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Abstract. The k-partition problem is as follows: Given a graph G and a positive integer k, partition the vertices of G into at most k parts A 1 , A 2 , . . . , A k , where it may be specified that A i induces a stable set, a clique, or an arbitrary subgraph, and pairs A i , A j (i = j) be completely nonadjacent, completely adjacent, or arbitrarily adjacent. The list k-partition problem generalizes the k-partition problem by specifying for each vertex x, a list L(x) of parts in which it is allowed to be placed. Many well-known graph problems can be formulated as list k-partition problems: e.g., 3-colorability, clique cutset, stable cutset, homogeneous set, skew partition, and 2-clique cutset. We classify, with the exception of two polynomially equivalent problems, each list 4-partition problem as either solvable in polynomial time or NP-complete. In doing so, we provide polynomial-time algorithms for many problems whose polynomial-time solvability was open, including the list 2-clique cutset problem. This also allows us to classify each list generalized 2-clique cutset problem and list generalized skew partition problem as solvable in polynomial time or NP-complete.

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A graph G is the k-leaf power of a tree T if its vertices are leaves of T such that two vertices are adjacent in G if and only if their distance in T is at most k. Then T is a k-leaf root of G. This notion was introduced and studied by Nishimura, Ragde, and Thilikos [2002], motivated by the search for underlying phylogenetic trees. Their results imply an O(n 3 )-time recognition algorithm for 4-leaf powers. Recently, Rautenbach [2006] as well as Dom et al. [2005] characterized 4-leaf powers without true twins in terms of forbidden subgraphs. We give new characterizations for 4-leaf powers and squares of trees by a complete structural analysis. As a consequence, we obtain a conceptually simple linear-time recognition of 4-leaf powers. ACM Reference Format: Brandstädt, A., Le, V. B., and Sritharan, R. 2008. Structure and linear-time recognition of 4-leaf powers.

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