Subclasses of Normal Helly Circular-Arc Graphs

A Helly circular-arc model M = (C,A) is a circle C together with a Helly family \A of arcs of C. If no arc is contained in any other, then M is a proper Helly circular-arc model, if every arc has the same length, then M is a unit Helly circular-arc model, and if there are no two arcs covering the circle, then M is a normal Helly circular-arc model. A Helly (resp. proper Helly, unit Helly, normal Helly) circular-arc graph is the intersection graph of the arcs of a Helly (resp. proper Helly, unit Helly, normal Helly) circular-arc model. In this article we study these subclasses of Helly circular-arc graphs. We show natural generalizations of several properties of (proper) interval graphs that hold for some of these Helly circular-arc subclasses. Next, we describe characterizations for the subclasses of Helly circular-arc graphs, including forbidden induced subgraphs characterizations. These characterizations lead to efficient algorithms for recognizing graphs within these classes. Finally, we show how do these classes of graphs relate with straight and round digraphs.Comment: 39 pages, 13 figures. A previous version of the paper (entitled Proper Helly Circular-Arc Graphs) appeared at WG'0

Interval Orders with Restrictions on the Interval Lengths

This thesis examines several classes of interval orders arising from restrictions on the permissible interval lengths. We first provide an accessible proof of the characterization theorem for the class of interval orders representable with lengths between 1 and k for each k in {1,2,...}. We then consider the interval orders representable with lengths exactly 1 and k for k in {0,1,...}. We characterize the class of interval orders representable with lengths 0 and 1, both structurally and algorithmically. To study the other classes in this family, we consider a related problem, in which each interval has a prescribed length. We derive a necessary and sufficient condition for an interval order to have a representation with a given set of prescribed lengths. Using this result, we provide a necessary condition for an interval order to have a representation with lengths 1 and 2

Patterns in permuted binary matrices

Reorganizing a dataset so that its hidden structure can be observed is useful in any data analysis task. For example, detecting a regularity in a dataset helps us to interpret the data, compress the data, and explain the processes behind the data. We study datasets that come in the form of binary matrices (tables with 0s and 1s). Our goal is to develop automatic methods that bring out certain patterns by permuting the rows and columns. We concentrate on the following patterns in binary matrices: consecutive-ones (C1P), simultaneous consecutive-ones (SC1P), nestedness, k-nestedness, and bandedness. These patterns reflect specific types of interplay and variation between the rows and columns, such as continuity and hierarchies. Furthermore, their combinatorial properties are interlinked, which helps us to develop the theory of binary matrices and efficient algorithms. Indeed, we can detect all these patterns in a binary matrix efficiently, that is, in polynomial time in the size of the matrix. Since real-world datasets often contain noise and errors, we rarely witness perfect patterns. Therefore we also need to assess how far an input matrix is from a pattern: we count the number of flips (from 0s to 1s or vice versa) needed to bring out the perfect pattern in the matrix. Unfortunately, for most patterns it is an NP-complete problem to find the minimum distance to a matrix that has the perfect pattern, which means that the existence of a polynomial-time algorithm is unlikely. To find patterns in datasets with noise, we need methods that are noise-tolerant and work in practical time with large datasets. The theory of binary matrices gives rise to robust heuristics that have good performance with synthetic data and discover easily interpretable structures in real-world datasets: dialectical variation in the spoken Finnish language, division of European locations by the hierarchies found in mammal occurrences, and co-occuring groups in network data. In addition to determining the distance from a dataset to a pattern, we need to determine whether the pattern is significant or a mere occurrence of a random chance. To this end, we use significance testing: we deem a dataset significant if it appears exceptional when compared to datasets generated from a certain null hypothesis. After detecting a significant pattern in a dataset, it is up to domain experts to interpret the results in the terms of the application.Aineiston uudelleenjärjestäminen paljastaa sen sisäisen rakenteen Elektroniset aineistot ovat usein suuria ja niiden sisältämät hahmot aluksi tuntemattomia, joten hahmojen löytämiseen tarvitaan tehokkaita tietokoneohjelmia. Hahmojen tunnistaminen auttaa kuvailemaan esimerkiksi nisäkäs- ja murresana-aineistojen sekä sosiaalisten verkostojen rakennetta. Parhaimmillaan tämä auttaa aineistoihin liittyvien tosimaailman ilmiöiden selittämisessä. Helsingin yliopistossa tarkastettava Esa Junttilan tietojenkäsittelytieteen alan väitöskirjatutkimus esittelee uusia automaattisia menetelmiä, jotka tunnistavat säännönmukaisuuksia aineistoissa. Uudet menetelmät perustuvat aineiston uudelleenjärjestämiseen, joka tuo aineiston sisältämän hahmon esiin. Aineistolla tarkoitetaan taulukkomuotoista dataa, joka sisältää vain ykkösiä ja nollia. Esimerkiksi ykköset nisäkkäiden levinneisyystaulukossa merkitsevät, että tietty nisäkäs elää tietyllä seudulla. Menetelmissä taulukon rivit ja sarakkeet järjestetään niin, että hahmo erottuu ihmisille mahdollisimman selvästi. Nisäkäsaineistolle sovellettuna kuvatut menetelmät voivat tuottaa esimerkiksi nisäkkäiden hierarkian, ryhmittymiä tai muun järjestyksen. Teoreettinen tarkastelu synnyttää hahmojen etsintään nopeita algoritmeja, jotka pystyvät käsittelemään tuhansia rivejä ja sarakkeita. Haasteena on menetelmien kyky sietää virheitä: esiintyvä hahmo on löydettävä silloinkin, kun aineiston laatu on kehno. Räätälöidyt tilastolliset testit kertovat lopulta löydetyn hahmon merkitsevyyden. Väittelijä on etsinyt kuvatuilla menetelmillä hahmoja esimerkiksi geneettisestä aineistosta, sosiaalisista verkostoista sekä nisäkkäiden, murresanojen ja fossiilien esiintymistä. Löydetty säännönmukaisuus vahvisti käsitystä tutkittujen aineistojen sisäisestä rakenteesta ja rohkaisee jatkotutkimuksiin vastaavilla tutkimusaloilla, kuten ekologiassa ja paleontologiassa. Esa Junttila väittelee matemaattis-luonnontieteellisessä tiedekunnassa 10.8.2011 kello 12 tietojenkäsittelytieteen alan tutkimuksellaan Patterns in Permuted Binary Matrices. Väitöstilaisuus järjestetään Yliopiston päärakennuksen salissa 13 (Fabianinkatu 33, 3. kerros). Vastaväittäjänä on professori Matti Nykänen (Itä-Suomen yliopisto) ja kustoksena professori Hannu Toivonen (Helsingin yliopisto). Lisätiedot: Esa Junttila, puhelin 040-8234987, [email protected]

Discrete Mathematics and Symmetry

Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group