15,634 research outputs found
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
Clique-Stable Set separation in perfect graphs with no balanced skew-partitions
Inspired by a question of Yannakakis on the Vertex Packing polytope of
perfect graphs, we study the Clique-Stable Set Separation in a non-hereditary
subclass of perfect graphs. A cut (B,W) of G (a bipartition of V(G)) separates
a clique K and a stable set S if and . A
Clique-Stable Set Separator is a family of cuts such that for every clique K,
and for every stable set S disjoint from K, there exists a cut in the family
that separates K and S. Given a class of graphs, the question is to know
whether every graph of the class admits a Clique-Stable Set Separator
containing only polynomially many cuts. It is open for the class of all graphs,
and also for perfect graphs, which was Yannakakis' original question. Here we
investigate on perfect graphs with no balanced skew-partition; the balanced
skew-partition was introduced in the proof of the Strong Perfect Graph Theorem.
Recently, Chudnovsky, Trotignon, Trunck and Vuskovic proved that forbidding
this unfriendly decomposition permits to recursively decompose Berge graphs
using 2-join and complement 2-join until reaching a basic graph, and they found
an efficient combinatorial algorithm to color those graphs. We apply their
decomposition result to prove that perfect graphs with no balanced
skew-partition admit a quadratic-size Clique-Stable Set Separator, by taking
advantage of the good behavior of 2-join with respect to this property. We then
generalize this result and prove that the Strong Erdos-Hajnal property holds in
this class, which means that every such graph has a linear-size biclique or
complement biclique. This property does not hold for all perfect graphs (Fox
2006), and moreover when the Strong Erdos-Hajnal property holds in a hereditary
class of graphs, then both the Erdos-Hajnal property and the polynomial
Clique-Stable Set Separation hold.Comment: arXiv admin note: text overlap with arXiv:1308.644
Refined dual stable Grothendieck polynomials and generalized Bender-Knuth involutions
The dual stable Grothendieck polynomials are a deformation of the Schur
functions, originating in the study of the K-theory of the Grassmannian. We
generalize these polynomials by introducing a countable family of additional
parameters, and we prove that this generalization still defines symmetric
functions. For this fact, we give two self-contained proofs, one of which
constructs a family of involutions on the set of reverse plane partitions
generalizing the Bender-Knuth involutions on semistandard tableaux, whereas the
other classifies the structure of reverse plane partitions with entries 1 and
2.Comment: 29 pages. Ancillary files contain an alternative version with
different exposition (including some material on the diamond lemma, which is
implicit in the main version of the paper). Comments are welcome
Complementary Algorithms For Tableaux
We study four operations defined on pairs of tableaux. Algorithms for the
first three involve the familiar procedures of jeu de taquin, row insertion,
and column insertion. The fourth operation, hopscotch, is new, although
specialised versions have appeared previously. Like the other three operations,
this new operation may be computed with a set of local rules in a growth
diagram, and it preserves Knuth equivalence class. Each of these four
operations gives rise to an a priori distinct theory of dual equivalence. We
show that these four theories coincide. The four operations are linked via the
involutive tableau operations of complementation and conjugation.Comment: 29 pages, 52 .eps files for figures, JCTA, to appea
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