On the Power of Isolation in Planar Graphs

Abstract

We study (deterministic) isolation for certain structures in directed and undirected planar graphs. The motivation for undertaking such a study comes from recent positive results on this topic. For example: Bourke et al. [2009] isolate a directed path in planar graphs and subsequently Datta et al. [2010b] isolate a perfect matching in bipartite planar graphs. Our first observation is that sufficiently strong (and plausible) isolations for certain structures in planar graphs would have strong consequences such as: NL ⊆ ⊕ L , Bipartite-Matching ∈ NC , and NP ⊆ ⊕ P . Our second observation is that although we do not yet have such strong isolations for arbitrary planar graphs, we do have them for bipartite planar graphs, that is, non-bipartiteness is the main bottleneck.