Planarity Allowing Few Error Vertices in Linear Time

We show that for every fixed k, there is a linear time algorithm that decides whether or not a given graph has a vertex set X of order at most k such that G-X is planar (we call this class of graphs k-apex), and if this is the case, computes a drawing of the graph in the plane after deleting at most k vertices. In fact, in this case, we shall determine the minimum value l ? k such that after deleting some l vertices, the resulting graph is planar. If this is not the case, then the algorithm gives rise to a minor which is not k-apex and is minimal with this property. This answers the question posed by Cabello and Mohar in 2005, and by Kawarabayashi and Reed (STOC´07), respectively. Note that the case k = 0 is the planarity case. Thus our algorithm can be viewed as a generalization of the seminal result by Hopcroft and Tarjan (J. ACM 1974), which determines if a given graph is planar in linear time. Our algorithm can be also compared to the algorithms by Mohar (STOC´96 and Siam J. Discrete Math 2001) for testing the embeddability of an input graph in a fixed surface in linear time, by Kawarabayashi and Mohar (STOC´08) for testing polyhedral embeddability of an input graph in a fixed surface in linear time, and by Kawarabayashi and Reed (STOC´07) for testing the fixed crossing number in linear time. Note that deciding the genus of k-apex graphs is NP-complete, even for k = 1, as shown by Mohar. Thus k-apex graphs are very different from bounded genus graphs in a sense. In addition, for any fixed c, k, we apply our algorithm to obtain a linear time approximation scheme for weighted TSP, and for minimum weighted c-edge-connected submultigraph, respectively, for k-apex graphs. (In this case, an embedding of a k-apex graph is not given in the input). The first result generalizes the recent planar result by Klein (FOCS´05), while the second result generalizes Czumaj et al. (SODA´04). We also extend several optimization results for planar graphs by Baker (J. ACM. 1994)- and others to k-apex graphs.