Graph theory

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In Graph Theory, a graph is defined to be a set of dots (called vertices or nodes) connected by links (called arcs or edges). A more formal definition is:

A simple graph is a set of nodes N and a set E of unordered pairs of distinct elements of N called edges.

 1 _____ 2 _____ 3
  \     /       /
   \   /       /    an example of graph with five vertices and six edges
    \ /       /
     5 _____ 4

Here, N = {1, 2, 3, 4, 5} and E = {(1,2), (1,5), (2,3), (2,5), (3,4), (4,5)}.

  • adjacent

Two nodes are considered adjacent if an edge exists between them. In the above graph, nodes 1 and 2 are adjecent, but nodes 2 and 4 are not.

  • neighbor

The set of neighbors for a node contains each node adjacent to it. In the above graph, node 1 has two neighbors: node 2 and node 5.

  • path

A path is a series of nodes in that each node is adjacent to both the preceding and succeding node. A path is considered simple if none of the nodes in the path are repeated. In the above graph, {1, 2, 5, 1, 2, 3} is a path, and {5, 2, 1} is a simple path.

If it is possible to establish a path from any vertex to any other vertex of a graph, the graph is said to be connected.

  • cycle

A cycle is a path that begins and ends with the same node. In the above graph, {1, 5, 2, 1} is a cycle. A path is acyclic if it is not a cycle. A simple cycle is one in which the beginning node only appears once more, as the ending node.

A connected graph without cycles is said a tree; equivalently, a tree on n vertices is a connected graph with exactly n-1 edges. Trees are largely used as data structures in computer science (see tree data structure).

  • complete

A complete graph is one in which every node is adjacent to every other node. The above graph is not complete. The complete graph on n vertices is often denoted by Kn. It has n(n-1)/2 edges (corresponding to all possible choices of pairs of vertices).

  • planar

A planar graph is one which can be drawn in the plane without any two edges intersecting. The above graph is planar; the complete graph on n vertices, for n> 4, is not planar.

Graph Problems

lots more stuff here when I get time!