While Going through Graph theory by West,I found a point where it is written I got the first point that
If a Graph is connected then it have a $uv$ path.
But have some confusions here -:
A Connection relation in a graph is an equivalence relation because it is
Reflexive Relation (take Path of length $0$)
Symmetric Relation (reversible path $\rightarrow$ obviously undirected one)
Transitive Relation-:reason given is
If a Graph has a $uv$ path and also $vw$ path then it will also contain $uw$ path
My Doubt starts here-:
If a Graph contains a $uv$ path and also $vw$ path ,then what is the gurantee that $uv$ path and $vw$ path have NO common Vertex .It Both the path wil have a single common vertex ,it wil no longer a path.
I am stuck here ,please help me out..!!!