A bounded search tree algorithm for parameterized face cover

Abstract

The parameterized complexity of the face cover problem is considered. The input to this problem is a plane graph G with n vertices. The question asked is whether, for a given parameter value k, there exists a set of k or fewer faces whose boundaries collectively cover (contain) every vertex in G. Early attempts achieved run times of O * (12 k ) or worse, by reducing the problem into a special form of dominating set, namely red-blue dominating set, restricted to planar graphs. Here, we consider a direct approach, where some surgical operation is performed on the graph at each branching decision. This paper builds on previous work of the authors and employs a structure theorem of Aksionov et al., with a detailed case analysis, to produce a face cover algorithm that runs in O (4.6056 k + n 2 ) time.We also point to the tight connections with red-blue dominating set on planar graphs via the annotated version of face cover that we consider in our search tree algorithm.Hence, we get both a O (4.6056 k + n 2 ) time algorithm for solving red-blue dominating set on planar graphs and a polynomial time algorithm for producing a linear kernel for annotated face cover.