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1.
REŠEVANJE LINEARNIH REKURZIVNIH ENAČB
Sonja Cank, 2010, undergraduate thesis

Abstract: V diplomski nalogi so predstavljene osnove kombinatorike, ki so potrebne za razumevanje rekurzije. Reševanje linearnih rekurzivnih enačb,lastnosti ter uporaba pa so jedro diplomskega dela.V uvodnem poglavju je razloºeno, kaj pomeni rekurzivno podajanje formule. Nato je na primeru razloºen postopek re²evanja linearnih rekurzivnih ena£b. Najprej si bomo pogledali homogene linearne rekurzivne enačbe s konstantnimi koeficienti, na to nehomogene enačbe in za konec še reševanje sistema linearnih rekurzivnih enačb. V vsakem poglavju so rešeni konkretni primeri.
Keywords: matematika, kombinatorika, linearne rekurzivne enačbe, karakteristični polinom, homogena enačba, splošna rešitev, partikularna rešitev.
Published: 03.02.2021; Views: 133; Downloads: 13
.pdf Full text (276,61 KB)

2.
Igra Križci in krožci
Martin Fras, 2019, master's thesis

Abstract: V magistrskem delu predstavljamo osnovne pojme kombinatoričnih iger na treh variacijah igre Križci in krožci: simetrične igre, igre tipa Izdelovalec-Lomilec in igre tipa berač. Obravnavamo igre Križci in krožci zrazličnimi dimenzijami igralnega polja in predstavimo različne strategije igralcev ter opazujemo odnose med spremembo dimenzije igralnega polja ter rezultatom igre. Ugotavimo, da v simetrični igri na polju velikosti $n^d$ z uporabo ustrezne strategije zmaga Prvi igralec, če je z uporabo iste strategije zmagal na polju $n^k,\ kKeywords: Pozicijska igra, Križci in krožci, strategija, šibka zmaga, močan remi
Published: 26.11.2019; Views: 588; Downloads: 45
.pdf Full text (1,24 MB)

3.
Contributions to the Study of Contemporary Domination Invariants of Graphs
2019, doctoral dissertation

Abstract: This doctoral dissertation is devoted to contemporary domination concepts, such as the Grundy domination, the convex domination, the isometric domination and the total domination. Our main focus is to study their structure and algorithmic properties. Four Grundy domination invariants are presented, namely the Grundy domination number, the Grundy total domination number, the Z-Grundy domination number, and the L-Grundy domination number. Some bounds and properties of Grundy domination invariants are proven. All four Grundy domination parameters are studied on trees, bipartite distance-hereditary graphs, split graphs, interval graphs, Sierpi\'nski graphs, Kneser graphs and $P_4$-tidy graphs. Graphs with equal total domination number and Grundy total domination number are investigated. Convex domination and isometric domination are studied on (weak) dominating pair graphs. For the chordal dominating pair graphs we present a polynomial algorithm to compute the convex domination number, and prove the NP-completeness of the corresponding decision problem for the chordal weak dominating pair graphs. For the isometric domination number of weak dominating pair graphs an efficient algorithm is presented. Total domination is studied on the Cartesian product of graphs. We dedicate ourselves to graphs for which the equality holds in Ho's theorem, which states that the total domination number of the Cartesian product of any two graphs without isolated vertices is at least one half of the product of their total domination numbers.
Keywords: Grundy domination, Grundy total domination, Z-Grundy domination, L-Grundy domination, convex domination, isometric domination, total domination, trees, split graphs, interval graphs, Sierpi\'nski graphs, Kneser graphs, modular decomposition, dominating pair graphs, Cartesian product
Published: 23.10.2019; Views: 563; Downloads: 11
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4.
Kombinatorična teorija iger
Nejc Babič, 2018, master's thesis

Abstract: V magistrskem delu obravnavamo izbrane vsebine s področja kombinatorične teorije iger. V uvodnih poglavjih predstavimo primere preprostih kombinatoričnih iger ter navedemo nekatere osnovne definicije in trditve, na katerih temelji teorija kombinatoričnih iger. Predstavimo dokaz Zermelovega izreka in podamo več primerov osnovnih strategij, ki jih igralca uporabljata pri igranju. Glavni poudarek je na igrah normalnega tipa, pri katerih zmaga tisti igralec, ki napravi zadnjo potezo. Ob tem obravnavamo pojme kot so: položaj in njegov tip, vsota položajev in ekvivalenca položajev, na katerih temelji kombinatorična teorija iger. V drugem delu predstavimo kombinatorično teorijo nepristranskih iger ob pomoči igre Nim in dokažemo Sprague-Grundyev izrek, ki nam omogoča celostno razumevanje ekvivalence pri nepristranskih igrah. Podobno predstavimo tudi njegovo različico za pristranske igre ob pomoči igre Hackenbush. Skozi celotno magistrsko delo povezujemo in odkrivamo zveze med obravnavanimi vsebinami, ki jih dopolnjujemo z rešenimi praktičnimi primeri.
Keywords: Zermelov izrek, strategije, igre normalnega tipa, nepristranske in pristranske igre, Sprague-Grundyev izrek.
Published: 08.01.2019; Views: 689; Downloads: 88
.pdf Full text (2,78 MB)

5.
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: 648; Downloads: 78
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6.
Guarded subgraphs and the domination game
Boštjan Brešar, Sandi Klavžar, Gašper Košmrlj, Douglas F. Rall, 2015, original scientific article

Abstract: V članku vpeljemo koncept zaščitenega podgrafa. Množica le-teh po definicji leži med množico konveksnih in 2-izometričnih podgrafov, hkrati pa ni primerljiva z množico izometričnimih podgrafov. Dokažemo nekatere metrične lastnosti zaščitenih podgrafov ter koncept uporabimo v dominacijski igri, v kateri dva igralca, Dominator in Zavlačevalka, izmenično izbirata vozlišča grafa, tako da vsako izbrano vozlišče poveča množico dominiranih vozlišč. Dominatorjev cilj je končati igro, tj. dominirati celoten graf, čim hitreje, medtem ko je Zavlačevalkin cilj odigrati čim več potez. Igralno dominacijsko število je število potez v igri, ko Dominator začne in oba igralca igrata optimalno. Kot glavni rezultat članka dokažemo, da igralno dominacijsko število grafa ni nikoli manjše, kot igralno dominacijsko število njegovega zaščitenega podgrafa. Predstavljenih je tudi več aplikacij tega rezultata.
Keywords: dominacijska igra, igralno dominacijsko številko, konveksni podgraf, (2-)izometrični podgraf
Published: 10.07.2017; Views: 680; Downloads: 90
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7.
How long can one bluff in the domination game?
Boštjan Brešar, Paul Dorbec, Sandi Klavžar, Gašper Košmrlj, 2017, original scientific article

Abstract: The domination game is played on an arbitrary graph ▫$G$▫ by two players, Dominator and Staller. The game is called Game 1 when Dominator starts it, and Game 2 otherwise. In this paper bluff graphs are introduced as the graphs in which every vertex is an optimal start vertex in Game 1 as well as in Game 2. It is proved that every minus graph (a graph in which Game 2 finishes faster than Game 1) is a bluff graph. A non-trivial infinite family of minus (and hence bluff) graphs is established. Minus graphs with game domination number equal to 3 are characterized. Double bluff graphs are also introduced and it is proved that Kneser graphs ▫$K(n,2)$▫, za ▫$n \ge 6$▫, are double bluff. The domination game is also studied on generalized Petersen graphs and on Hamming graphs. Several generalized Petersen graphs that are bluff graphs but not vertex-transitive are found. It is proved that Hamming graphs are not double bluff.
Keywords: domination game, game domination number, bluff graphs, minus graphs, generalized Petersen graphs, Kneser graphs, Cartesian product of graphs, Hamming graphs
Published: 09.05.2017; Views: 669; Downloads: 361
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8.
Tree-like isometric subgraphs of hypercubes
Boštjan Brešar, Wilfried Imrich, Sandi Klavžar, 2003, original scientific article

Abstract: Tree-like isometric subgraphs of hypercubes, or tree-like partial cubes as we call them, are a generalization of median graphs. Just as median graphs they capture numerous properties of trees, but may contain larger classes of graphs that may be easier to recognize than the class of median graphs. We investigate the structure of tree-like partial cubes, characterize them, and provide examples of similarities with trees and median graphs. For instance, we show that the cube graph of tree-like partial cube is dismantlable. This in particular implies that every tree-like partial cube ▫$G$▫ contains a cube that is invariant under every automorphism of ▫$G$▫. We also show that weak retractions preserve tree-like partial cubes, which in turn implies that every contraction of a tree-like partial cube fixes a cube. The paper ends with several Frucht-type results and a list of open problems.
Keywords: mathematics, graph theory, Isometric embeddings, partial cubes, expansion procedures, trees, median graphs, graph automorphisms, automorphism groups, dismantlable graphs
Published: 31.03.2017; Views: 827; Downloads: 293
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9.
The periphery graph of a median graph
Boštjan Brešar, Manoj Changat, Ajitha R. Subhamathi, Aleksandra Tepeh, 2010, original scientific article

Abstract: The periphery graph of a median graph is the intersection graph of its peripheral subgraphs. We show that every graph without a universal vertex can be realized as the periphery graph of a median graph. We characterize those median graphs whose periphery graph is the join of two graphs and show that they are precisely Cartesian products of median graphs. Path-like median graphs are introduced as the graphs whose periphery graph has independence number 2, and it is proved that there are path-like median graphs with arbitrarily large geodetic number. Peripheral expansion with respect to periphery graph is also considered, and connections with the concept of crossing graph are established.
Keywords: mathematics, graph theory, median graph, Cartesian product, geodesic, periphery, peripheral expansion
Published: 31.03.2017; Views: 855; Downloads: 308
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10.
On Vizing's conjecture
Boštjan Brešar, 2001, original scientific article

Abstract: A dominating set ▫$D$▫ gor a graph ▫$G$▫ is a subset ▫$V(G)$▫ such that any vertex in ▫$V(G)-D$▫ has a neighbor in ▫$D$▫, and a domination number ▫$\gamma(G)$▫ is the size of a minimum dominating set for ▫$G$▫. For the Cartesian product ▫$G \Box H$▫ Vizing's conjecture states that ▫$\gamma(G \Box H) \ge \gamma(G)\gamma(H)$▫ for every pair of graphs ▫$G,H$▫. In this paper we introduce a new concept which extends the ordinary domination of graphs, and prove that the conjecture holds when ▫$\gamma(G) = \gamma(H) = 3$▫.
Keywords: mathematics, graph theory, graph, Cartesian product, domination number
Published: 31.03.2017; Views: 757; Downloads: 80
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