Increase by one, Shortest path, changes the edges or not?

Question

as i read the following text :

"Let P be a shortest path from some vertex s to some other vertex t in a graph. If the weight of each edge in the graph is increased by one, P will still be a shortest path from s to t"

Solution: False. the shortest path would change if 1 was added to every edge weight.

I ran into a new question:

Suppose we have a Graph G in which weight of all edges is >1 (positive). If we increase weight of all edges by one, the shortest path between two specific vertex has the same edges.

I doubt about this question. would u please anyone clarify me?

Answer 1

Let $G$ be the complete graph on three vertices $A,B,$ and $C$. Let edges $AB$ and $BC$ have weight $2$, and edge $AC$ have weight $4.5$.

Then the shortest path from $A$ to $C$ is via $B$. But if you increase the weights to $3,3,$ and $5.5$, the shortest path is the edge $AC$.

If you want integer weights, you can do it with four points, with edge weights $AB=BC=CD=2$, $AD=7$.

Answer 2

Hint:

A path with less edges has its weight increase less under the add-1-to-each-edge scenario than a path with more edges. So a path with initially higher weight could be surpassed by a lower weight path with more edges.