By Kulikov A. S.

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1 Parameter really small? 1, the parameter “size of the vertex cover set” appeared as a natural choice, and, for general graphs, it seems plausible in many cases to assume that the size of the vertex cover set is signiﬁcantly smaller than the total number of graph vertices. Hence this can be considered as a useful parameterization of Vertex Cover. The situation deﬁnitely changes when we turn our attention to Vertex Cover restricted to planar graphs. Observing the fact that planar graphs with n vertices have at most 3n − 6 edges and an average vertex degree of less than six, it is no longer clear that the vertex cover sizes for planar graphs are really “small” compared with the total number of graph vertices.

This is also where the “W” stems from—it refers to the weft of Boolean circuits, that is, the maximum number of unbounded fan-in gates on any path from the input variables to the output gate of the (decision) circuit. The weft has also been interpreted as the “logical depth” of a problem. Clearly, by deﬁnition, Short Turing Machine Acceptance is W [1 ]-complete. Other problems that are W [1 ]-complete 3 That is, the complement graph has the same set of vertices and there is an edge between a pair of vertices iﬀ there is no edge between them in the original graph.

For the W [1 ]-completeness of Independent Set, the dual problem of Vertex Cover, refer to Downey and Fellows (1999). The four-color theorem for planar graphs is due to Appel and Haken (1977a) and Appel and Haken (1977b); refer to Robertson et al. (1997) for later improvements. The polynomial-time algorithm to four-color a graph is given in Robertson et al. (1996). The idea of parameterizing above guaranteed values was put forward in Mahajan and Raman (1999), there restricted to the problems Maximum Satisfiability and Maximum Cut.

### A 2E4-time algorithm for MAX-CUT by Kulikov A. S.

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