Basic concepts of graph theory

The classification of several common graphs:

Type	Side	Allow multiple edges	Allow ring
Simple diagram	No direction	Whether	Whether
Multiple graphs	No direction	Is	Whether
Pseudo graph	No direction	Is	Is
A map of the direction	Have to	Whether	Is
To a multi-map	Have to	Is	Is

Full Picture: A complete graph on n vertices is a simple graph with exactly one edge between each pair of vertices.

Dichotomy: If the vertex set of the simple graph G is divided into two disjoint non-empty sets V1 and V2, each edge of the diagram is connected to a vertex in the V1 and a vertex in the V2 (so no edge in G is connected to the two vertices in V1 or two vertices in V2), Then G is called a binary graph (as can be seen from the definition, the binary graph must have no ring, but can have multiple edges).

It is easy to judge whether a graph is a two-part diagram by using the labeling method:

The first one is labeled A, a vertex adjacent to the vertex of Mark A is labeled B, a vertex adjacent to the vertex labeled B is marked with a, and the continuation of this method, if the process can be completed so that no adjacent vertices are labeled with the same letter, then the graph is a binary graph, otherwise it is not a binary graph.

Another: If the tag succeeds, you can separate two vertex subsets X from the vertices in Y to form a same composition as the original, so that you can see directly that it is a binary graph.

Complete dichotomy: Set G to be a binary graph, if G is a simple graph, and each vertex in X is contiguous with each vertex in Y, then the G is a complete binary graph.

Easy to know, Total binary chart | x| | Y| the edge of the bar.

Representation of graphs and isomorphism of graphs:

Common diagram representations: Edge table, adjacent matrix (the adjacent matrix of a simple graph is a symmetric 0-1 matrix, when a ring or multiple edges, the matrix symmetric but not the 0-1 matrix, at this time the first (I, j) is the number of sides associated with {VI, VJ}. In addition, the adjacent matrix of the graph does not have to be symmetric, and the associative matrix (vertex and Edge n*m matrices, which represent multiple edges with a column of equal entries, exactly one column representing a loop equal to 1).

Isomorphism: When two simple graphs are isomorphic, the vertices of the two graphs have one by one correspondence to maintain the neighboring relationship (note: Isomorphic graphs are equivalent plots, but only in different ways).

It is easy to see that the necessary conditions for isomorphism of two graphs are: equal number of vertices, equal number of edges, equal number of vertices with the same degree (the total vertex degree is more equal). Unfortunately, there is no set of invariants to determine whether simple graphs are isomorphic.

Connectivity:

Pathway: The path from the vertex u to V in the these drawings.

Loop: If a path begins and ends at the same vertex, it is a loop.

Simple Path (loop): If a path (or loop) does not contain the same edges repeatedly, it is simple.

undirected graph Connectivity: If there is a path between each pair of vertices in the undirected graph, the graph is called connected.

Directed graph connectivity: depending on whether the direction of the edge is considered: The directed graph consists of two connectivity concepts.

Strong connectivity: Whenever A and b are a vertex of a graph, there are paths from A to B and from B to a.

Weak connectivity: There is a path between any two vertices in the basemap of the directed graph (the undirected graph drawn after the directional graph ignores the direction of the edge).

Obviously, any strongly connected undirected graphs are also weakly interconnected.

Connected Branch: The disconnected graph is the sum of two or more connected sub-graphs, each of which has no common vertices, and these disjoint connected sub-graphs are called connected branches of graphs.

Several concepts that are caused by connectivity:

Cut point: If you delete a vertex and its associated edges, the connected branch of the graph increases, and such vertices are called cut points.

Trimming: The edge of a sub-graph that produces more connected branches than the original as soon as it is deleted is called a cut edge (or bridge).

Ouratonlo and Hamilton Access:

Oraton Road (or loop): A simple path (or loop) that contains each edge of Figure G.

Theorem: A connected multi-graph has a Euler loop when and only if it has even degrees for each vertex.

Theorem: A connected multi-graph has a Euler path without a Euler loop, when and only if it has exactly two odd vertex vertices.

The proofs of the above two theorems are shown here.

Hamiltonian paths and Hamiltonian circuits: simple paths and loops that contain exactly once each vertex of the graph (relatively speaking: Euler's and Euler's circuits contain multiple graphs each with exactly one edge).

Basic concepts of graph theory