Dissertations / Theses on the topic 'Number stables'

Le calcul du nombre chromatique et la détermination d'une colo- ration optimale des sommets d'un graphe sont des problèmes NP- difficiles en général. Ils peuvent cependant être résolus en temps po- lynomial dans les graphes parfaits. Par ailleurs, la perfection d'un graphe peut être décidée efficacement. Les graphes parfaits sont caractérisés par la structure de leur poly- tope des stables : les facettes non-triviales sont définies exclusivement par des inégalités de cliques. Réciproquement, une structure similaire des facettes du polytope des stables détermine-t-elle des propriétés combinatoires et algorithmiques intéressantes? Un graphe est h-parfait si les facettes non-triviales de son polytope des stables sont définies par des inégalités de cliques et de circuits impairs. On ne connaît que peu de résultats analogues au cas des graphes parfaits pour la h-perfection, et on ne sait pas si les problèmes sont NP-difficiles. Par exemple, les complexités algorithmiques de la re- connaissance des graphes h-parfaits et du calcul de leur nombre chro- matique sont toujours ouvertes. Par ailleurs, on ne dispose pas de borne sur la différence entre le nombre chromatique et la taille maxi- mum d'une clique d'un graphe h-parfait. Dans cette thèse, nous montrons tout d'abord que les opérations de t-mineurs conservent la h-perfection (ce qui fournit une extension non triviale d'un résultat de Gerards et Shepherd pour la t-perfection). De plus, nous prouvons qu'elles préservent la propriété de décompo- sition entière du polytope des stables. Nous utilisons ce résultat pour répondre négativement à une question de Shepherd sur les graphes h-parfaits 3-colorables. L'étude des graphes minimalement h-imparfaits (relativement aux t-mineurs) est liée à la recherche d'une caractérisation co-NP com- binatoire de la h-perfection. Nous faisons l'inventaire des exemples connus de tels graphes, donnons une description de leur polytope des stables et énonçons plusieurs conjectures à leur propos. D'autre part, nous montrons que le nombre chromatique (pondéré) de certains graphes h-parfaits peut être obtenu efficacement en ar- rondissant sa relaxation fractionnaire à l'entier supérieur. Ce résultat implique notamment un nouveau cas d'une conjecture de Goldberg et Seymour sur la coloration d'arêtes. Enfin, nous présentons un nouveau paramètre de graphe associé aux facettes du polytope des couplages et l'utilisons pour donner un algorithme simple et efficace de reconnaissance des graphes h- parfaits dans la classe des graphes adjoints
Computing the chromatic number and finding an optimal coloring of a perfect graph can be done efficiently, whereas it is an NP-hard problem in general. Furthermore, testing perfection can be carried- out in polynomial-time. Perfect graphs are characterized by a minimal structure of their sta- ble set polytope: the non-trivial facets are defined by clique-inequalities only. Conversely, does a similar facet-structure for the stable set polytope imply nice combinatorial and algorithmic properties of the graph ? A graph is h-perfect if its stable set polytope is completely de- scribed by non-negativity, clique and odd-circuit inequalities. Statements analogous to the results on perfection are far from being understood for h-perfection, and negative results are missing. For ex- ample, testing h-perfection and determining the chromatic number of an h-perfect graph are unsolved. Besides, no upper bound is known on the gap between the chromatic and clique numbers of an h-perfect graph. Our first main result states that the operations of t-minors keep h- perfection (this is a non-trivial extension of a result of Gerards and Shepherd on t-perfect graphs). We show that it also keeps the Integer Decomposition Property of the stable set polytope, and use this to answer a question of Shepherd on 3-colorable h-perfect graphs in the negative. The study of minimally h-imperfect graphs with respect to t-minors may yield a combinatorial co-NP characterization of h-perfection. We review the currently known examples of such graphs, study their stable set polytope and state several conjectures on their structure. On the other hand, we show that the (weighted) chromatic number of certain h-perfect graphs can be obtained efficiently by rounding-up its fractional relaxation. This is related to conjectures of Goldberg and Seymour on edge-colorings. Finally, we introduce a new parameter on the complexity of the matching polytope and use it to give an efficient and elementary al- gorithm for testing h-perfection in line-graphs

Strongly stable ideals are important in algebraic geometry, commutative algebra, and combinatorics. Prompted, for example, by combinatorial approaches for studying Hilbert schemes and the existence of maximal total Betti numbers among saturated ideals with a given Hilbert polynomial, three algorithms are presented. Each of these algorithms produces all strongly stable ideals with some prescribed property: the saturated strongly stable ideals with a given Hilbert polynomial, the almost lexsegment ideals with a given Hilbert polynomial, and the saturated strongly stable ideals with a given Hilbert function. Bounds for the complexity of our algorithms are included. Also included are some applications for these algorithms and some estimates for counting strongly stable ideals with a fixed Hilbert polynomial.

The objective of this dissertation is to study the structure and behavior of the Atmospheric Boundary Layer (ABL) in stable conditions. This type of boundary layer is not completely well understood yet, although it is very important for many practical uses, from forecast modeling to atmospheric dispersion of pollutants. We analyzed data from the SABLES98 experiment (Stable Atmospheric Boundary Layer Experiment in Spain, 1998), and compared the behaviour of this data using Monin-Obukhov's similarity functions for wind speed and potential temperature. Analyzing the vertical profiles of various variables, in particular the thermal and momentum fluxes, we identified two main contrasting structures describing two different states of the SBL, a traditional and an upside-down boundary layer. We were able to determine the main features of these two states of the boundary layer in terms of vertical profiles of potential temperature and wind speed, turbulent kinetic energy and fluxes, studying the time series and vertical structure of the atmosphere for two separate nights in the dataset, taken as case studies. We also developed an original classification of the SBL, in order to separate the influence of mesoscale phenomena from turbulent behavior, using as parameters the wind speed and the gradient Richardson number. We then compared these two formulations, using the SABLES98 dataset, verifying their validity for different variables (wind speed and potential temperature, and their difference, at different heights) and with different stability parameters (zita or Rg). Despite these two classifications having completely different physical origins, we were able to find some common behavior, in particular under weak stability conditions.

This work develops optimal assembly sequences for modular building blocks. The underlying concept is that an automated device could take a virtual shape such as a CAD file, and automatically decide how to physically build the shape using simple, identical building blocks. This entails deciding where to place blocks inside the shape and generating an efficient assembly sequence that a robot could use to build the shape. The blocks are defined in a general, parameterized manner such that the model can be easily modified in the future. The primary focus of this work is the development of methods for generating assembly sequences in a time-feasible manner that ensure static stability at each step of the assembly. Most existing research focuses on complete enumeration of every possible assembly sequence and evaluation of many possible sequences. This, however, is not practical for systems with a large number of parts for two reasons: (1) the number of possible assembly sequences is exponential in the number of parts, and (2) each static stability test is very time-consuming. The approach proposed here is to develop a multi-hierarchical rule-based approach to assembly sequences. This is accomplished by formalizing and justifying both high-level and mid-level assembly rules based on static considerations. Application of these rules helps develop assembly sequences rapidly. The assembly sequence is developed in a time-feasible manner according to the geometry of the structure, rather than evaluating statics along the way. This work only evaluates the static stability of each step of the assembly once. The behavior of the various rules is observed both numerically and through theory, and guidelines are developed to suggest which rules to apply. A secondary focus of this work is to introduce methods by which the inside of the structure can be optimized. This structure optimization research is implemented by genetic algorithms that solve the multi-objective optimization problem in two dimensions, and can be extended to three dimensions.

We apply the semidefinite programming method to obtain a new upper bound on the cardinality of codes made of subspaces of a linear vector space over a finite field. Such codes are of interest in network coding.Next, with the same method, we prove an upper bound on the cardinality of sets avoiding one distance in the Johnson space, which is essentially Schrijver semidefinite program. This bound is used to improve existing results on the measurable chromatic number of the Euclidean space.We build a new hierarchy of semidefinite programs whose optimal values give upper bounds on the independence number of a graph. This hierarchy is based on matrices arising from simplicial complexes. We show some properties that our hierarchy shares with other classical ones. As an example, we show its application to the problem of determining the independence number of Paley graphs.

The Generalized Finite Element Method (GFEM) is essentially a partition of unity based method (PUM) that explores the Partition of Unity (PoU) concept to match a set of functions chosen to efficiently approximate the solution locally. Despite its well-known advantages, the method may present some drawbacks. For instance, increasing the approximation space through enrichment functions may introduce linear dependences in the solving system of equations, as well as the appearance of blending elements. To address the drawbacks pointed out above, some improved versions of the GFEM were developed. The Stable GFEM (SGFEM) is a first version hereby considered in which the GFEM enrichment functions are modified. The Higher Order SGFEM proposes an additional modification for generating the shape functions attached to the enriched patch. This research aims to present and numerically test these new versions recently proposed for the GFEM. In addition to highlighting its main features, some aspects about the numerical integration when using the higher order SGFEM, in particular are also addressed. Hence, a splitting rule of the quadrilateral element area, guided by the PoU definition itself is described in detail. The examples chosen for the numerical experiments consist of 2-D panels that present favorable geometries to explore the advantages of each method. Essentially, singular functions with good properties to approximate the solution near corner points and polynomial functions for approximating smooth solutions are examined. Moreover, a comparison among the conventional FEM and the methods herein described is made taking into consideration the scaled condition number and rates of convergence of the relative errors on displacements. Finally, the numerical experiments show that the Higher Order SGFEM is the more robust and reliable among the versions of the GFEM tested.
O Método dos Elementos Finitos Generalizados (MEFG) é essencialmente baseado no método da partição da unidade, que explora o conceito de partição da unidade para compatibilizar um conjunto de funções escolhidas para localmente aproximar de forma eficiente a solução. Apesar de suas vantagens bem conhecidas, o método pode apresentar algumas desvantagens. Por exemplo, o aumento do espaço de aproximação por meio das funções de enriquecimento pode introduzir dependências lineares no sistema de equações resolvente, assim como o aparecimento de elementos de mistura. Para contornar as desvantagens apontadas acima, algumas versões aprimoradas do MEFG foram desenvolvidas. O MEFG Estável é uma primeira versão aqui considerada na qual as funções de enriquecimento do MEFG são modificadas. O MEFG Estável de ordem superior propõe uma modificação adicional para a geração das funções de forma atreladas ao espaço enriquecido. Esta pesquisa visa apresentar e testar numericamente essas novas versões do MEFG recentemente propostas. Além de destacar suas principais características, alguns aspectos sobre a integração numérica quando usado o MEFG Estável de ordem superior, em particular, são também abordados. Por exemplo, detalha-se uma regra de divisão da área do elemento quadrilateral, guiada pela própria definição de sua partição da unidade. Os exemplos escolhidos para os experimentos numéricos consistem em chapas com geometrias favoráveis para explorar as vantagens de cada método. Essencialmente, examinam-se funções singulares com boas propriedades de aproximar a solução nas vizinhanças de vértices de cantos, bem como funções polinomiais para aproximar soluções suaves. Ademais, uma comparação entre o MEF convencional e os métodos aqui descritos é feita levando-se em consideração o número de condição do sistema escalonado e as razões de convergência do erro relativo em deslocamento. Finalmente, os experimentos numéricos mostram que o MEFG Estável de ordem superior é a mais robusta e confiável entre as versões do MEFG testadas.

Evapotranspiration is a nexus for planetary energy and carbon cycles, as yet poorly constrained. Here I use stable isotopes of oxygen and hydrogen to partition flux of water due to plant transpiration from the direct evaporative flux from soils, water bodies and plant. The study areas, Langat and Kelantan watersheds represent examples of domains dominated by the respective Southwest and Northeast monsoons on the two sides of the main orographic barrier (Titiwangsa mountain range). Mean annual rainfall for the Langat watershed, obtained from 30 years of hydrological data, is 2145 ± 237 mm. Tentatively, 48% of this precipitation returns to the atmosphere via transpiration (T), with 33% partitioned into discharge (Q), 8% into interception (In), and 11% into evaporation (Ed). In the Kelantan watershed, the mean annual rainfall, also based on the 30 year hydrological data, is 2383 ± 120 mm. Similar to Langat, the T accounts for 43% of precipitation (P), 45% is discharged into South China Sea (Q), 12% partitioned into interception (In) and tentatively 0% for evaporation (Ed). Ed for the Langat watershed represents only a small proportion in terms of volumetric significance, up to almost ~11% with strong effect on the isotopic fingerprints of waters associated with the summer Southwest Monsoon (SWM). Note, however, that insignificant Ed for the Kelantan watershed may be an artefact of rain and river water sampling at only coastal downstream portion of the watershed. High humidity (80%) also was recorded for the Malaysian Peninsula watershed. T appropriates about half of all solar energy absorbed by the continents, here ~1000*103 g H2O m-2 yr-1 similar to other tropical regions at 900-1200*103 g H2O m-2 yr-1. The associated carbon fluxes are ~ 1300 g C m-2yr-1, independent of P. Vegetation responses to solar irradiance, via T and photosynthesis reflects the importance of stomatal regulation of the water and carbon fluxes. In order to maintain high transpiration in the tropical region, “constant” water supply is required for continuous pumping of water that delivers nutrients to the plant, suggesting that water and carbon cycle are co-driven by the energy of the sun. The existence of the water conveyor belt may be precondition for nutrient delivery, hence operation of the carbon cycle. Potentially, this may change our perspective on the role that biology plays in the water cycle. In such perspective, the global water cycle is the medium that redistributes the incoming solar energy across the planet, and the anatomical structures of plants then help to optimize the loop of energy transfer via evaporation and precipitation in the hydrologic cycle. The main features of aquatic geochemistry of the Langat and Kelantan rivers inferred from the Principal Component Analysis are controlled by three components that explain 80% and 82% of total variances. These components are reflecting of the geogenic factor with superimposed pollution, the latter particularly pronounced in urbanized sections of the Langat river and dominant in downstream of the Kelantan river. There is no correlation between seasonal variations in major ion chemistry and environmental variables such as precipitation, discharge, temperature or solar activity.

This thesis focuses mainly on the Maximum Independent Set (MIS) problem. Some related graph theoretical combinatorial problems are also considered. As these problems are generally NP-hard, we study their complexity in hereditary graph classes, i.e. graph classes defined by a set F of forbidden induced subgraphs. We revise the literature about the issue, for example complexity results, applications, and techniques tackling the problem. Through considering some general approach, we exhibit several cases where the problem admits a polynomial-time solution. More specifically, we present polynomial-time algorithms for the MIS problem in: + some subclasses of $S_{2;j;k}$-free graphs (thus generalizing the classical result for $S_{1;2;k}$-free graphs); + some subclasses of $tree_{k}$-free graphs (thus generalizing the classical results for subclasses of P5-free graphs); + some subclasses of $P_{7}$-free graphs and $S_{2;2;2}$-free graphs; and various subclasses of graphs of bounded maximum degree, for example subcubic graphs. Our algorithms are based on various approaches. In particular, we characterize augmenting graphs in a subclass of $S_{2;k;k}$-free graphs and a subclass of $S_{2;2;5}$-free graphs. These characterizations are partly based on extensions of the concept of redundant set [125]. We also propose methods finding augmenting chains, an extension of the method in [99], and finding augmenting trees, an extension of the methods in [125]. We apply the augmenting vertex technique, originally used for $P_{5}$-free graphs or banner-free graphs, for some more general graph classes. We consider a general graph theoretical combinatorial problem, the so-called Maximum -Set problem. Two special cases of this problem, the so-called Maximum F-(Strongly) Independent Subgraph and Maximum F-Induced Subgraph, where F is a connected graph set, are considered. The complexity of the Maximum F-(Strongly) Independent Subgraph problem is revised and the NP-hardness of the Maximum F-Induced Subgraph problem is proved. We also extend the augmenting approach to apply it for the general Maximum Π -Set problem. We revise on classical graph transformations and give two unified views based on pseudo-boolean functions and αff-redundant vertex. We also make extensive uses of α-redundant vertices, originally mainly used for $P_{5}$-free graphs, to give polynomial solutions for some subclasses of $S_{2;2;2}$-free graphs and $tree_{k}$-free graphs. We consider some classical sequential greedy heuristic methods. We also combine classical algorithms with αff-redundant vertices to have new strategies of choosing the next vertex in greedy methods. Some aspects of the algorithms, for example forbidden induced subgraph sets and worst case results, are also considered. Finally, we restrict our attention on graphs of bounded maximum degree and subcubic graphs. Then by using some techniques, for example ff-redundant vertex, clique separator, and arguments based on distance, we general these results for some subclasses of $S_{i;j;k}$-free subcubic graphs.

Thesis (M.S.)--Worcester Polytechnic Institute.
Keywords: crown; splash; spreading; oscillatory; droplets; microgravity; viscosity; map; stability; solid surface; surface tension; gravity. Includes bibliographical references (p. 111-113).

碩士
國立交通大學
應用數學系
87
This work investigates the global mosaic patterns and computes the number of stable equilibria associate to Dirichlet , Neumann and Periodic boundary conditions respectly in one dimensional Cellular Neural Network (CNN) , we will demonstrate a general method for calculating the number as above and show you how we generalized what Patrick.Thiran approached in 1997.

A maximum stable set is a stable set with the largest possible size, for a given graph G. This size is called the stability number of G, and it is denoted α(G). The problem of determining the stability number of an arbitrary graph, is a NP-complete optimization problem. As such, it is unlikely that there is a polynomial algorithm for finding a maximum stable set of a graph. The main purpose of this thesis is the achievement of recognition algorithms for graphs with convex quadratic stability number that are graphs whose stability number is equal to the optimal value of a convex quadratic program associated to the corresponding adjacency matrix . For that, results that relate the eigenvalues of the adjacency matrix and maximum stable sets are established and recognition algorithms are derived from those results. Such algorithms are applied to several well known problems such as efficient domination and the determination of graphs with perfect matchings and Hamiltonian cycles.
Um conjunto estável máximo num grafo G é um conjunto estável com cardinalidade máxima. A cardinalidade de um conjunto estável máximo chama-se número de estabilidade do grafo e denota-se α(G). O problema da determinação do número de estabilidade de um grafo arbitrário é um problema de optimização NP-completo e, como tal, não se conhecem algoritmos polinomiais capazes dessa determinação. O objectivo desta tese é a construção de algoritmos de reconhecimento para grafos com número de estabilidade quadrático convexo, que são grafos cujo número de estabilidade é igual ao valor óptimo de um programa quadrático convexo associado à respectiva matriz de adjacência. Com esse objectivo, apresentam-se resultados que relacionam os valores próprios da matriz de adjacência com a existência de estáveis máximos e descrevem-se algoritmos de reconhecimento baseados em tais resultados. Os algoritmos são posteriormente aplicados a vários problemas clássicos como o da dominação eficiente e da existência de emparelhamentos perfeitos e de ciclos de Hamilton.
Programa Doutoral em Matemática

碩士
國立交通大學
環境工程系所
104
In this study, a Shaker-Fluidized bed-Atomizer disperser (SFA) powder dispersion system consist of an atomizer, powder storage container, supporting base and orbital shaker with 0.5 L/min dispersion flow rate and 35 psi compressed air was used to generate long-term stable concentration and well-dispersed multi-wall carbon nanotube (MWCNT) and purified multi-wall carbon nanotube (P-MWCNT). The physicochemical properties both of MWCNT and P-MWCNT dispersed by SFA such as number distribution, mass distribution, morphology, real-time mass concentration and surface area concentration were measured before animal inhalation exposure test. In addition, the purity and functional groups of MWCNT and P-MWCNT were also measured. The result of mass distribution shows bimodal distribution with 0.26、3.05 µm and 0.18、1.28 µm of MMAD for MWCNT and P-MWCNT, respectively. Also the carbonyl group (-COOH) appeared on the surface of P-MWCNT after purification. To investigate biological effects of MWCNT and P-MWCNT on animal, the SFA dispersion system was combined with Nanoparticle Exposure Chamber System (NECS) in NHRI to conduct whole-body inhalation exposure at high dose MWCNT (MWCNT-HD), low dose MWCNT (MWCNT-LD) and low dose P-MWCNT (P-MWCNT-LD) exposure groups for 4 hour per day and 5 days per test. Results indicated that the severity of pulmonary function impairment from high to low was MWCNT-HD, MWCNT-LD and P-MWCNT-LD. The results of histopathology also showed obvious inflammation, alveolar wall thicken and interstitial pneumonia-like histopathological outcomes caused by MWCNT-HD exposure. Due to better dispersion, P-MWCNT showed the highest deposition amounts in lung but did not present the most serious biological responses. The study proposed protein corona adsorbed on P-MWCNT eliminating acute phase biological responses and the short-term observation after one-week exposure was not able to present chronic effects.

With increasing application of DNA transfer technology in human gene therapy and heterologous protein production, various methods of introducing DNA into ex vivo or in vivo cells have been developed in a wide variety of cells. Recently, growing interest has focused on insect cells for production of heterologous proteins. Insect expression systems have been recognized as an easy and rapid system for production of a large quantity of recombinant or therapeutic proteins. Liposome-mediated transfection is an effective non-viral delivery method. However, optimization is essential to achieve maximal production of the heterologous proteins. The goal of this project is to optimize the transgene expression in Sf9 insect cells using lipofection. Transfection variables involved in the efficiency of transgene uptake by cells were examined for optimization. The type of cationic liposomes and the liposome to DNA ratios were the most significant determinants of efficient DNA uptake. The transient expression of the B-galactosidase reporter gene could be improved by 7.8 fold by optimization. Southern blot analyses revealed that the integrated plasmids were predominantly arranged into head-to-tail concatemers in all the stably transformed cell lines except for one. The concatemeric integration was speculated to occur by extrachromosomal homologous recombination between plasmid DNAs and a subsequent illegitimate recombination into the chromosomal DNA by a double-strand break repair pathway. Up to a 27-fold difference in the integrated copy number was observed among 19 stably transformed cell lines by quantitative southern blot analyses. The reporter gene expression in stably transformed cells could be correlated with the integrated copy number in general. In addition to the copy number, the positional effect of the integration loci is likely to play a role in controlling the expression level. A large amount of DNA uptake reflected on the high transient expression is necessary, but not sufficient for high expression in stable cells. However, there appeared to be higher probability of obtaining high-producer cell lines with high transient expression. Therefore, the use of a lipofection protocol, producing high transient expression, is advantageous in producing high producer stable cell lines.

This thesis focuses mainly on the Maximum Independent Set (MIS) problem. Some related graph theoretical combinatorial problems are also considered. As these problems are generally NP-hard, we study their complexity in hereditary graph classes, i.e. graph classes defined by a set F of forbidden induced subgraphs. We revise the literature about the issue, for example complexity results, applications, and techniques tackling the problem. Through considering some general approach, we exhibit several cases where the problem admits a polynomial-time solution. More specifically, we present polynomial-time algorithms for the MIS problem in: + some subclasses of $S_{2;j;k}$-free graphs (thus generalizing the classical result for $S_{1;2;k}$-free graphs); + some subclasses of $tree_{k}$-free graphs (thus generalizing the classical results for subclasses of P5-free graphs); + some subclasses of $P_{7}$-free graphs and $S_{2;2;2}$-free graphs; and various subclasses of graphs of bounded maximum degree, for example subcubic graphs. Our algorithms are based on various approaches. In particular, we characterize augmenting graphs in a subclass of $S_{2;k;k}$-free graphs and a subclass of $S_{2;2;5}$-free graphs. These characterizations are partly based on extensions of the concept of redundant set [125]. We also propose methods finding augmenting chains, an extension of the method in [99], and finding augmenting trees, an extension of the methods in [125]. We apply the augmenting vertex technique, originally used for $P_{5}$-free graphs or banner-free graphs, for some more general graph classes. We consider a general graph theoretical combinatorial problem, the so-called Maximum -Set problem. Two special cases of this problem, the so-called Maximum F-(Strongly) Independent Subgraph and Maximum F-Induced Subgraph, where F is a connected graph set, are considered. The complexity of the Maximum F-(Strongly) Independent Subgraph problem is revised and the NP-hardness of the Maximum F-Induced Subgraph problem is proved. We also extend the augmenting approach to apply it for the general Maximum Π -Set problem. We revise on classical graph transformations and give two unified views based on pseudo-boolean functions and αff-redundant vertex. We also make extensive uses of α-redundant vertices, originally mainly used for $P_{5}$-free graphs, to give polynomial solutions for some subclasses of $S_{2;2;2}$-free graphs and $tree_{k}$-free graphs. We consider some classical sequential greedy heuristic methods. We also combine classical algorithms with αff-redundant vertices to have new strategies of choosing the next vertex in greedy methods. Some aspects of the algorithms, for example forbidden induced subgraph sets and worst case results, are also considered. Finally, we restrict our attention on graphs of bounded maximum degree and subcubic graphs. Then by using some techniques, for example ff-redundant vertex, clique separator, and arguments based on distance, we general these results for some subclasses of $S_{i;j;k}$-free subcubic graphs.