‎Recently‎, ‎some techniques such as adding whiskers and attaching graphs to vertices of a given graph‎, ‎have been proposed for constructing a new vertex decomposable graph‎. ‎In this paper‎, ‎we present a new method for constructing vertex decomposable graphs‎. ‎Then we use this construction to generalize the result due to Cook and Nagel‎.

We introduce two numerical conjugacy invariants for dynamical systems – the complexity and weak complexity indices – which are well-suited for the study of “completely integrable” Hamiltonian systems. These invariants can be seen as “slow entropies”, they describe the polynomial growth rate of the number of balls (for the usual “dynamical” distances) of coverings of the ambient space. We then d...

Using a similarity transformation that maps the Calogero model into N decoupled quantum harmonic oscillators, we construct a set of mutually commuting conserved operators of the model and their simultaneous eigenfunctions. The simultaneous eigenfunction is a deformation of the symmetrized number state (bosonic state) and forms an orthogonal basis of the Hilbert (Fock) space of the model. This o...

We describe a new algorithm for compiling conjunctive normal form (CNF) into Deterministic Decomposable Negation Normal (d-DNNF), which is a tractable logical form that permits model counting in polynomial time. The new implementation is based on latest techniques from both the SAT and OBDD literatures, and appears to be orders of magnitude more efficient than previous algorithms for this purpo...

We investigate different notions of linear independence and of matrix rank that are relevant for max-plus or tropical semirings. The factor rank and tropical rank have already received attention, we compare them with the ranks defined in terms of signed tropical determinants or arising from a notion of linear independence introduced by Gondran and Minoux. To do this, we revisit the symmetrizati...

In this paper we give the first characterisation of graphs with linear clique-width at most 3, and we give a polynomial-time recognition algorithm for such graphs. In addition, we give a new characterisation of graphs with linear clique-width at most 2 and a new layout characterisation of linear clique-width in general. Among our results is also a decomposition scheme that preserves the linear ...

Drawing inspiration from the theory of linear “decomposable systems”, we provide a method, based on linear matrix inequalities (LMIs), which makes it possible to prove the convergence (or consensus) of a set of interacting agents with polynomial dynamic. We also show that the use of a generalised version of the famous Kalman-Yakubovic-Popov lemma allows the development of an LMI test whose size...

In this paper we present decomposable priors, a family of priors over structure and parameters of tree belief nets for which Bayesian learning with complete observations is tractable, in the sense that the posterior is also decomposable and can be completely determined ana­ lytically in polynomial time. This fol­ lows from two main results: First, we show that factored distributions over spanni...

We consider a general nonsymmetric second-order linear elliptic PDE in the framework of Lax-Milgram lemma. formulate and analyze an adaptive finite element algorithm with arbitrary polynomial degree that steers mesh-refinement inexact iterative solution arising systems. More precisely, solver employs, as outer loop, so-called Zarantonello iteration to symmetrize system and, inner uniformly cont...

Let g(x) be a fixed non-constant complex polynomial. It was conjectured by Schinzel that if g(h(x)) has boundedly many terms, then h(x) ∈ C[x] must also have boundedly many terms. Solving an older conjecture raised by Rényi and by Erdös, Schinzel had proved this in the special cases g(x) = x; however that method does not extend to the general case. Here we prove the full Schinzel’s conjecture (...