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.

A graph G = (V, E) is a unipolar graph if there exists a partition V = V 1 ∪ V 2 such that, V 1 is a clique and V 2 induces the disjoint union of cliques. The complement-closed class of generalized split graphs contains those graphs G such that either G or the complement of G is unipolar. Generalized split graphs are a large subclass of perfect graphs. In fact, it has been shown that almost all C 5 -free (and hence, almost all perfect graphs) are generalized split graphs. In this paper we present a recognition algorithm for unipolar graphs that utilizes a minimal triangulation of the given graph, and produces a partition when one exists. Our algorithm has running time O(nm + nm F ), where m F is the number of edges added in a minimal triangulation of the given graph. Generalized split graphs can be recognized via this algorithm in O(n 3 ) time. We give algorithms on unipolar graphs for finding a maximum independent set and a minimum clique cover in O(n + m) time and for finding a maximum clique and a minimum proper coloring in O(n 2.5 / log n) time, when a unipolar partition is given. These algorithms yield algorithms for the four optimization problems on generalized split graphs that have the same worst-case time bounds. We also report that the perfect code problem is NP-Complete for chordal unipolar graphs.

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Let G be a finite undirected graph. A vertex dominates itself and all its neighbors in G. A vertex set D is an efficient dominating set (e.d. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d. in G, is known to be NP-complete even for very restricted graph classes such as P 7 -free chordal graphs. The ED problem on a graph G can be reduced to the Maximum Weight Independent Set (MWIS) problem on the square of G. The complexity of the ED problem is an open question for P 6 -free graphs and was open even for the subclass of P 6 -free chordal graphs. In this paper, we show that squares of P 6 -free chordal graphs that have an e.d. are chordal; this even holds for the larger class of (P 6 , house, hole, domino)-free graphs. This implies that ED/WeightedED is solvable in polynomial time for (P 6 , house, hole, domino)-free graphs; in particular, for P 6 -free chordal graphs. Moreover, based on our result that squares of P 6 -free graphs that have an e.d. are hole-free and some properties concerning odd antiholes, we show that squares of (P 6 , house)-free graphs ((P 6 , bull)-free graphs, respectively) that have an e.d. are perfect. This implies that ED/WeightedED is solvable in polynomial time for (P 6 , house)-free graphs and for (P 6 , bull)-free graphs (the time bound for (P 6 , house, hole, domino)-free graphs is better than that for (P 6 , house)-free graphs). The complexity of the ED problem for P 6 -free graphs remains an open question.

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In a finite undirected graph G, a vertex dominates itself and all its neighbors in G. A vertex set D is an efficient dominating set (e.d. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d. in G, is known to be NP-complete even for very restricted graph classes such as P 7 -free chordal graphs; its complexity was an open question for P 6 -free graphs and was open even for the subclass of P 6 -free chordal graphs. Recently, Lokshtanov et al., and independently, Brandstädt and Mosca showed that ED is solvable in polynomial time for P 6 -free graphs.The ED problem on a graph G can be reduced to the Maximum Weight Independent Set problem on the square of G. In this paper, we show that squares of P 6 -free chordal graphs that have an e.d. are chordal; this even holds for the larger class of (P 6 , house, hole, domino)-free graphs. Thus, ED/WeightedED is solvable in polynomial time for (P 6 , house, hole, domino)-free graphs, and in particular, for P 6 -free chordal graphs. Also, the time bound achieved for ED on this class is much better than in the P 6 -free case.Moreover, we show that squares of P 6 -free graphs that have an e.d. are hole-free. Based on this result, we show that ED is solvable in polynomial time for (P 6 , net)-free graphs; again, the time bound for ED on this class is much better than in the P 6 -free case.

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