An Application of Duality to Edge-Deletion Problems

Abstract

For a property $\pi $ on graphs, the corresponding edge-deletion problem $P_{{\text{ED}}} (\pi )$ (on planar graphs) is stated as follows: given a (planar) graph G, find a set of edges of minimum cardinality whose deletion results in a graph satisfying $\pi $. We show that the edge-deletion problem $P_{{\text{ED}}} (\pi )$ on planar graphs is NP-hard if $\pi $ is nontrivial and is determined by the weighted 3-connected components. As a corollary, the edge-deletion problem $P_{{\text{ED}}} (\pi )$ on planar graphs is NP-complete for several properties $\pi $, such as $\pi = $ “series-parallel,” “outerplanar,” “without cocycles of cardinality $ \geqq 3$,” etc.