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1.
Nekatere s pakiranji povezane lastnosti grafov
Dragana Božović, 2020, doctoral dissertation

Abstract: V disertaciji se ukvarjamo z različnimi problemi, povezanimi s pakiranji. Disertacija je sestavljena iz štirih delov. Prvi del je namenjen grafom, ki imajo enolično pakirno množico največje moči. Najprej predstavimo nekatere lastnosti teh grafov. Nato podamo še dve karakterizaciji dreves z enolično pakirno množico. V drugem delu vpeljemo pojem dimenzije incidenčnosti, ki je neposredno povezana z 2-pakirnim številom grafa, in določimo formulo za njen izračun. Dokažemo, da je problem iskanja incidenčne dimenzije grafa v splošnem NP-poln. Tretji del namenimo pakirnemu kromatičnemu številu leksikografskega produkta grafov. Določimo njegovo spodnjo in zgornjo mejo ter izboljšano zgornjo mejo za primer, ko je prvi faktor v produktu izomorfen poti. V zadnjem delu se posvetimo učinkoviti odprti dominaciji produktov digrafov. Okarakteriziramo učinkovito odprto dominirane direktne in leksikografske produkte digrafov. Pri kartezičnem produktu okarakteriziramo tiste, kjer je prvi faktor usmerjena pot, usmerjen cikel ali zvezda z enim izvorom. Predstavimo tudi karakterizacijo učinkovito odprto dominiranega krepkega produkta, katerega temeljni graf obeh faktorjev je monocikličen graf.
Keywords: pakirna množica, enolično največje pakiranje, dimenzija incidenčnosti, generator incidenčnosti, pakirno kromatično število, leksikografski produkt grafov, učinkovita odprta dominacija, usmerjeni grafi, produkti usmerjenih grafov
Published: 27.11.2020; Views: 231; Downloads: 59
.pdf Full text (753,30 KB)

2.
Diskretne strukture
Iztok Peterin, 2020

Abstract: V učbeniku so predstavljene nekatere veje diskretne matematike, ki so še posebej uporabne v računalništvu. Tako se sprehodimo skozi logiko, s posebnim poudarkom na dokazu. Sledijo teorije, pri katerih igra poglavitno vlogo matematična indukcija oziroma bolj splošno induktivna posplošitev. Spoznamo osnove kombinatorike in teorije števil. Predstavljene so rekurzivne relacije, s katerimi lahko opišemo ponavljajoče se procese. To nam omogoča tudi vrednotenje algoritmov glede na čas potreben za njegovo izvedbo. Relacije, ki so podmnožice kartezičnega produkta poljubnih množic, predstavljajo širok vir presenetljivih rezultatov. Eden izmed njih rezultira v mrežah in njihovih posebnih predstavnikih Booleovih algebrah. Končamo z grafi, ki predstavljajo neverjetno uporaben matematični model za simuliranje procesov iz realnega življenja.
Keywords: izjavni račun, indukcija, kombinatorika, rekurzivna relacija, časovna zahtevnost, teorija števil, relacija, mreža, Booleova algebra, graf
Published: 27.10.2020; Views: 225; Downloads: 90
.pdf Full text (5,40 MB)

3.
Hereditarnia 2019
2019, other monographs and other completed works

Abstract: The booklet contains the abstracts of the talks given at the 22th Hereditarnia Workshop on Graph Properties that was held at the Faculty of Electrical Engineering and Computer Science in Maribor on 21st and 22nd of June, 2019. The workshop attracted 22 participants from 8 countries. All of the participants are researchers in di˙erent areas of graph theory, but at this event they all presented topics connected with (hereditary) graph properties. Themes of the talks encompass a wide range of contemporary graph theory research, notably, various types of graph colorings, graph domination, some graph dimensions matchings and graph products. Beside the abstracts of the plenary speaker (Roman Sotak) and three invited speakers (Tanja Gologranc, Michael A. Henning and Ismael G. Yero), the booklet also contains the abstracts of 7 contributed talks given at the event.
Keywords: mathematics, graph theory, Hereditarnia, Maribor, Slovenia
Published: 13.12.2019; Views: 341; Downloads: 119
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4.
Nekatere lastnosti posplošenih grafov Sierpińskega
Teja Bezgovšek, 2019, master's thesis

Abstract: V magistrskem delu so obravnavane in s slikovnimi zgledi predstavljene nekatere lastnosti posplošenih grafov Sierpińskega, zgrajenih na poljubnem baznem grafu G. V prvem poglavju so povzete osnovne definicije iz teorije grafov, ki so pomembne pri razumevanju magistrskega dela. Nato so predstavljeni grafi Sierpińskega in definirani posplošeni grafi Sierpińskega. Tretje poglavje obravnava popolno kromatično število obravnavanih grafov, med drugim tudi za konkretne primere baznih grafov, in sicer graf hiše, kolo, cikel in hiperkocko. V četrtem poglavju so z zgledi podane formule za izračun števila listov, število vozliščnega pokritja in neodvisno število v posplošenih grafih Sierpińskega. V poglavju je tudi dokazano, da sta kromatično in klično število teh grafov enaka kot v bazi. V nadaljevanju je podana zgornja meja dominacijskega števila obravnavanih grafov in tudi točno dominacijsko število teh grafov z dotičnimi lastnostmi. V zadnjem poglavju je dokazana spodnja meja krepke metrične dimenzije posplošenih grafov Sierpińskega in podana je formula za izračun te lastnosti v obravnavanih grafih, v katerih je vsako notranje vozlišče presečno vozlišče.
Keywords: posplošeni grafi Sierpińskega, popolno kromatično število, število vozliščnega pokritja, dominacijsko število, krepka metrična dimenzija.
Published: 04.03.2019; Views: 489; Downloads: 63
.pdf Full text (627,83 KB)

5.
Učinkovita odprta in zaprta dominacija na drevesih
Uroš Gašpar, 2018, master's thesis

Abstract: V magistrskem delu smo predstavili učinkovito odprte in zaprte dominacije. Omenjena pojma posebej obravnavamo na drevesih. V nadaljevanju magistrskega dela se posvetimo preseku obeh razredov, ki ga imenujemo učinkovito odprto-zaprto dominirana drevesa. Zelo zanimivo je dejstvo, da je za izgradnjo učinkovito odprto-zaprto dominiranih dreves potrebnih le pet operacij, ki jih podrobneje dokažemo v magistrskem delu. V prvem delu magistrskega dela smo podali osnovne pojme in definicije, ki jih nato uporabljamo skozi celotno magistrsko delo. V drugem poglavju definiramo in podamo lastnosti učinkovito odprto dominiranih dreves. V tretjem poglavju podrobneje pogledamo učinkovito zaprto dominirana drevesa. V zadnjem četrtem poglavju na začetku podamo lastnosti, ki veljajo za učinkovito odprto-zaprto dominirane grafe ter se nato posebej posvetimo samo učinkovito odprto-zaprto dominiranim drevesom. Podamo vseh pet operacij, ki so značilne za izgradnjo omenjenih dreves.
Keywords: učinkovito odprto dominirana množica, učinkovito zaprto dominirana množica, učinkovito odprto-zaprto dominirana množica, drevo
Published: 24.09.2018; Views: 340; Downloads: 46
.pdf Full text (426,49 KB)

6.
Ljubljana-Leoben graph theory seminar
2017, other monographs and other completed works

Abstract: The booklet contains the abstracts of the talks given at the 30th Ljubljana-Leoben Graph Theory Seminar that was held at the Faculty of Natural Sciences and Mathematics in Maribor between 13-15 September, 2017. The seminar attracted more than 30 participants from eight countries, all of which are researchers in different areas of graph theory. The topics of the talks encompass a wide range of contemporary graph theory research, notably, various types of graph colorings (b-coloring, packing coloring, edge colorings), graph domination (rainbow domination, Grundy domination, graph security), distinguishing problems, algebraic graph theory, graph algorithms, chemical graph theory, coverings, matchings and also some classical extremal problems. Beside the abstracts of the four invited speakers (Csilla Bujtás, Premysl Holub, Jakub Przybyło, Zsolt Tuza), the booklet contains also the abstracts of 18 contributed talks given at the event.
Keywords: mathematics, discrete mathematics, graph theory, Ljubljana-Leoben seminar, Maribor, Slovenia
Published: 08.12.2017; Views: 607; Downloads: 70
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7.
Wiener index of strong product of graphs
Iztok Peterin, Petra Žigert Pleteršek, 2018, original scientific article

Abstract: The Wiener index of a connected graph ▫$G$▫ is the sum of distances between all pairs of vertices of ▫$G$▫. The strong product is one of the four most investigated graph products. In this paper the general formula for the Wiener index of the strong product of connected graphs is given. The formula can be simplified if both factors are graphs with the constant eccentricity. Consequently, closed formulas for the Wiener index of the strong product of a connected graph ▫$G$▫ with a cycle are derived.
Keywords: Wiener index, graph product, strong product
Published: 30.11.2017; Views: 754; Downloads: 303
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8.
Partitioning the vertex set of ▫$G$▫ to make ▫$G \Box H$▫ an efficient open domination graph
Tadeja Kraner Šumenjak, Iztok Peterin, Douglas F. Rall, Aleksandra Tepeh, 2016, original scientific article

Abstract: A graph is an efficient open domination graph if there exists a subset of vertices whose open neighborhoods partition its vertex set. We characterize those graphs ▫$G$▫ for which the Cartesian product ▫$G \Box H$▫ is an efficient open domination graph when ▫$H$▫ is a complete graph of order at least 3 or a complete bipartite graph. The characterization is based on the existence of a certain type of weak partition of ▫$V(G)$▫. For the class of trees when ▫$H$▫ is complete of order at least 3, the characterization is constructive. In addition, a special type of efficient open domination graph is characterized among Cartesian products ▫$G \Box H$▫ when ▫$H$▫ is a 5-cycle or a 4-cycle.
Keywords: efficient open domination, Cartesian product, vertex labeling, total domination
Published: 10.07.2017; Views: 405; Downloads: 78
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9.
Open $k$-monopolies in graphs: complexity and related concepts
Dorota Kuziak, Iztok Peterin, Ismael G. Yero, 2016, original scientific article

Abstract: Closed monopolies in graphs have a quite long range of applications in several problems related to overcoming failures, since they frequently have some common approaches around the notion of majorities, for instance to consensus problems, diagnosis problems or voting systems. We introduce here open ▫$k$▫-monopolies in graphs which are closely related to different parameters in graphs. Given a graph ▫$G=(V,E)$▫ and ▫$X \subseteq V$▫, if ▫$\delta_X(v)$▫ is the number of neighbors ▫$v$▫ has in ▫$X$▫, ▫$k$▫ is an integer and ▫$t$▫ is a positive integer, then we establish in this article a connection between the following three concepts: (1) Given a nonempty set ▫$M\subseteq V$▫ a vertex ▫$v$▫ of ▫$G$▫ is said to be ▫$k$▫-controlled by ▫$M$▫ if ▫$\delta_M(v)\ge \frac{\delta_V(v)}{2}+k$▫. The set ▫$M$▫ is called an open ▫$k$▫-monopoly for ▫$G$▫ if it ▫$k$▫-controls every vertex ▫$v$▫ of ▫$G$▫. (2) A function ▫$f: V\rightarrow \{-1,1\}$▫ is called a signed total ▫$t$▫-dominating function for ▫$G$▫ if ▫$f(N(v))=\sum_{v\in N(v)}f(v)\geq t$▫ for all ▫$v\in V$▫. (3) A nonempty set ▫$S\subseteq V$▫ is a global (defensive and offensive) ▫$k$▫-alliance in ▫$G$▫ if ▫$\delta_S(v)\ge \delta_{V-S}(v)+k$▫ holds for every ▫$v\in V$▫. In this article we prove that the problem of computing the minimum cardinality of an open ▫$0$▫-monopoly in a graph is NP-complete even restricted to bipartite or chordal graphs. In addition we present some general bounds for the minimum cardinality of open ▫$k$▫-monopolies and we derive some exact values.
Keywords: open k-monopolies, k-signed total domination, global defensive k-alliance, global offensive k-alliance
Published: 10.07.2017; Views: 489; Downloads: 65
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10.
Efficient open domination in graph products
Dorota Kuziak, Iztok Peterin, Ismael G. Yero, 2014, original scientific article

Abstract: A graph ▫$G$▫ is an efficient open domination graph if there exists a subset ▫$D$▫ of ▫$V(G)$▫ for which the open neighborhoods centered in vertices of ▫$D$▫ form a partition of ▫$V(G)$▫. We completely describe efficient open domination graphs among lexicographic, strong, and disjunctive products of graphs. For the Cartesian product we give a characterization when one factor is ▫$K_2$▫.
Keywords: graph theory, efficient open domination, graph products, total domination
Published: 10.07.2017; Views: 520; Downloads: 72
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