Obstruction characterization of co-TT graphs

Threshold tolerance graphs and their complement graphs ( known as co-TT graphs) were introduced by Monma, Reed and Trotter[24]. Introducing the concept of negative interval Hell et al.[19] defined signed-interval bigraphs/digraphs and have shown that they are equivalent to several seemingly different classes of bigraphs/digraphs. They have also shown that co-TT graphs are equivalent to symmetric signed-interval digraphs. In this paper we characterize signed-interval bigraphs and signed-interval graphs respectively in terms of their biadjacency matrices and adjacency matrices. Finally, based on the geometric representation of signed-interval graphs we have setteled the open problem of forbidden induced subgraph characterization of co-TT graphs posed by Monma, Reed and Trotter in the same paper.Comment: arXiv admin note: substantial text overlap with arXiv:2206.0591

Linear-Time Recognition of Double-Threshold Graphs

A graph G=(V, E) is a double-threshold graph if there exist a vertex-weight function w:V→ℝ and two real numbers lb, ub ∈ ℝ such that uv ∈ E if and only if lb ≤ w(u)+w(v) ≤ ub. In the literature, those graphs are studied also as the pairwise compatibility graphs that have stars as their underlying trees. We give a new characterization of double-threshold graphs that relates them to bipartite permutation graphs. Using the new characterization, we present a linear-time algorithm for recognizing double-threshold graphs. Prior to our work, the fastest known algorithm by Xiao and Nagamochi [Algorithmica 2020] ran in O(n³ m) time, where n and m are the numbers of vertices and edges, respectively

Lonesum (0,1)-matrices and poly-Bernoulli numbers of negative index

This thesis shows that the number of (0,1)-matrices with n rows and k columns uniquely reconstructible from their row and column sums are the poly-Bernoulli numbers of negative index, B[subscript n superscript ( -k)] . Two proofs of this main theorem are presented giving a combinatorial bijection between two poly-Bernoulli formula found in the literature. Next, some connections to Fermat are proved showing that for a positive integer n and prime number p B[subscript n superscript ( -p) congruent 2 superscript n (mod p),] and that for all positive integers (x, y, z, n) greater than two there exist no solutions to the equation: B[subscript x superscript ( -n)] + B[subscript y superscript ( -n)] = B[subscript z superscript ( -n)]. In addition directed graphs with sum reconstructible adjacency matrices are surveyed, and enumerations of similar (0,1)-matrix sets are given as an appendix

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]

Short Proofs For Interval Digraphs

. We give short proofs of the adjacency matrix characterizations of interval digraphs and unit interval digraphs. 1. INTRODUCTION An intersection representation of a graph assigns each vertex a set so that vertices are adjacent if and only if the corresponding sets intersect. Beineke and Zamfirescu [1] introduced the analogous concept of intersection digraph, under the name "connection digraph". Let f(S v ; T v )g be a collection of ordered pairs of sets indexed by a set V ; we call S v the source set and T v the sink set for v. The intersection digraph of this collection is the digraph with vertex set V having an edge from u to v if and only if Su " T v 6= Ø. The pairs of sets form an intersection representation. Harary, Kabell, and McMorris [3] defined an equivalent intersection model for bipartite graphs. Treating the partite sets as source vertices and sink vertices, we represent each vertex by one set and take the intersection graph, but we ignore intersections between source se..