Decision Mathematics 1 definitions

Cycle

A closed path, ie the end node of the last arc is the start node of the first arc.

Tree

A connected graph with no cycles.

Eulerian Cycle

A cycle in which every node of the graph is visited only once. A ........ graph cannot have any odd arcs.

Path

A finite sequence of arcs, such that the end node of one arc in the sequence is the start node of the next, and in which no node appears more then once.

Network (Weighted graph)

A graph with a number associated with each arc (usually called its weight).

Simple graph

A graph with no loops and no more than one arc connecting any pair of nodes.

Matching

A pairing of some or all of the elements of one set, X, with elements of a second set, Y.

Minimum spanning tree (minimum connector)

A spanning tree such that the total length of its arcs is as small as possible.

Spanning tree

A subgraph of a graph, G, which includes all the nodes of G and is also a tree.

Connected graph

All its nodes have paths between them (are connected).

Loop

An arc that starts and finishes at the same node.

Complete matching

Every member of X is paired with a member of Y of the matching.

Complete graph

Every vertex is connected to every other vertex

Semi-Eulerian graph

Has exactly two odd vertices

Incidence Matrix

Matrix of connections that can be drawn up to illistrate a graph. Non digraph matrices will be symmetrical.

Subgraph

Part of a graph, G, each of whose nodes belongs to G and each of whose arcs belongs to G.

Graph

Points (vertices or nodes) which are connected by lines (edges or arcs).

Digraph

The arcs of a graph have a direction associated with them.

Plannar graph

The graph can be drawn without crossing any arcs

Valency (degree)

The number of arcs incident to the nodes. A node is odd (even) if it has odd (even) degree.

Bipartite graph

Two sets of nodes X and Y. The arcs only join nodes in X to nodes in Y, not nodes within a set.