776 research outputs found
Torus orbits on homogeneous varieties and Kac polynomials of quivers
In this paper we prove that the counting polynomials of certain torus orbits
in products of partial flag varieties coincides with the Kac polynomials of
supernova quivers, which arise in the study of the moduli spaces of certain
irregular meromorphic connections on trivial bundles over the projective line.
We also prove that these polynomials can be expressed as a specialization of
Tutte polynomials of certain graphs providing a combinatorial proof of the
non-negativity of their coefficients
Rational exponents in extremal graph theory
Given a family of graphs , the extremal number is the largest for which there exists a graph with
vertices and edges containing no graph from the family as a
subgraph. We show that for every rational number between and , there
is a family of graphs such that . This solves a longstanding problem in the area of extremal
graph theory.Comment: 11 pages. arXiv admin note: text overlap with arXiv:1411.085
On retracts, absolute retracts, and folds in cographs
Let G and H be two cographs. We show that the problem to determine whether H
is a retract of G is NP-complete. We show that this problem is fixed-parameter
tractable when parameterized by the size of H. When restricted to the class of
threshold graphs or to the class of trivially perfect graphs, the problem
becomes tractable in polynomial time. The problem is also soluble when one
cograph is given as an induced subgraph of the other. We characterize absolute
retracts of cographs.Comment: 15 page
Coalition structure generation over graphs
We give the analysis of the computational complexity of coalition structure generation over graphs. Given an undirected graph G = (N,E) and a valuation function v : P(N) → R over the subsets of nodes, the problem is to find a partition of N into connected subsets, that maximises the sum of the components values. This problem is generally NP-complete; in particular, it is hard for a defined class of valuation functions which are independent of disconnected members — that is, two nodes have no effect on each others marginal contribution to their vertex separator. Nonetheless, for all such functions we provide bounds on the complexity of coalition structure generation over general and minor free graphs. Our proof is constructive and yields algorithms for solving corresponding instances of the problem. Furthermore, we derive linear time bounds for graphs of bounded treewidth. However, as we show, the problem remains NP-complete for planar graphs, and hence, for any Kk minor free graphs where k ≥ 5. Moreover, a 3-SAT problem with m clauses can be represented by a coalition structure generation problem over a planar graph with O(m2) nodes. Importantly, our hardness result holds for a particular subclass of valuation functions, termed edge sum, where the value of each subset of nodes is simply determined by the sum of given weights of the edges in the induced subgraph
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