The Complexity of the List Partition Problem for Graphs

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 A1, A2, . . . , Ak, where it may be specified that Ai induces a stable set, a clique, or an arbitrary subgraph, and pairs Ai, Aj (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

Recognition of some perfectly orderable graph classes

AbstractThis paper presents new algorithms for recognizing several classes of perfectly orderable graphs. Bipolarizable and P4-simplicial graphs are recognized in O(n3.376) time, improving the previous bounds of O(n4) and O(n5), respectively. Brittle and semi-simplicial graphs are recognized in O(n3) time using a randomized algorithm, and O(n3log2n) time if a deterministic algorithm is required. The best previous time bound for recognizing these classes of graphs is O(m2). Welsh–Powell opposition graphs are recognized in O(n3) time, improving the previous bound of O(n4). HHP-free graphs and maxibrittle graphs are recognized in O(mn) and O(n3.376) time, respectively

Consecutive-ones: handling lattice planarity efficiently

A concept lattice may have a size exponential in the number of objects it models. Polynomial-size lattices and/or compact representations are thus desirable. This is the case for planar concept lattices, which has both polynomial size and representation without edge crossing, but a generic process for drawing them efficiently is yet to be found. Recently, it has been shown that when the relation has the consecutive-ones property (i.e, the matrix of the relation can be rapidly reorderd so that the 1s are consecutive in every row), the number of concepts is polynomial and these can be efficiently generated. In this paper we show that a consecutive-ones relation |R | has a planar lattice which can be drawn in O(|R|) time. We also present a hierarchical classification of polynomial-size lattices based on structural properties of the relation R, its associated graphs Gbip and GR, and its concept lattice L(R)