Graph (Network - Nodes and edges)

> (Data|State) Management and Processing > (Data Type|Data Structure) > Graph (Network - Nodes and edges)

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).

Except of the special graph that a tree is, the data structure of a graph is non-hierarchical.

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

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3 - Application

Are mostly graphs:

4 - Type

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5 - Example

5.1 - Map

Tube_map

London Tube Map 1908 Lond Tube Map 1933

5.2 - Directed Graph

6 - 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.

6.1 - Grow Algorithm

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

6.2 - Shrink Algorithm

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

7 - Path

7.1 - Definition

7.2 - Cycle

A x-to-x path is called a cycle

7.3 - 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.

7.4 - Forest

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

8 - Problem

8.1 - Minimum spanning forest

Definition:

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.
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8.1.1 - 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.

8.1.2 - 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.

9 - Documentation / Reference