Math Problem Statement

if g is 2 connected then G\e is connected for every edge

Solution

Yes, that’s correct! If a graph GG is 2-connected, then GeG \setminus e (the graph GG with any edge ee removed) is still connected.

Explanation:

A graph GG is 2-connected if:

  1. GG is connected, meaning there is a path between any two vertices.
  2. GG remains connected even if any one vertex is removed, which implies that GG has no articulation points (a vertex whose removal would disconnect the graph).

Given GG is 2-connected, this means:

  • Removing any single edge ee from GG will not disconnect the graph because GG would need to lose a vertex to become disconnected (which it cannot, by definition).
  • GG has at least two disjoint paths between any two vertices, so the removal of an edge ee will still leave at least one path between any pair of vertices, thus keeping the graph connected.

Therefore, for a 2-connected graph GG, removing any edge ee will leave the graph GeG \setminus e connected.

Would you like to explore more about 2-connected graphs or have specific questions about graph connectivity? Here are some related questions you might find interesting:

  1. What are the characteristics of a 3-connected graph?
  2. How can you prove a graph is 2-connected?
  3. Is a complete graph always 2-connected?
  4. What is the relationship between 2-connected graphs and cycles?
  5. Can a graph be 2-connected if it contains a cut vertex?

Tip: Remember, if a graph has a cycle between any two vertices, removing any single edge in that cycle won’t disconnect the graph. This is a handy property to remember when dealing with connectivity.

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Math Problem Analysis

Mathematical Concepts

Graph Theory
Connectivity
2-Connected Graphs

Formulas

-

Theorems

2-Connected Graph Theorem
Edge Connectivity in Graphs

Suitable Grade Level

Grades 11-12, College level