ACM Midatlantic programming contest 2014, Problem F

Thoughts:

Pseudocode:

1. Seek a path P from i,j to k,l.  
   If none: Print "Yes", exit.  
   Else continue.

2. Let S be the set of points in P other than i,j and k,l.
   While S is not empty  
     Pick a point (x,y) in S. //Should we choose in a certain order?
     Place an (additional) obstacle at x,y.
     Seek a path Q from i,j to k,l. //Should we randomize the search a bit?
     If none: Print "Yes", exit.
     Else S = S intersect the points of Q. //Refines a little the class discussion.
     Undo the obstacle at x,y.
   Print "No".

Details needed:
How to search for a path with given set of obstacles?
How to represent sets of points to have easy intersection and selection.
Representation of the board (with obstacle placement).