# Graph (Network)

### Table of Contents

## 1 - About

A graph is a set of vertices connected by edges. See Graph - (Property Graph) Model

Data representation that naturally captures complex relationships is a graph (or network).

Points are called nodes, links are called edges. A link can only connect two nodes (only binary relationship ?)

Each edge has two endpoints, the nodes it connects. The endpoints of an edge are neighbors.

See also: (Graph|Network) - Database

Code, Data Modeling, Social Interaction are mostly graphs

## 2 - Articles Related

## 3 - Example

## 4 - Dominating set

A dominating set in a graph is a set S of nodes such that every node is in S or a neighbor of a node in S.

Neither algorithm is guaranteed to find the smallest solution.

### 4.1 - Grow Algorithm

initialize S = 0; while S is not a dominating set, add a node to S.

### 4.2 - Shrink Algorithm

initialize S = all nodes while there is a node x such that S −{x} is a dominating set, remove x from S

## 5 - Path

### 5.1 - Definition

A sequence of edges <math>[\{x_1, x_2\}, \{x_2, x_3\}, \{x_3, x_4\}, \dots , \{x_{k_1}, x_k\}]</math> is called a <math>x_1-to-x_k</math> path.

Example “Main Quad”-to-”Gregorian Quad” paths in the graph:

- one goes through “Wriston Quad” ,
- one goes through “Keeney Quad”

### 5.2 - Cycle

A x-to-x path is called a cycle

### 5.3 - Spanning

A set S of edges is spanning for a graph G if, for every edge {x, y} of G, there is an x-to-y path consisting of edges of S.

### 5.4 - Forest

A set of edges of G is a forest if the set includes no cycles.

## 6 - Problem

### 6.1 - Minimum spanning forest

Definition:

- input: a graph G, and an assignment of real-number weights to the edges of G.
- output: a minimum-weight set S of edges that is spanning and a Tree - Forest (Set of Tree).

Application: Design hot-water delivery network for the university campus:

- Network must achieve same connectivity as input graph.
- An edge represents a possible pipe.
- Weight of edge is cost of installing the pipe.
- Goal: minimize total cost.

#### 6.1.1 - Grow

def Grow(G) S := 0; consider the edges in increasing order # Increasing order: 2, 3, 4, 5, 6, 7, 8, 9. for each edge e: if e’s endpoints are not yet connected add e to S.

#### 6.1.2 - Shrink

def Shrink(G) S = {all edges} consider the edges in order, from highest-weight to lowest-weight # Decreasing order: 9, 8, 7, 6, 5, 4, 3, 2. for each edge e: if every pair of nodes are connected via S -{e}: remove e from S.