Figure 1: The two non-trivial **perfect** **graphs** dealt with in Table 1: the first has a BSP, the second one does not.
2. Definitions
We first need to introduce trigraphs: this is a generalization of **graphs** where a new kind of adjacency between vertices is defined: the semi-adjacency. The intuitive meaning of a pair of semi-adjacent vertices, also called a switchable pair, is that in some situations, the vertices are considered as adjacent, and in some other situations, they are considered as non-adjacent. This implies to be very careful about terminology, for example in a trigraph two vertices are said adjacent if there is a “real” edge between them but also if they are semi-adjacent. What if we want to speak about “really adjacent” vertices, in the old-fashioned way? The dedicated terminology is strongly adjacent, adapted to strong neighborhood, strong clique and so on.

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union of G and H is a graph G ∪ H whose vertex set is V (G) ∪ V (H) and
whose edge set is E(G) ∪ E(H). The disjoint union is clearly an associative operation, and for each nonnegative integer t we will denote by tG the disjoint union of t copies of G. The join of G and H is a graph G+H whose vertex set is V (G)∪V (H) and whose edge set is E(G)∪E(H)∪{vw : v ∈ V (G), w ∈ V (H)}. A graph is bipartite if its vertex set can be partitioned into two (possibly empty) stable sets. A graph is chordal if every cycle of length at least 4 has at least one chord. A comparability graph is a graph that admits a transitive acyclic orientation of its edges. Bipartite and chordal **graphs** can be recognized in linear time and comparability **graphs** can be recognized in polynomial time [65, 70]. Bipartite, chordal and comparability **graphs** are subclasses of **perfect** **graphs**.

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Trigraphs were introduced by one of us in [3], because of difficulties that arose from the decom- position theorem for **perfect** **graphs**. In particular, for some of the decompositions used in [5], the graph is best viewed as being decomposed into smaller trigraphs rather than into smaller **graphs**; and trigraphs naturally arise in the context of decomposition theorems for Berge **graphs**. Thus no doubt we will eventually need an algorithm to find a balanced skew partition in a Berge trigraph; and so let us sketch here what needs to be modified to make the algorithms of this paper work for trigraphs.

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free **graphs** is polynomial-time solvable, and NP-hard or co-NP-hard otherwise. However, for the other ﬁve variants we still have a number of missing cases to solve.
Finally, we aim to determine a dichotomy with respect to H-free **graphs** for the variant (π ∈ {α, ω, χ}), where S consists of other graph operations, for instance when S consists of an edge deletion. This variant has been less studied than the vertex deletion and edge contraction variant. The reason for this is that most classes of H-free **graphs** are not closed under edge deletion, whereas such classes are closed under vertex deletion, and in the case when H is a linear forest, under edge contraction as well. For π = χ we are close to a dichotomy. Recall that an edge of a graph is critical or contraction-critical if its deletion or contraction, respectively, reduces the chromatic number of G by 1. It is known that an edge is contraction-critical if and only if it is critical [36]. Hence by Theorem 25 we only need to consider the cases where H ⊆ i P 4 or H ⊆ i P 1 ⊕ P 3 . Bazgan et al. [2] showed that

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Strongly **perfect** **graphs** have been studied by several authors (e.g. Berge and Duchet [ 1 ], Ravindra [ 12 ], Wang [ 14 ]). In a series of two papers, the current paper being the first one, we investigate a fractional relaxation of strong perfection. Motivated by a wireless networking problem, we consider claw-free **graphs** that are fractionally strongly **perfect** in the complement. We obtain a forbidden induced subgraph characterization and display graph-theoretic properties of such **graphs**. It turns out that the forbidden induced subgraphs that characterize claw-free that are fractionally strongly **perfect** in the complement are precisely the cycle of length 6, all cycles of length at least 8, four particular **graphs**, and a collection of **graphs** that are constructed by taking two **graphs**, each a copy of one of three particular **graphs**, and joining them in a certain way by a path of arbitrary length. Wang [ 14 ] gave a characterization of strongly **perfect** claw- free **graphs**. As a corollary of the results in this paper, we obtain a characterization of claw-free **graphs** whose complements are strongly **perfect**.

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imperfect line **graphs** (Section 3). Since cliques and odd holes are clearly near-bipartite, Corollary 9 shows that Conjecture 1 is true for line **graphs**.
In the following, we will discuss a reformulation of Conjecture 1.
As superclass of a-**perfect** **graphs**, joined a-**perfect** **graphs** were introduced in [7] as those **graphs** whose only facet-defining subgraphs are complete joins of a clique and prime antiwebs (an antiweb A k n is prime if k + 1 and n are relatively prime integers). The inequalities obtained from complete joins of antiwebs, called joined antiweb constraints, are of the form

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2 2-sided near-triangulations
In this paper we consider plane **graphs** without loops nor multiple edges. In a plane graph there is an infinite face, called the outer face, and the other faces are called inner faces. A near-triangulation is a plane graph such that every inner face is a triangle. In a plane graph G, a chord is an edge not incident to the outer face but that links two vertices of the outer face. A separating triangle of G is a cycle of length three such that both regions delimited by this cycle (the inner and the outer region) contain some vertices. It is well known that a triangulation is 4-connected if and only if it contains no separating triangle. Given a vertex v on the outer face, the inner-neighbors of v are the neighbors of v that are not on the outer face. We define here 2-sided near-triangulations (see Figure 1) whose structure will be useful in the inductions of the proofs of Theorem 2, and Theorem 7.

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1.3 From undirected **graphs** to oriented **graphs**
To avoid any confusion, let us recall that an orientation D of an undirected graph G is obtained when every edge uv of G is oriented either from u to v (resulting in the arc (u, v)) or conversely (resulting in the arc (v, u)). An oriented graph D is a directed graph that is an orientation of a simple graph. Note that when G is simple, D cannot have two vertices u, v such that (u, v) and (v, u) are arcs. Such symmetric arcs are allowed in digraphs, which is the main dierence between oriented **graphs** and digraphs. Throughout this paper, when simply referring to a graph, we mean an undirected graph.

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Clearly, MD( P ) = 1; however, the orientation D of P obtained by making every vertex v 2k+1
( k = 0, ..., n − 1) become a source (i.e., orienting its incident edges away) verifies MD(D) = n. As shown in this paper, this phenomenon occurs for strong orientations as well.
In [4], the authors proved that, for every positive integer k, there exist infinitely many **graphs** for which the metric dimension of any of its strongly-connected orientations is exactly k. They have also proved that there is no constant k such that the metric dimension of any tournament is at most k.

5 Comparison with the classical **perfect** sampler
As discussed earlier, our new algorithm allows for **perfect** sampling of Jackson networks with finite and infinite buffers. This was not possible with the classical **perfect** sampling algorithms that requires a finite state space. As such, our algorithm broadens the scope of **perfect** sampling techniques. In addition, this new approach also provides time improvements in the case where all buffers are finite, because the time complexity of the new approach is essentially independent of the capacities. In that case, it reduces the sampling time by a factor corresponding to the ratio between the maximum capacity of the buffers over the expected size of the queues under the stationary law of the bounding process, at least in the acyclic case (the comparison in the cyclic case is not as striking because of the squared term but behaves essentially in the same way).

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k 3.
The reasons for this diculty are that **perfect** sampling techniques, while not requiring monotonicity structures in the Markov transition, work better under such an assumption, and that exhibiting such monotonicity in the mixture model requires hard work. One of the key features of Hobert et al.' (1999) solution, along with the specic representation of the Dirichlet distribution in terms of basic exponential random variables, is to exploit the Duality Principle established by Diebolt and Robert (1994) for latent variable models.

(1 + i,j z i z j ) i,j , (2.19)
where i,j = 1 if (w i , w j ) ∈ {(≺, 0 ), ( ≺ 0 , )} and i,j = −1 otherwise. 3 Bijective sampling of Schur processes
Exact sampling algorithms based on partition processes for classes of tilings including plane and skew plane partitions [2] (see also [4] for a general approach and [3] for an application of this approach to a restricted class of **graphs** including, but not limited to, the Aztec diamond graph) and Aztec diamonds (see [12] and remark below) have been proposed before. There is also the coupling from the past approach of Propp and Wilson [26]. In this section we give an overall encompassing algorithm for Schur processes (different from, but similar to, that of [2]) which under appropriate specializations gives exact random sampling of plane (and skew plane) partitions, Aztec diamonds, pyramid partitions and more generally, steep tilings.

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PROCESSES
L. DECREUSEFOND, I. FLINT, AND K.C. LOW
Abstract. Determinantal point processes (DPP) serve as a practicable mod- eling for many applications of repulsive point processes. A known approach for simulation was proposed in [5], which generate the desired distribution point wise through rejection sampling. Unfortunately, the size of rejection could be very large. In this paper, we investigate the application of **perfect** simulation via coupling from the past (CFTP) on DPP. We give a general framework for **perfect** simulation on DPP model. It is shown that the limiting sequence of the time-to-coalescence of the coupling is bounded by

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Keywords — non-linear stiffness modeling, parallel manipula-
tors, compliance errors, non-**perfect** manipulators.
I. I NTRODUCTION
N modern industrial robotics, stiffness becomes one of the most important performance measures that defines poten- tial accuracy of the manipulator. This problem has been the focus of numerous works [1-5], where different solutions for serial and parallel manipulators have been proposed assum- ing that the manipulator geometry perfectly corresponds to the nominal one. However in practice, parallel manipulators are usually composed of non-**perfect** serial chains, whose geometrical parameters differ from the nominal values. It is evident that these manufacturing errors may generate essen- tial internal forces and have effects on the manipulator stiff- ness behavior. However, this problem has attracted very li- mited attention in robotics.

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Figure 9: Reflection |R| against θ for a perforated rigid film, e/h = 0.1 (ϕ = 0.5 and kh = 1). Same representation as in figure 4.
5. Conclusion
We have studied the scattering properties of perforated films with a focus on thin films. The conditions under which **perfect** transmissions are possible are modified when ultrathin devices are considered; in particular an extraordinary transmission observed for a thick film can disappear when reducing its thickness, and the reverse situation is possible as illustrated at the begining of this paper in figure 2. In all cases, a shift in the Brewster incidence θ ∗ has been exhibited : for film materials lighter than that of the surrounding matrix, θ ∗ increases with the thickness up to θ B , for film material heavier, θ ∗ decreases with the

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In this section, we collect some data plots of our computations of **perfect** unary forms. We explain these results by relating unary forms to continued fractions in Section 5.
We first plot the number of forms by discriminant. See Figure 4. Notice that on average there are far fewer **perfect** forms for discriminants that are D = 4d, where d ≡ 2, 3 mod 4. Because of this, we compute more examples and parameterize by d. Indeed the different cases are much more similar when viewed as functions of d. See Figure 4. The **perfect** forms for F = Q( √ d) are computed for d ≤ 200000. This consists of 10,732,735 **perfect** forms divided among 121,580 fields.

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over κ X ⊗ k κ op
X i . Note that freely irreducible morphisms are particular cases of strongly irreducible
morphisms introduced in [5]. In particular, when X is indecomposable, f is freely irreducible if the division algebra EndA(X)/rad(X, X) is trivial. When k is an algebraically closed field, the irreducible morphism in Theorem B is automatically freely irreducible. In this setting, Statements (1) and (2) of Theorem B where proved in [4] as well as the particular case ` = n of Statement (3). Actually, it is possible to derive from the above theorem extensions of the results in [4] (there, k is an algebraically closed field) to finite dimensional algebras over **perfect** fields. For instance

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We study lattices, in Euclidean space, whose minimal vectors form a spherical 4-design. Such lattices are called strongly **perfect** lattices. They form an interesting class of lattices. In particular strongly **perfect** lattices are **perfect** lattices in the sense of Voronoi (cf. [Mar]). In small dimensions there are only few such lattices and a complete classification was known

Secondly, the congruence in b), (in which we may think of the modu- lus x 2 + x + 1 as the analogue of the number 3 ∈ Z ) seems insufficient to bar polynomials from being **perfect**. This is in contrast to the case of integers. Indeed, already in 1937, Steuerwald [6] proved by a clever use of congruences modulo 3 that there are no odd **perfect** numbers of the form n = p 4k+1 p 2

High-R&D pooling equilibria are also characterized by a signicant R&D activity, but the process of granting patents cannot rely on self-screening and only rests on the imperfect exam- ination. So, some obvious projects obtain unwarranted patent protection that is, they would not be patented given the standard σ ∗ if the examination technology were **perfect**. A marginal increase in R&D induces a shift in the probability of getting a patent from h(1) for obvious projects to 1 for non-obvious ones. In equilibrium, marginal benet equals marginal cost of R&D: γ 0 (π P ) = v(1 − h(1)) , provided the non-obviousness standard is low enough for such an eort to be more protable than simply not investing at all in R&D. Moreover, the application fee has to be small as well, so that applying for protection is attractive even for obvious projects. Therefore, pooling equilibria exist within the domain D P ≡ {(σ ∗ , f ) ∈ R 2 + ; a(σ ∗ ) ≤ v(1 − h(1)), f ≤ h(1)v)} of low application fees and low non-obviousness standard (see Figure 1).

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