Math Problem Statement
if g is 2 connected then G\e is connected for every edge
Solution
Yes, that’s correct! If a graph is 2-connected, then (the graph with any edge removed) is still connected.
Explanation:
A graph is 2-connected if:
- is connected, meaning there is a path between any two vertices.
- remains connected even if any one vertex is removed, which implies that has no articulation points (a vertex whose removal would disconnect the graph).
Given is 2-connected, this means:
- Removing any single edge from will not disconnect the graph because would need to lose a vertex to become disconnected (which it cannot, by definition).
- has at least two disjoint paths between any two vertices, so the removal of an edge will still leave at least one path between any pair of vertices, thus keeping the graph connected.
Therefore, for a 2-connected graph , removing any edge will leave the graph 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:
- What are the characteristics of a 3-connected graph?
- How can you prove a graph is 2-connected?
- Is a complete graph always 2-connected?
- What is the relationship between 2-connected graphs and cycles?
- 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
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Theorems
2-Connected Graph Theorem
Edge Connectivity in Graphs
Suitable Grade Level
Grades 11-12, College level