Here's a question relating to graph theory that was asked in the [International Kangaroo Maths Contest 2017](http://ikmc.kangaroo.org.pk/). I am a college student and have a very less knowhow of graph theory. The question goes like this: 

We have $10$ islands that have been connected by $15$ bridges.

[![](https://i.stack.imgur.com/y62HO.png)](https://i.stack.imgur.com/y62HO.png)

What is the smallest number of bridges that may be eliminated so that it becomes impossible to get from $A$ to $B$ by bridge?

**Answer:** The answer given in the key is $3$, but I can't see how that might be.

**What I did:**

First of all I transferred the problem into graphical terms.

*We have $10$ vertices connected by $15$ edges.*

[![](https://i.stack.imgur.com/UOjkL.png)](https://i.stack.imgur.com/UOjkL.png)

*What is the smallest possible number of edges that when removed to make it impossible to find a path from $A$ to $B$?*

So I considered that we can do this by 'isolating' either $A$ or $B$. This may be done by removing all the edges corresponding to either $A$ or $B$. The number of edges corresponding to $A$ is $4$ while those with $B$ are $5$. So the smallest number that can be the answer is $4$. So what is wrong with this method? How can be the answer, $3$, be reached? 

Thanks for the attention.