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Connecting face hitting sets in planar graphs. (English) Zbl 1259.05048

Summary: 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.

MSC:

05C10 Planar graphs; geometric and topological aspects of graph theory
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References:

[1] Bodlaender, H. L.; Feremans, C.; Grigoriev, A.; Penninkx, E.; Sitters, R.; Wolle, T., On the minimum corridor connection problem and other generalized geometric problems, Computational Geometry: Theory and Applications, 42, 9, 939-951 (2009) · Zbl 1200.05215
[2] Bodlaender, H. L.; Penninkx, E., A linear kernel for planar feedback vertex set, (IWPEC. IWPEC, Lecture Notes in Computer Science, vol. 5018 (2008), Springer), 160-171 · Zbl 1142.68451
[3] Borradaile, G.; Klein, P.; Mathieu, C., An \(O(n \log n)\) approximation scheme for Steiner tree in planar graphs, ACM Transactions on Algorithms, 5, 3, 1-31 (2009) · Zbl 1300.05294
[4] Demaine, E. D.; Hajiaghayi, M. T., The bidimensionality theory and its algorithmic applications, The Computer Journal, 51, 3, 292-302 (2007)
[5] Garey, M. R.; Johnson, D. S., Computers and Intractability: A Guide to the Theory of NP-Completeness (1979), W.H. Freeman · Zbl 0411.68039
[6] Grigoriev, A.; Sitters, R., Connected feedback vertex set in planar graphs, (Graph-Theoretic Concepts in Computer Science, 35th International Workshop, WG 2009, Revised Papers (2009), Springer), 143-153 · Zbl 1273.68409
[7] Koutsonas, A.; Thilikos, D. M., Planar feedback vertex set and face cover: Combinatorial bounds and subexponential algorithms, (Graph-Theoretic Concepts in Computer Science, 34th International Workshop, WG 2008, Revised Papers (2008), Springer), 264-274 · Zbl 1202.68284
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