1.1 Basic concepts of graphs

(Reference books: 2018 data structure King of the Postgraduate examination) definition of Diagram

Figure G consists of a fixed point set V and a side set E

recorded as G= (V,e)

which V (g) is a finite non-empty set of vertices in g

E (g) is the G Middle edge (vertex relationship) set and

| v| represents The number of vertices in G, also known as the order of graphs

e={(U, v) | U, v are vertices} | e| represents Number of bars on the middle side of G

Note: The figure cannot be empty, the edge can be empty, but the vertex must be non-empty

a map of the direction

if E is a directional edge (also called an arc ), when the graph is a directed graph

Remember to do <u,v > U for Arc Head V for Arc tail, arc called U to V, u adjacency to V, v adjacent to U arc

G = (E, V)

E = {A-c}

V = {<1, 2>, <1, 3>, <2, 3>, <3, 2>}

graph without Direction

E is a non-forward edge (abbreviation edge ), then the graph is a graph without direction

(U, v) or (V, u) at this time (U, v) = = (V, u) because there is no direction

G = (V, E)

V = {1, 4, 5, 6}

E = {(1, 6), (1, 5), (5, 6), (4, 6)}

Simple diagram

The following conditions are met:

1. No duplicate edges

2. No vertex to its own edge, the above two diagram is a simple diagram

Multiple graphs

In contrast to a simple diagram:

1. Duplicate edges exist

2. Allow vertices to connect to themselves through an edge

Full picture

No direction full picture: An edge exists between any two points

Directed full graph: Two arcs in the opposite direction between any two points

Sub-chart

g = (V, e) g ' = (v ', E ')

V ', E ' is a subgroup of V, E, then G ' is a sub-graph of G, if V (g ') = V (g), then G ' is the generation of G-sub-graph

connectivity, connected graphs and connected components (undirected graphs)

The vertex v to Vertex W has a path, which is said V and W are connected. If any two vertices in Figure g are connected, the G is a connected graph , otherwise called a non-connected graph .

The maximal connected sub-graph of undirected graphs is called connected component

Maximal connectivity Sub-graph: The connected component of undirected graphs, requiring the sub-graph to contain all the edges.

Minimal connected sub-graph: a sub-graph that remains connected and has a minimum number of edges.

Strongly connected graphs, strongly connected components (undirected graphs)

V->w w->v are all by path, the two vertices are strongly connected.

If any pair of vertices in the graph are strongly connected, then the graph is a strongly connected graph

The maximal connected sub-graph is called the strongly connected component of the graph.

Spanning tree, spanning forest (connected graph)

Spanning Tree of connected graphs: a very small connected sub-graph containing all the vertices on the way.

Non-connected graph, spanning tree of connected components forming non-connected graph generation Forest

degrees, degrees, and degrees of vertices

Set E as Edge (edge), V as the number of edges attached to a vertex

Graph without direction: Sum (TD (vi)) = 2e

Forward graph: TD (v) = ID (v) + OD (v), SUM (ID (vi)) + SUN (OD (vi)) = E

Right and net of the edge

In the diagram, each edge can be labeled with a value that has a meaning, which is called the weight of the edge

A graph with weights on the edge is called a weighted graph (net)

Dense graphs, sparse graphs

A graph with a few sides, called a dense graph.

The number of edges at this time, usually satisfying: | e| < | v| * log | v|

path, path length, and loop

Path: Refers to the vertex sequence of vertices to vertices in the graph

Path length: Number of top of path

Loop, Ring: The same path as the first vertex and the last vertex

Simple path, simple loop

Simple path: No duplicate vertices in this path

Simple loop: Vertex repetition only, other vertices no duplicates

Distance

Shortest path from u to V

U to v does not exist path do infinity

There is a direction to the tree

There is a vertex with a degree of 0, and other graphs with a vertex-in degree of 1 are called a tree.

1.1 Basic concepts of graphs