A graph $G=(V,E)$ is connected and its edge $e$ lies on a cycle. Prove that, then the graph $G^{'} = (V,E)$ is connected, that is deleting edge of the cycle doesn't violate the connectivity.
How to prove this theorem. I need hint/resource recommendation.
Definition of Connectivity
The following is similar to the definition of connectivity found in many textbooks:
Some Remarks on the Definition of Connectivity
Path and walks begin at a dot (a vertex). You then travel to a neighboring dot. You repeat this process until arriving at some destination dot.
The definition of connectivity is bad if the definition talks about paths.
This is because a graph is connected if and only if there exists a walk between every pair of nodes.
Suppose that a graph can be divided into two separate pieces $L$ and $R$ so that no edge (line segment) crosses the gap between $L$ and $R$
If you cannot get to vertex $w$ from vertex $v$, then it does not matter if you repeat nodes on your journey or not. That is, it does not matter if we talk about paths, or walks, when it comes to connectivity.
The Relationship between a Cycle and Two paths
A cycle is can be viewed as two paths which have been glued end-to-end.
A cycle is the same as two paths which have been glued end-to-end.
If you delete one of the paths in a cycle, then there is still one path remaining.
Sketch of a Proof
Let $G$ be a graph $G$ such that graph $G$ contains a cycle.
Suppose that $v$ and $w$ are nodes on the cycle and that $e = \{v, w\}$ is an edge in the cycle.
We now destroy an edge $\{v, w\}$ in the cycle to form graph $G-\{v, w\}$
Consider any walk in graph $G$ from a node $a$ to a node $b$
Suppose that the walk from $a$ to $b$ contained edge $\{v, w\}$.
After edge $e$ is deleted, how do we get from node $a$ to node $b$?
Edge $\{v, w\}$ can be viewed as a very short path.
A cycle is the union of two non-overlapping paths.
Thus, we replace edge $\{v, w\}$ with the path from the cycle which does not contain ede $\{v, w\}$.
This gives us a new walk with the same starting and ending vertices.
If the old walk was a path (did not repeat vertices), then the new walk might repeat vertices.
It is okay if the new walk repeats vertices. A graph is connected if and only if there exists a walk between every pair of vertices. It is okay for there to exist at least one vertex in the graph which is visited more than once on the walk.
Definition of Walk
Suppose that $m$ is a whole number, such as 1, 2, 3, 964, etc...
A walk in graph $G$ is a mapping $F$ from $\{n \in \mathbb{N}: 1 \leq n \leq m\}$ to dots (vertices) in graph $G$ such that if $k$ is a whole number between $1$ and $m$ then there is an edge/line between dot $F(k)$ and dot $F(k+1)$ in graph $G$.
A walk must contain at least one node.
It is possible, but unusual, for a walk contain only one node. Usually, there are many nodes in the walk under discussion.