A graph G is a collection of two sets: V, a set of vertices and E, a set of edges that connect the vertices

{v1, v2} - graph in which each edge is associated with an ordered pair of vertices

graph in which each edge is associated with an unordered pair of vertices

a vertex v1 in a graph G is adjacent to v2 if there is an edge between v1 to v2

sequence of vertices v1, v2, v3,...vn where there is an edge from vi to vi+1

path in which all vertices are distinct

a path in which first and last vertices are the same

a cycle that begins and ends at the same vertex and does not pass through other vertices more than once

connected, acyclic graph with a specially designated vertex called the root

there is a path between any two vertices (undirected)

Strongly Connected Graph

there is a path between any two vertices (digraph)

Weakly Connected Graph

there is a path from vertex I to vertex J or from vertex J to vertex I

a graph with each pair of distinct vertices having an edge between them (graph with n vertices and exactly n(n-1)/2 edges)

Complete Directed Graph

a directed graph with exactly n(n-1) edges

figure which has multiple occurrences of the same edge (two or more edges between two vertices)

graph in which each edge has an associated positive numerical weight

linear ordering of the vertices in a digraph in which vertex x precedes vertex y if there is a directed edge from x to y in the graph

a subgraph of a connected undirected graph that contains all of its vertices and enough of its edges to form a tree

spanning tree created by executing a depth-first search of a graph's vertices

Minimal Spanning Tree

spanning tree of a weighted, connected, undirected graph which has minimal edge-weight sum

a path in an undirected graph that begins at a vertex v, passes through every edge in the graph exactly once, and terminates at v

a path that begins at a vertex v, passes through every vertex in the graph exactly once, and terminates at v.

Traveling Salesman Problem

A weighted digraph must be traversed with minimal cost. It is in the set NP; the solution can be guessed and then checked in polynomial time. It is also NPC.

Since the proving of the theorem, efficient algorithms have been found for 4-coloring maps requiring only O(n^2) time, where n is the number of vertices, found in 1996

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Tree Data Structures

Vocab associated with trees

Term	Definition
Graph	A graph G is a collection of two sets: V, a set of vertices and E, a set of edges that connect the vertices
Digraph	{v1, v2} - graph in which each edge is associated with an ordered pair of vertices
Undirected Graph	graph in which each edge is associated with an unordered pair of vertices
Adjacent	a vertex v1 in a graph G is adjacent to v2 if there is an edge between v1 to v2
Path	sequence of vertices v1, v2, v3,...vn where there is an edge from vi to vi+1
Simple Path	path in which all vertices are distinct
Cycle	a path in which first and last vertices are the same
Simple Cycle	a cycle that begins and ends at the same vertex and does not pass through other vertices more than once
Acyclic Graph	graph with no cycles
Tree	connected, acyclic graph with a specially designated vertex called the root
Connected Graph	there is a path between any two vertices (undirected)
Strongly Connected Graph	there is a path between any two vertices (digraph)
Weakly Connected Graph	there is a path from vertex I to vertex J or from vertex J to vertex I
Complete Graph	a graph with each pair of distinct vertices having an edge between them (graph with n vertices and exactly n(n-1)/2 edges)
Complete Directed Graph	a directed graph with exactly n(n-1) edges
Multigraph	figure which has multiple occurrences of the same edge (two or more edges between two vertices)
Network	graph in which each edge has an associated positive numerical weight
Topological Order	linear ordering of the vertices in a digraph in which vertex x precedes vertex y if there is a directed edge from x to y in the graph
Spanning Tree	a subgraph of a connected undirected graph that contains all of its vertices and enough of its edges to form a tree
DFS Spanning Tree	spanning tree created by executing a depth-first search of a graph's vertices
Minimal Spanning Tree	spanning tree of a weighted, connected, undirected graph which has minimal edge-weight sum
Euler Circuit	a path in an undirected graph that begins at a vertex v, passes through every edge in the graph exactly once, and terminates at v
Hamilton Circuit	a path that begins at a vertex v, passes through every vertex in the graph exactly once, and terminates at v.
Traveling Salesman Problem	A weighted digraph must be traversed with minimal cost. It is in the set NP; the solution can be guessed and then checked in polynomial time. It is also NPC.
Four Color Problem	Since the proving of the theorem, efficient algorithms have been found for 4-coloring maps requiring only O(n^2) time, where n is the number of vertices, found in 1996
Three Utility Problem

Created by: lb3s

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