2020/04/19 by Jansen, Bart M. P., de Kroon, Jari J. H. · 1 citation
#05C50 #05C85 #68R10 #Combinatorics (math.CO) #Data Structures and Algorithms (cs.DS) #F.2.2 #FOS: Computer and information sciences #FOS: Mathematics #G.2.2
paper · doi:10.48550/arxiv.2004.08818
We consider the Π-free Deletion problem parameterized by the size of a vertex cover, for a range of graph properties Π. Given an input graph G, this problem asks whether there is a subset of at most k vertices whose removal ensures the resulting graph does not contain a graph from Π as induced subgraph. Many vertex-deletion problems such as Perfect Deletion, Wheel-free Deletion, and Interval Deletion fit into this framework. We introduce the concept of characterizing a graph property Π by low-rank adjacencies, and use it as the cornerstone of a general kernelization theorem for Π-Free Deletion parameterized by the size of a vertex cover. The resulting framework captures problems such as AT-Free Deletion, Wheel-free Deletion, and Interval Deletion. Moreover, our new framework shows that the vertex-deletion problem to perfect graphs has a polynomial kernel when parameterized by vertex cover, thereby resolving an open question by Fomin et al. [JCSS 2014]. Our main technical contribution shows how linear-algebraic dependence of suitably defined vectors over \mathbbF2 implies graph-theoretic statements about the presence of forbidden induced subgraphs.