Kernelization hardness of connectivity problems in d-degenerate graphs

A graph is d-degenerate if every subgraph contains a vertex of degree at most d. For instance, planar graphs are 5-degenerate. Inspired by the recent work by Philip, Raman and Sikdar, who have shown the existence of a polynomial kernel for DOMINATING SET in d-degenerate graphs, we investigate the kernelization complexity of problems that include a connectivity requirement in this class of graphs. Our main contribution is the proof that CONNECTED DOMINATING SET does not admit a polynomial kernel in d-degenerate graphs for d >= 2 unless NP subset of coNP/poly, which is known to cause a collapse of the polynomial hierarchy to its third level. We prove this using a problem that originates from bioinformatics - COLOURFUL GRAPH MOTIF - analysed and proved to be NP-hard by Fellows et al. This problem nicely encapsulates the hardness of the connectivity requirement in kernelization. Our technique also yields an alternative proof that, under the same complexity assumption, STEINER TREE does not admit a polynomial kernel. The original proof, via a reduction from SET COVER, is due to Dom, Lokshtanov and Saurabh. We extend our analysis by showing that, unless NP subset of coNP/poly, there do not exist polynomial kernels for STEINER TREE, CONNECTED FEEDBACK VERTEX SET and CONNECTED ODD CYCLE TRANSVERSAL in d-degenerate graphs for d >= 2. On the other hand, we show a polynomial kernel for CONNECTED VERTEX COVER in graphs that do not contain the biclique K-ij as a subgraph. As a d-degenerate graph cannot contain Kd+1.d+.1 as a subgraph, the results holds also for graphs of bounded degeneracy. (C) 2012 Elsevier BM. All rights reserved.