Completion of the mixed unit interval graphs hierarchy

We describe the missing class of the hierarchy of mixed unit interval graphs, generated by the intersection graphs of closed, open and one type of half-open intervals of the real line. This class lies strictly between unit interval graphs and mixed unit interval graphs. We give a complete characterization of this new class, as well as quadratic-time algorithms that recognize graphs from this class and produce a corresponding interval representation if one exists. We also mention that the work in arXiv:1405.4247 directly extends to provide a quadratic-time algorithm to recognize the class of mixed unit interval graphs.Comment: 17 pages, 36 figures (three not numbered). v1 Accepted in the TAMC 2015 conference. The recognition algorithm is faster in v2. One graph was not listed in Theorem 7 of v1 of this paper v3 provides a proposition to recognize the mixed unit interval graphs in quadratic time. v4 is a lot cleare

Individual Professional Practice in the Company

Tato bakalářská práce popisuje mou praxi u společnosti EDIacademy, s.r.o. Společnost se zabývá tvorbou výukových aplikací pro žáky základních škol. Mým hlavním úkolem bylo vytvoření aplikace Zvířátka dědy Lesoně, která slouží k výuce matematiky. Práce popisuje technologie, které byly použity ve vývoji a řešení praktických problémů.This bachelor thesis describes my work experience at EDIacademy, s.r.o. The company focuses on creation of interactive teaching applications intended for pupils of elementary school. My main task was creating an application called Zvířátka dědy Lesoně, that assists teaching of mathematics. Contents of this thesis describe the technology used in development of this application and solutions to implementation challenges.460 - Katedra informatikyvýborn

String graphs. I. The number of critical nonstring graphs is infinite

AbstractString graphs (intersection graphs of curves in the plane) were originally studied in connection with RC-circuits. The family of string graphs is closed in the induced minor order, and so it is reasonable to study critical nonstring graphs (nonstring graphs such that all of their proper induced minors are string graphs). The question of whether there are infinitely many nonisomorphic critical nonstring graphs has been an open problem for some time. The main result of this paper settles this question. In a later paper of this series we show that recognizing string graphs is NP-hard