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In Section 4 we will show that Minimum 2-Core is still in FPT with parameter t, using the preceding FPT result for Multiple Weigthed Hypernode Hitting Set with m = 2. Prior to this final contribution, Section 3 provides some preliminary results on the hardness of Minimum d-Core. In particular, our FPT result for d = 2 is best possible in the sense that case d = 3 is not in FPT (unconditional). Here it is worth mentioning another related hardness result: Finding a chordless cycle through two target vertices is W[1]-complete [8].

Furthermore we write [n] = {1, . . , n}. The sentence is defined as ∃u1 , . . , uk , e1 , . . , ek , a1 , . . , ak , b1 , . . , bk (φk ∧ ∀v φk ) φk = k +k +k ≤k where φk and φk are defined below. φk is the conjunction of the following clauses (1). . (4) that ensure that u1 , . . , uk represent deleted vertices, e1 , . . ek represent deleted edges, and ai , bi , 1 ≤ i ≤ k represent ends of added edges. Note that since added edges are not present in the given structure we need to express them in terms vertex pairs.

COCOA 2008, LNCS 5165, pp. 32–42, 2008. c Springer-Verlag Berlin Heidelberg 2008 Multiple Hypernode Hitting Sets and Smallest Two-Cores with Targets 33 Multiple Weighted Hypernode Hitting Set. The Multiple Weighted Hitting Set problem is the special case where the the hypernodes are singleton sets, one for each node in V . In Multiple Weighted Hypernode Hitting Set we may assume without loss of generality that all hyperedges are pairwise disjoint: If they are not, we “disjunctify” them as follows.

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