Connecting face hitting sets in planar graphs

Authors:
Pascal Schweitzer;Patrick Schweitzer
Affiliations:
Max-Planck-Institute for Computer Science, Campus E1 4, D-66123 Saarbrücken, Germany;University of Luxembourg, Interdisciplinary Centre for Security, Reliability and Trust, 6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg
Venue:
Information Processing Letters
Year:
2010

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Abstract

We show that any face hitting set of size n of a connected planar graph with a minimum degree of at least 3 is contained in a connected subgraph of size 5n-6. Furthermore we show that this bound is tight by providing a lower bound in the form of a family of graphs. This improves the previously known upper and lower bound of 11n-18 and 3n respectively by Grigoriev and Sitters. Our proof is valid for simple graphs with loops and generalizes to graphs embedded in surfaces of arbitrary genus.