155 research outputs found

    On regular and new types of codes for location-domination

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    Identifying codes and locating-dominating codes have been designed for locating irregularities in sensor networks. In both cases, we can locate only one irregularity and cannot even detect multiple ones. To overcome this issue, self-identifying codes have been introduced which can locate one irregularity and detect multiple ones. In this paper, we define two new classes of locating-dominating codes which have similar properties. These new locating-dominating codes as well as the regular ones are then more closely studied in the rook’s graphs and binary Hamming spaces.In the rook’s graphs, we present optimal codes, i.e., codes with the smallest possible cardinalities, for regular location-domination as well as for the two new classes. In the binary Hamming spaces, we present lower bounds and constructions for the new classes of codes; in some cases, the constructions are optimal. Moreover, one of the obtained lower bounds improves the bound of Honkala et al. (2004) on codes for locating multiple irregularities.Besides studying the new classes of codes, we also present record-breaking constructions for regular locating-dominating codes. In particular, we present a locating-dominating code in the binary Hamming space of length 11 with 320 vertices improving the earlier bound of 352; the best known lower bound for such code is 309 by Honkala et al. (2004).</p

    On location, domination and information retrieval

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    The thesis is divided into two main branches: identifying and locatingdominating codes, and information retrieval. The former topics are motivated by the aim to locate objects in sensor networks (or other similar applications) and the latter one by the need to retrieve information in memories such as DNA data storage systems. Albeit the underlying applications, the study on these topics mainly belongs to discrete mathematics; more specically, to the elds of coding and graph theory. The sensor networks are usually represented by graphs where vertices represent the monitored locations and edges the connections between the locations. Moreover, the locations of the sensors are determined by a code. Furthermore, the desired properties of the sensor network are deeply linked with the properties of the underlying code. The number of errors in reading the data is abundant in the DNA data storage systems. In particular, there can occur more errors than a reasonable error-correcting code can handle. However, this problem is somewhat oset by the possibility to obtain multiple approximations of the same information from the data storage. Hence, the information retrieval process can be modelled by the Levenshtein's channel model, where a message is sent through multiple noisy channels and multiple outputs are received. In the rst two papers of the thesis, we introduce and study the new concepts of self- and solid-locating-dominating codes as a natural analogy to self-identifying codes with respect to locating-dominating codes. The rst paper introduces these new codes and considers them in some graphs such as the Hamming graphs. Then, in the second paper, we broaden our view on the topic by considering graph theoretical questions. We give optimal codes in multiple dierent graph classes and some more general results using concepts such as the Dilworth number and graph complements. The third paper focuses on the q-ary Hamming spaces. In particular, we disprove a conjecture proposed by Goddard and Wash related to identifying codes. In the fourth paper, we return to self- and solid-locating-dominating codes and give optimal codes in some graph classes and consider their densities in innite graphs. In the fth paper, we consider information retrieval in memories; in particular, the Levenshtein's channel model. In the channel model, we transmit some codeword belonging to the binary Hamming space through multiple identical channels. With the help of multiple dierent outputs, we give a list of codewords which may have been sent. In the paper, we study the number of channels required to have a rather small (constant) list size when the properties of the channels, the code and the dimension of the Hamming space are xed. In particular, we give an exact relation between the number of channels and the asymptotic value of the maximum list size.Väitöskirja käsittelee kahta aihetta: identioivia ja paikantavia peittokoodeja sekä tiedon noutamista muistista. Ensimmäisen aiheen motivaationa on objektien paikantaminen sensoriverkoista (sekä muut samankaltaiset sovellukset) ja jälkimmäisen tiedonnouto DNA-muisteista. Näiden aiheiden tutkimus kuuluu diskreettiin matematiikkaan, täsmällisemmin koodaus- ja graa-teoriaan. Sensoriverkkoja kuvataan yleensä graafeilla, joissa solmut esittävät tarkkailtuja kohteita ja viivat yhteyksiä näiden kohteiden välillä. Edelleen sensorien paikat määräytyvät annetun koodin perusteella. Tästä johtuen sensoriverkon halutut ominaisuudet pohjautuvat vahvasti alla olevaan koodiin. Luettaessa tietoa DNA-muisteista tapahtuvien virheiden määrä saattaa olla erittäin suuri; erityisesti suurempi kuin kiinnitetyn virheitä korjaavan koodin korjauskyky. Toisaalta tilanne ei ole aivan näin ongelmallinen, sillä DNA-muisteista voidaan saada useita eri arvioita muistiin tallennetusta tiedosta. Näistä syistä johtuen tietojen noutamista DNA-muisteista voidaan mallintaa käyttäen Levenshteinin kanavamallia. Kanavamallissa yksi viesti lähetetään useiden häiriöisten kanavien kautta ja näin vastaanotetaan useita viestejä (yksi jokaisesta kanavasta). Väitöskirjan kahdessa ensimmäisessä julkaisussa esitellään ja tutkitaan uusia paikantavien peittokoodien luokkia, jotka pohjautuvat aiemmin tutkittuihin itse-identioiviin koodeihin. Ensimmäisessä julkaisussa on esitelty nämä koodiluokat sekä tutkittu niitä joissain graafeissa kuten Hammingin graafeissa. Tämän jälkeen toisessa julkaisussa käsitellään yleisiä graa-teoreettisia kysymyksiä. Julkaisussa esitetään optimaaliset koodit useille graaperheille sekä joitain yleisempiä tuloksia käyttäen mm. Dilworthin lukua sekä graakomplementteja. Kolmas julkaisu keskittyy q-arisiin Hammingin avaruuksiin. Erityisesti julkaisussa todistetaan vääräksi Goddardin ja Washin aiemmin esittämä identioivia koodeja koskeva otaksuma. Neljäs artikkeli käsittelee jo kahdessa ensimmäisessä artikkelissa esiteltyjä paikantavien peittokoodien luokkia. Artikkeli esittää optimaalisia koodeja useille graaperheille sekä käsittelee äärettömiä graafeja. Viides artikkeli käsittelee tiedonnoutoa ja erityisesti Levenshteinin kanavamallia. Kanavamallissa binääriseen Hammingin avaruuteen kuuluva koodisana lähetetään useiden identtisten kanavien läpi. Näistä kanavista vastaanotetaan useita eri arvioita lähetetystä koodisanasta ja rakennetaan lista mahdollisesti lähetetyistä sanoista. Artikkelissa tutkitaan kuinka monta kanavaa tarvitaan, jotta tämän listan koko on pieni (vakio), kun kanavien ominaisuudet, koodi ja Hammingin avaruuden dimensio on kiinnitetty. Erityisesti löydetään täsmällinen suhde kanavien lukumäärän ja asymptoottisesti maksimaalisen listan koon välille

    The Structure and Properties of Clique Graphs of Regular Graphs

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    In the following thesis, the structure and properties of G and its clique graph clt (G) are analyzed for graphs G that are non-complete, regular with degree δ , and where every edge of G is contained in a t -clique. In a clique graph clt (G), all cliques of order t of the original graph G become the clique graph’s vertices, and the vertices of the clique graph are adjacent if and only if the corresponding cliques in the original graph have at least 1 vertex in common. This thesis mainly investigates if properties of regular graphs are carried over to clique graphs of regular graphs. In particular, the first question considered is whether the clique graph of a regular graph must also be regular. It is shown that while line graphs, cl2(G), of regular graphs are regular, the degree difference of the clique graph cl3(R) can be arbitrarily large using δ -regular graphs R with δ ≥ 3. Next, the question of whether a clique graph can have a large independent set is considered (independent sets in regular graphs can be composed of half the vertices in the graph at the most). In particular, the relation between the degree difference and the independence number of clt (G) will be analyzed. Lastly, we close with some further questions regarding clique graphs

    Graph Theory for the Secondary School Classroom.

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    After recognizing the beauty and the utility of Graph Theory in solving a variety of problems, the author decided that it would be a good idea to make the subject available for students earlier in their educational experience. In this thesis, the author developed four units in Graph Theory, namely Vertex Coloring, Minimum Spanning Tree, Domination, and Hamiltonian Paths and Cycles, which are appropriate for high school level

    Matchings and Covers in Hypergraphs

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    In this thesis, we study three variations of matching and covering problems in hypergraphs. The first is motivated by an old conjecture of Ryser which says that if \mcH is an rr-uniform, rr-partite hypergraph which does not have a matching of size at least ν+1\nu +1, then \mcH has a vertex cover of size at most (r1)ν(r-1)\nu. In particular, we examine the extremal hypergraphs for the r=3r=3 case of Ryser's conjecture. In 2014, Haxell, Narins, and Szab{\'{o}} characterized these 33-uniform, tripartite hypergraphs. Their work relies heavily on topological arguments and seems difficult to generalize. We reprove their characterization and significantly reduce the topological dependencies. Our proof starts by using topology to show that every 33-uniform, tripartite hypergraph has two matchings which interact with each other in a very restricted way. However, the remainder of the proof uses only elementary methods to show how the extremal hypergraphs are built around these two matchings. Our second motivational pillar is Tuza's conjecture from 1984. For graphs GG and HH, let νH(G)\nu_{H}(G) denote the size of a maximum collection of pairwise edge-disjoint copies of HH in GG and let τH(G)\tau_{H}(G) denote the minimum size of a set of edges which meets every copy of HH in GG. The conjecture is relevant to the case where H=K3H=K_{3} and says that τ(G)2ν(G)\tau_{\triangledown}(G) \leq 2\nu_{\triangledown}(G) for every graph GG. In 1998, Haxell and Kohayakawa proved that if GG is a tripartite graph, then τ(G)1.956ν(G)\tau_{\triangledown}(G) \leq 1.956\nu_{\triangledown}(G). We use similar techniques plus a topological result to show that τ(G)1.87ν(G)\tau_{\triangledown}(G) \leq 1.87\nu_{\triangledown}(G) for all tripartite graphs GG. We also examine a special subclass of tripartite graphs and use a simple network flow argument to prove that τ(H)=ν(H)\tau_{\triangledown}(H) = \nu_{\triangledown}(H) for all such graphs HH. We then look at the problem of packing and covering edge-disjoint K4K_{4}'s. Yuster proved that if a graph GG does not have a fractional packing of K4K_{4}'s of size bigger than ν(G)\nu_{\boxtimes}^{*}(G), then τ(G)4ν(G)\tau_{\boxtimes}(G) \leq 4\nu_{\boxtimes}^{*}(G). We give a complementary result to Yuster's for K4K_{4}'s: We show that every graph GG has a fractional cover of K4K_{4}'s of size at most 92ν(G)\frac{9}{2}\nu_{\boxtimes}(G). We also provide upper bounds on τ\tau_{\boxtimes} for several classes of graphs. Our final topic is a discussion of fractional stable matchings. Tan proved that every graph has a 12\frac{1}{2}-integral stable matching. We consider hypergraphs. There is a natural notion of fractional stable matching for hypergraphs, and we may ask whether an analogous result exists for this setting. We show this is not the case: Using a construction of Chung, F{\"{u}}redi, Garey, and Graham, we prove that, for all n \in \mbN, there is a 33-uniform hypergraph with preferences with a fractional stable matching that is unique and has denominators of size at least nn

    The potential of the comet competence diagnostic model for the assessment and development of occupational competence and commitment in technical vocational education and training

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    This study sets out to examine the potential of COMET occupational competence diagnostic model to assess and develop occupational competence in TVET. Holistic problem solving competence grounded in the COMET three-dimensional model, eight COMET criteria and Work Process Knowledge is a key element in this study. TVET is frequently seen as an instrument to address socio-economic and political challenges at a national and global level. The significance of a well-defined occupational competence assessment tool to advance quality assurance in TVET is accentuated by the high expectations of TVET to mitigate global concerns. Central to the debate around high youth unemployment is the issue of student occupational competence and workplace readiness to access the opportunities created by government policies and strategies. A mixed methodological approach consisting of qualitative and quantitative research methods was adopted. Cross-sectional data as well as elements of longitudinal data was collected from 715 participants comprising of students and artisans from six Public TVET Colleges, two Private TVET Institutions and three Industry Academies. The highest percentage of students (34,4%) ranged between ages 21-24. COMET Large-Scale Open-ended Test Tasks were conducted in measuring the occupational competence of TVET students to solve domain specific, complex problems, holistically. Four context questionnaires and semi-structured focus group interviews complete the assessment methods for data collection. COMET psychometric model and IBM SPSS version 22 were predominantly used to analyse the data collected. Findings of this study provide strong evidence that qualifications do not guarantee the development of occupational competence amongst students. The typical applied curriculum appears to be insufficient in preparing TVET students for the 21st century, modern work where reflective, divergent thinking and holistic problem solving competence has become the norm. 60.4% of students are at risk at Nominal Competence level and merely 11.4% achieved Holistic Shaping Competence. Functional Competence was achieved by 14.9% and 13.1% of students achieved Processual Competence. Students in the pilot dual system apprenticeship programme (DSAP) exposed to a real work situation, acquired higher occupation competence levels in comparison to sole College based peers. Stagnation in the development of occupational competence over three years of training is a major finding. Occupational commitment and motivation of students was high contrary to poor holistic problem solving competence. Arising from these findings, COMET occupational competence diagnostic model is proposed to position TVET to assess and develop value related occupational competence in TVET. This study supports COMET pedagogy of accumulative occupational competence in developing students from a novice to an expert skilled professional

    Roman Domination in Complementary Prism Graphs

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    A Roman domination function on a complementary prism graph GGc is a function f : V [ V c ! {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. The Roman domination number R(GGc) of a graph G = (V,E) is the minimum of Px2V [V c f(x) over such functions, where the complementary prism GGc of G is graph obtained from disjoint union of G and its complement Gc by adding edges of a perfect matching between corresponding vertices of G and Gc. In this paper, we have investigated few properties of R(GGc) and its relation with other parameters are obtaine
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