Deciding Relaxed Two-Colorability - A Hardness Jump

Robert Berke, Tibor Szabó

Abstract: A coloring is proper if each color class induces connected components of order one (where the order of a graph is its number of vertices). Here we study relaxations of proper two-colorings, such that the order of the induced monochromatic components in one (or both) of the color classes is bounded by a constant. In a (C 1 , C 2 )-relaxed coloring of a graph G every monochromatic component induced by vertices of the first (second) color is of order at most C 1 (C 2, resp.). We are mostly concerned with (1,C)-relaxed colorings, in other words when/how is it possible to break up a graph into small components with the removal of an independent set.