Discrete Math Notes:
Chapter 6: Graphs and Trees
6.5 Graph connectivity
Connected, The two vertices, v and w
Disconnected, A graph is said to be connected if every pair of vertices in the graph is connected
connected component a maximal set of vertices that is connected.
isolated vertex, A vertex that is not connected with any other vertex
k-vertex-connected, An undirected graph G
vertex connectivity, a graph is the largest k such that the graph is k-vertex-connected.
k-edge-connected, undirected graph that remains connected after any k - 1 edges are removed from the graph
edge connectivity, the largest k such that the graph is k-edge-connected
overview:
If there is a path from vertex v to vertex w in an undirected graph, there is also a path from w to v. It is claimed that the two vertices, v and w, are connected. A vertex is always thought to be related to another vertex. If the graph resembles a road or communication network, connecting every pair of vertices is highly desirable. More than one connected component can be found in a disconnected graph. A linked component is a maximally connected set of vertices. If any vertex is added to a connected component, the collection of vertices will no longer be connected, as the term ftmaximalft implies. Isolated vertex refers to a vertex that is not connected to any other vertex and so is a connected component with only one vertex.
6.7 Introduction to trees
Tree, an undirected graph that is connected and has no cycles.
free tree, no particular organization of the vertices and edges.
rooted tree, the vertex at the top of the drawing is designated as the root of the tree.
Level, a vertex is its distance from the root
Height, a tree is the highest level of any vertex.
Parent, the parent of vertex v is the first vertex after v encountered along the path from v to the root.
Ancestor, every vertex along the path from v to the root (except for the vertex v itself)
Child, v is the parent of vertex u, then u is a child of vertex v.
Descendant, if u is an ancestor of v, then v is a descendant of u
Leaf, a vertex which has no children.
Siblings, if they have the same parent.
Subtree, at vertex v is the tree consisting of v and all v's descendants.
Overview:
Example of a Free and Rooted Tree
Free Tree.
Rooted Tree
root
level4 vertices
Maximum level is 4 The tree has height 4
Captions ^
1. In a free tree, the vertices are not ordered. To make a free tree into a rooted tree, select a vertex to be the root and place the root at the top 2. The root is the only level 0 vertex. 3.The level of each of the remaining vertices is its distance from the root. In the given example the maximum level is 4, so the tree has height 4.
The tree on the lef