Minimal disconnected cuts in planar graphs

The problem of finding a disconnected cut in a graph is NP‐hard in general but polynomial‐time solvable on planar graphs. The problem of finding a minimal disconnected cut is also NP‐hard but its computational complexity was not known for planar graphs. We show that it is polynomial‐time solvable on 3‐connected planar graphs but NP‐hard for 2‐connected planar graphs. Our technique for the first result is based on a structural characterization of minimal disconnected cuts in 3‐connected K 3 , 3 ‐free‐minor graphs and on solving a topological minor problem in the dual. In addition we show that the problem of finding a minimal connected cut of size at least 3 is NP‐hard for 2‐connected apex graphs. Finally, we relax the notion of minimality and prove that the problem of finding a so‐called semi‐minimal disconnected cut is still polynomial‐time solvable on planar graphs.