3,214 research outputs found
On the linear extension complexity of stable set polytopes for perfect graphs
We study the linear extension complexity of stable set polytopes of perfect graphs. We make use of known structural results permitting to decompose perfect graphs into basic perfect graphs by means of two graph operations: 2-join and skew partitions. Exploiting the link between extension complexity and the nonnegative rank of an associated slack matrix, we investigate the behaviour of the extension complexity under these graph operations. We show bounds for the extension complexity of the stable set polytope of a perfect graph G depending linearly on the size of G and involving the depth of a decomposition tree of G in terms of basic perfect graphs
On the linear extension complexity of stable set polytopes for perfect graphs
We study the linear extension complexity of stable set polytopes of perfect graphs. We make use of known structural results permitting to decompose perfect graphs into basic perfect graphs by means of two graph operations: 2-joins and skew partitions. Exploiting the link between extension complexity and the nonnegative rank of an associated slack matrix, we investigate the behavior of the extension complexity under these graph operations. We show bounds for the extension complexity of the stable set polytope of a perfect graph G depending linearly on the size of G and involving the depth of a decomposition tree of G in terms of basic perfect graphs
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
Matrix partitions of perfect graphs
AbstractGiven a symmetric m by m matrix M over 0,1,*, the M-partition problem asks whether or not an input graph G can be partitioned into m parts corresponding to the rows (and columns) of M so that two distinct vertices from parts i and j (possibly with i=j) are non-adjacent if M(i,j)=0, and adjacent if M(i,j)=1. These matrix partition problems generalize graph colourings and homomorphisms, and arise frequently in the study of perfect graphs; example problems include split graphs, clique and skew cutsets, homogeneous sets, and joins.In this paper we study M-partitions restricted to perfect graphs. We identify a natural class of ‘normal’ matrices M for which M-partitionability of perfect graphs can be characterized by a finite family of forbidden induced subgraphs (and hence admits polynomial time algorithms for perfect graphs). We further classify normal matrices into two classes: for the first class, the size of the forbidden subgraphs is linear in the size of M; for the second class we only prove exponential bounds on the size of forbidden subgraphs. (We exhibit normal matrices of the second class for which linear bounds do not hold.)We present evidence that matrices M which are not normal yield badly behaved M-partition problems: there are polynomial time solvable M-partition problems that do not have finite forbidden subgraph characterizations for perfect graphs. There are M-partition problems that are NP-complete for perfect graphs. There are classes of matrices M for which even proving ‘dichotomy’ of the corresponding M-partition problems for perfect graphs—i.e., proving that these problems are all polynomial or NP-complete—is likely to be difficult
Decomposing Berge graphs and detecting balanced skew partitions
We prove that the problem of deciding whether a graph has a balanced skew partition is NP-hard. We give an O(n9)-time algorithm for the same problem restricted to Berge graphs. Our algorithm is not constructive : it certifies that a graph has a balanced skew partition if it has one. It relies on a new decomposition theorem for Berge graphs, that is more precise than the previously known theorems and implies them easily. Our theorem also implies that every Berge graph can be decomposed in a first step by using only balanced skew partitions, and in a second step by using only 2-joins. Our proof of this new theorem uses at an essential step one of the decomposition theorems of Chudnovsky.Perfect graph, Berge graph, 2-join, balanced skew partition, decomposition, detection, recognition.
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