On the complexity of covering vertices by faces in a planar graph
SIAM Journal on Computing
An efficient fixed-parameter algorithm for 3-hitting set
Journal of Discrete Algorithms
Exact algorithms for finding minimum transversals in rank-3 hypergraphs
Journal of Algorithms
Refined memorization for vertex cover
Information Processing Letters
Deeply asymmetric planar graphs
Journal of Combinatorial Theory Series B
A refined search tree technique for Dominating Set on planar graphs
Journal of Computer and System Sciences
Parametric Duality and Kernelization: Lower Bounds and Upper Bounds on Kernel Size
SIAM Journal on Computing
Parameterized Complexity
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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.