Large Independent Sets in Subquartic Planar Graphs

Abstract

By the famous four color theorem, every planar graph admits an independent set that contains at least one quarter of its vertices. This lower bound is tight for infinitely many planar graphs, and finding maximum independent sets in planar graphs is \mathsf {np}\mathsf {np}-hard. A well-known open question in the field of parameterized complexity asks whether the problem of finding a maximum independent set in a given planar graph is fixed-parameter tractable, for parameter the “gain” over this tight lower bound. This open problem has been posed many times [4, 8, 10, 13, 17, 20, 31, 32, 35, 38].we show fixed-parameter tractability of the independent set problem parameterized above tight lower bound in planar graphs with maximum degree at most 4, in subexponential time