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On obstructions to small face covers in planar graphs. (English) Zbl 0781.05014
It is shown that every plane graph contains a set $$V$$ of vertices no two of which are on the same face and a set $$F$$ of faces covering all vertices such that the cardinality of $$F$$ is at most 27 times that of $$V$$. The authors introduce the class $$F(k)$$ of graphs which cannot be embedded in the plane with $$k$$ or fewer faces and which are minor minimal with this property. It is shown that there exists a constant $$c$$ such that every graph in $$F(k)$$ has at most $$ck^ 3$$ vertices.

##### MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
##### Keywords:
small face covers; plane graph
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##### References:
 [1] Ando, K; Enomoto, H; Saito, A, Contractible edges in 3-connected graphs, J. combin. theory ser. B, 42, 89-93, (1987) · Zbl 0611.05037 [2] Bienstock, D; Monma, C.L, On the complexity of covering vertices by faces in a planar graph, SIAM J. comput., 17, 53-76, (1988) · Zbl 0646.68085 [3] Bienstock, D, Linear-time test for small face covers in any fixed surface, SIAM J. comput., 19, 907-911, (1990) · Zbl 0711.68056 [4] Erdös, P; Pósa, L, On independent circuits contained in a graph, Canad. J. math, 17, 347-352, (1965) · Zbl 0129.39904 [5] Erickson, R.E; Monma, C.L; Vienott, A.F, Send-and-split method for minimum-concave-cost network flows, Math. oper. res., 12, 634-664, (1987) · Zbl 0667.90036 [6] Fellows, M; Hickling, F; Syslo, M, Topological parameterization and hard graph problems, (1986), University of New Mexico, Extended abstract · Zbl 0654.05027 [7] Frank, A, Edge-disjoint paths in planar graphs, J. combin. theory ser. B, 39, 164-178, (1985) · Zbl 0583.05036 [8] Garey, M.R; Johnson, D.S, () [9] Halin, R, Untersuchungen über minimale n-fach zusammenhängende graphen, Math. ann., 182, 175-188, (1969) · Zbl 0172.25804 [10] Harary, F, () [11] Lovász, L, () [12] Okamura, H; Seymour, P.D, Multicommodity flows in planar graphs, J. combin. theory ser. B, 31, 75-81, (1981) · Zbl 0465.90029 [13] Ota, K; Saito, A, Non-separating induced cyclin 3-connected graphs, (1986), manuscript [14] Robertson, N; Seymour, P.D, Graph minors. VI. disjoint paths across a disk, J. combin. theory ser. B, 41, 115-138, (1986) · Zbl 0598.05042 [15] Robertson, N; Seymour, P.D, Graph minors. VII. disjoint paths on a surface, J. combin. theory ser. B, 45, 212-254, (1988) · Zbl 0658.05044 [16] Robertson, N; Seymour, P.D, Graph minors. VIII. A Kuratowski theorem for general surfaces, J. combin. theory ser. B, 48, 255-288, (1990) · Zbl 0719.05033 [17] Robertson, N; Seymour, P.D, Graph minors. XV. Wagner’s conjecture, (1988), manuscript · Zbl 1061.05088 [18] Seymour, P.D, Disjoint paths in graphs, Discrete math., 29, 293-309, (1980) · Zbl 0457.05043 [19] \scP. D. Seymour, Personal communication, 1988. [20] \scA. Schrijver, Disjoint homotopic paths and trees in a planar graph, Discrete Comput. Geom., to appear. · Zbl 0755.05033 [21] Steinitz, E, Polyheder und raumeinteilungen, Enzykl. math. wiss. 3 (geometrie), 12, 1-139, (1922), Part 3AB [22] Thomassen, C, Planarity and duality of finite and infinite graphs, J. combin. theory ser. B, 29, 244-271, (1980) · Zbl 0441.05023 [23] Whitney, H, Two-isomorphic graphs, Trans. amer. math. soc., 34, 339-362, (1932) · JFM 58.0608.01
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