(Original post by Arsey)
Definitions for the D1 Exam (and what they really mean!)
A graph G consists of points (vertices or nodes) which are connected by lines (edges or arcs). A graph is just some points joined together with lines.
A subgraph of G is a graph, each of whose vertices belongs to G and each of whose edges belongs to G. A subgraph is just part of a bigger graph.
If a graph has a number associated with each edge (usually called its weight) then the graph is called a weighted graph or network. A weighted graph is one with values, eg distance, assigned to the edges.
The degree or valency of a vertex is the number of edges incident to it. A vertex is odd (even) if it has odd (even) degree. The degree of a vertex is how many edges go into it.
A path is a finite sequence of edges, such that the end vertex of one edge in the sequence is the start vertex of the next, and in which no vertex appears more than once. A path describes a ‘journey’ around a graph, where no vertex is visited more than once.
A walk is a path where you may visit vertices more than once.
A cycle (circuit) is a closed path, i.e. the end vertex of the last edge is the start vertex of the first edge.
Two vertices are connected if there is a path between them. A graph is connected if all its vertices are connected. Two vertices are connected if you can walk along edges from one to the othe, in a connected graph all vertices are connected.
If the edges of a graph have a direction associated with them they are known as directed edges and the graph is known as a digraph. A digraph will have arrows on the edges.
A tree is a connected graph with no cycles.
A spanning tree of a graph G is a subgraph which includes all the vertices of G and is also a tree.
A minimum spanning tree (MST) is a spanning tree such that the total length of its arcs is as small as possible. (MST is sometimes called a minimum connector.)
A graph in which each of the n vertices is connected to every other vertex is called a complete graph. In a complete graph every vertex is directly connected to every other edge.
A bipartite graph consists of two sets of vertices X and Y. The edges only join vertices in X to vertices in Y, not vertices within a set. (If there are r vertices in X and s vertices in Y then this graph is Kr,s.) A bipartite graph has two sets of nodes. Nodes of one set can only be matched to nodes of the other set.
A matching is the pairing of some or all of the elements of one set, X, with elements of a second set, Y. If every member of X is paired with a member of Y the matching is said to be a complete matching. In a matching some of one set are joined to the other. In a complete matching they’re all joined.
An Alternating Path starts at an unconnected vertex in one set and ends at an unconnected vertex in the other set. Edges alternate between those ‘not in’ and ‘in’ the initial matching.