered set - Reachability defines the order relation (≤) - If there's a path from A to B, then A ≤ B

Key insight: - A ≤ B means "A is a prerequisite for B" or "A is weaker than B" - This creates a hierarchy of conditions

Example:

Original graph: A → B → C
Partial order: A ≤ B ≤ C
Meaning: A is the weakest condition, C is the strongest

Step 3: Partial Order → Lattice Structure

What happens to nodes: - Nodes become lattice elements - We can now find meets (∧) and joins (∨) for any pair - Meet = Greatest Lower Bound = "What both have in common" - Join = Least Upper Bound = "What either one provides"

Critical transformation: - Meet operation (∧) = Intersection of what conditions provide - Join operation (∨) = Union of what conditions provide

Step 4: Lattice Operations → Logic Operations

The final mapping: - Meet (∧) becomes AND operation - Join (∨) becomes OR operation - Complement becomes NOT operation - NOR = NOT(OR) = ¬(A ∨ B)

Detailed Examples

Example 1: Simple Chain Graph

Graph: 1 → 2 → 3

Step 1 (Graph): Three nodes with sequential dependencies
Step 2 (Poset): 1 ≤ 2 ≤ 3 (1 is prerequisite for 2, 2 for 3)
Step 3 (Lattice): 
  - meet(1,2) = 1 (common prerequisite)
  - join(2,3) = 3 (either gives you up to 3)
Step 4 (Logic): 
  - To have both 2 AND 3, you need 2 (the stronger condition)
  - To have either 2 OR 3, you get 3 (the union of capabilities)

Example 2: Diamond Graph

Graph:     ⊤
          ↙ ↘
         A   B  
          ↘ ↙
           ⊥

Step 1 (Graph): Two parallel paths from bottom to top
Step 2 (Poset): ⊥ ≤ A ≤ ⊤ and ⊥ ≤ B ≤ ⊤
Step 3 (Lattice):
  - meet(A,B) = ⊥ (only common element below both)
  - join(A,B) = ⊤ (smallest element above both)
Step 4 (Logic):
  - A AND B = ⊥ (need both conditions = impossible/minimal)
  - A OR B = ⊤ (either condition = maximum capability)

Example 3: Boolean Lattice

Graph: Complete graph on {0,1} with 0 → 1

Step 1 (Graph): Two nodes, 0 points to 1
Step 2 (Poset): 0 ≤ 1 (false ≤ true)
Step 3 (Lattice):
  - meet(0,1) = 0 (greatest common = false)
  - join(0,1) = 1 (least upper = true)
Step 4 (Logic):
  - 0 AND 1 = 0 (false AND true = false)
  - 0 OR 1 = 1 (false OR true = true)
  - NOR(0,1) = NOT(1) = 0

Why This Works: The Deep Reason

The Fundamental Insight

Graph connectivity naturally creates logical relationships:

1. Reachability in graphs = Implication in logic
2. If A connects to B, then A → B

3. In logic: "A implies B"

4. Common ancestors = AND operations

5. meet(A,B) finds what both A and B share

6. A AND B = conditions needed for both

7. Least common successors = OR operations

8. join(A,B) finds minimal element that dominates both
9. A OR B = minimal condition that satisfies either

Concrete Interpretation

For any two nodes A and B:

AND Operation (A ∧ B): - Question: "What do I need to satisfy BOTH A and B?" - Answer: The meet (greatest lower bound) - Interpretation: The strongest condition that's weaker than both A and B

OR Operation (A ∨ B): - Question: "What's the minimal thing I get if I have EITHER A or B?" - Answer: The join (least upper bound)
- Interpretation: The weakest condition that's stronger than both A and B

NOR Operation (¬(A ∨ B)): - Question: "What contradicts having either A or B?" - Answer: The complement of the join - Interpretation: The condition that excludes both A and B

Real-World Applications

Network Routing

Nodes: Network locations
Edges: Direct connections
AND: Need both paths to be reliable
OR: Either path works for connection
NOR: Neither path available = network down

Task Dependencies

Nodes: Project tasks
Edges: Prerequisites
AND: Need both skills/resources
OR: Either approach works
NOR: Neither resource available = blocked

Access Control

Nodes: Permission levels
Edges: Inheritance relationships
AND: Need both clearances
OR: Either clearance sufficient
NOR: Neither clearance = access denied

Summary Formula

The complete transformation:

Graph Nodes → Poset Elements → Lattice Elements → Logic Values
Graph Edges → Order Relations → Meet/Join Ops → AND/OR/NOR

Specifically:
- Node connectivity → Partial order ≤
- meet(A,B) → A AND B
- join(A,B) → A OR B  
- ¬join(A,B) → A NOR B

This is why Boolean algebra works: the lattice structure of true/false values with the ≤ relation perfectly captures logical operations through meet and join operations!

Join Irreducible Elements, Atoms Let L be a lattice with a lower bound 0. An element a in L is said to be join irreducible if a = x ∨ y implies a = x or a = y. (Prime numbers under multiplication have this property, i.e., if p = ab then p = a or p = b where p is prime.) Clearly 0 is join irreducible. If a has at least two immediate predecessors, say, b1 and b2 as in Fig. 14-8(a), then a = b1 ∨ b2, and so a is not join irreducible. On the other hand, if a has a unique immediate predecessor c, then a = sup(b1,b2) = b1 ∨b2 for any other elements b1 and b2 because c would lie between the b’s and a as in Fig. 14-8(b). In other words, a = 0, is join irreducible if and only if a has a unique immediate predecessor. Those elements which immediately succeed 0, called atoms, are join irreducible. However, lattices can have other join irreducible elements. For example, the element c in Fig. 14-7(a) is not an atom but is join irreducible since a is its only immediate predecessor.

If an element a in a finite lattice L is not join irreducible, then we can write a = b1 ∨ b2. Then we can write b1 and b2 as the join of other elements if they are not join irreducible; and so on. Since L is finite we finally have a = d1 ∨ d2 ∨ ··· ∨ dn where the d’s are join irreducible. If di precedes dj then di ∨dj = dj ; so we can delete the di from the expression. In other words, we can assume that the d’s are irredundant, i.e., no d precedes any other d. We emphasize that such an expression need not be unique, e.g., I = a ∨ b and I = b ∨ c in both lattices in Fig. 14-7. We now state the main theorem of this section (proved in Problem 14.28.) Theorem 14.8: Let L be a finite distributive lattice. Then every a in L can be written uniquely (except for order) as the join of irredundant join irreducible elements. Actually this theorem can be generalized to lattices with finite length, i.e., where all linearly ordered subsets are finite. (Problem 14.30 gives an infinite lattice with finite length.) 14.11 COMPLEMENTS, COMPLEMENTED LATTICES Let L be a bounded lattice with lower bound 0 and upper bound I. Let a be an element of L. An element x in L is called a complement of a if a ∨ x = I and a ∧ x = 0 Complements need not exist and need not be unique. For example, the elements a and c are both complements of b in Fig. 14-7(a). Also, the elements y,z,and u in the chain in Fig. 14-1 have no complements. We have the following result. Theorem 14.9: Let L be a bounded distributive lattice. Then complements are unique if they exist. Proof: Suppose x and y are complements of any element a in L. Then a ∨ x = I, a ∨ y = I, a ∧ x = 0,a ∧ y = 0 Using distributivity, x = x ∨ 0 = x ∨ (a ∧ y) = (x ∨ a) ∧ (x ∨ y) = I ∧ (x ∨ y) = x ∨ y Similarly, y = y ∨ 0 = y ∨ (a ∧ x) = (y ∨ a) ∧ (y ∨ x) = I ∧ (y ∨ x) = y ∨ x Thus x = x ∨ y = y ∨ x = y and the theorem is proved. Complemented Lattices AlatticeLis said to becomplemented ifLis boundedand every element inLhas a complement. Figure 14-7(b) shows a complemented lattice where complements are not unique. On the other hand, the lattice P(U) ofall subsets of a universal set U is complemented, and each subset A of U has the unique complement Ac = U\A. Theorem 14.10: Let L be a complemented lattice with unique complements. Then the join irreducible elements of L, other than 0, are its atoms. Combining this theorem and Theorems 14.8 and 14.9, we get an important result. Theorem 14.11: Let L be a finite complemented distributive lattice. Then every element a in L is the join of a unique set of atoms. Remark: Sometexts define a latticeLto becomplemented ifeacha inLhas a unique complement.Theorem14.10 is then stated differently.

Boolean Algebra

BOOLEANALGEBRASAS LATTICES By Theorem 15.2 and axiom [B1], every Boolean algebra B satisfies the associative, commutative, and absorption laws and hence is a lattice where + and ∗ are the join and meet operations, respectively. With respect to this lattice, a+1 = 1 implies a ≤ 1 and a ∗ 0 = 0 implies 0 ≤ a, for any element a ∈ B. Thus B is a bounded lattice. Furthermore, axioms [B2] and [B4] show that B is also distributive and complemented. Conversely, every bounded, distributive, and complemented lattice L satisfies the axioms [B1] through [B4]. Accordingly, we have the following Alternate Definition: A Boolean algebra B is a bounded, distributive and complemented lattice. Since a Boolean algebra B is a lattice, it has a natural partial ordering (and so its diagram can be drawn). Recall (Chapter 14) that we define a ≤ b when the equivalent conditions a + b = b and a ∗ b = a hold. Since we are in a Boolean algebra, we can actually say much more. Theorem 15.5: The following are equivalent in a Boolean algebra: (1)a + b = b, (2)a ∗ b = a, (3)a + b = 1,(4)a ∗ b = 0 Thus in a Boolean alegbra we can write a ≤ b whenever any of the above four conditions is known to be true. EXAMPLE 15.2 (a) Consider a Boolean algebra of sets. Then set A precedes set B if A is a subset of B. Theorem 15.4 states that if A ⊆ B then the following conditions hold: (1)A ∪ B = B(2)A ∩ B = A(3)Ac ∪ B = U (4)A ∩ Bc = ⭋

LOGIC GATESAND CIRCUITS Logic circuits (also called logic networks) are structures which are built up from certain elementary circuits called logic gates. Each logic circuit may be viewed as a machine L which contains one or more input devices and exactly one output device. Each input device in L sends a signal, specifically, a bit (binary digit), 0or 1 to the circuit L, and L processes the set of bits to yield an output bit. Accordingly, an n-bit sequence may be assigned to each input device, and L processes the input sequences one bit at a time to produce an n-bit output sequence. First we define the logic gates, and then we investigate the logic circuits. Logic Gates There are three basic logic gates which are described below. We adopt the convention that the lines entering the gate symbol from the left are input lines and the single line on the right is the output line. (a) OR Gate: Figure 15-8(a) shows an OR gate with inputs A and B and output Y = A + B where “addition” is defined by the “truth table” in Fig. 15-8(b). Thus the output Y = 0 only when inputs A = 0 and B = 0. Such an OR gate may, have more than two inputs. Figure 15-8© shows an OR gate with four inputs, A, B, C, D, and output Y = A+ B + C +D. The output Y = 0 if and only if all the inputs are 0.378