A (2 +)-vertex kernel for the dual coloring problem

Wenjun Li, Yang Ding, Yongjie Yang, Guozhen Rong

Abstract: Given a graph G of n vertices and an integer k, the Dual Coloring problem determines if G is (n−k)<math><mo stretchy="false" is="true">(</mo><mi is="true">n</mi><mo linebreak="badbreak" linebreakstyle="after" is="true">−</mo><mi is="true">k</mi><mo stretchy="false" is="true">)</mo></math>-colorable, i.e., if we can color vertices of G with at most n−k<math><mi is="true">n</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mi is="true">k</mi></math> colors so that every vertex obtains exactly one color and every two adjacent vertices have different colors. We derive a kernelization for the Dual Coloring problem with respect to the parameter k. In particular, for any fixed ϵ>0<math><mi is="true">ϵ</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">&gt;</mo><mn is="true">0</mn></math>, our kernelization yields a kernel of at most (2+ϵ)k<math><mo stretchy="false" is="true">(</mo><mn is="true">2</mn><mo linebreak="badbreak" linebreakstyle="after" is="true">+</mo><mi is="true">ϵ</mi><mo stretchy="false" is="true">)</mo><mi is="true">k</mi></math> vertices, improving the currently best result 3k−3<math><mn is="true">3</mn><mi is="true">k</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mn is="true">3</mn></math>.