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81.
Domination game: extremal families of graphs for 3/5-conjectures
Boštjan Brešar, Sandi Klavžar, Gašper Košmrlj, Douglas F. Rall, 2013, izvirni znanstveni članek

Opis: Igralca, Dominator in Zavlačevalka, izmenoma izbirata vozlišča grafa ▫$G$▫, takoda vsako izbrano vozlišče poveča množico do sedaj dominiranih vozlišč. Cilj Dominatorja je končati igro čim hitreje, medtem ko je Zavlačevalkin cilj ravno nasprotno. Igralno dominacijsko število ▫$gamma_g(G)$▫ je skupno število izbranih vozlišč v igri, ko Dominator naredi prvo potezo in oba igralca igrata optimalno. Postavljena je bila domneva [W.B. Kinnersley, D.B. West, R. Zemani, Extremal problems for game domination number, Manuscript, 2012], da velja ▫$gamma_g(G) leq frac{3|V(G)|}{5}$▫ za poljuben graf ▫$G$▫ brez izoliranih vozlišč. V posebnem je domneva odprta tudi, ko je ▫$G$▫ gozd. V tem članku predstavimo konstrukcije, ki nam dajo velike družine dreves, ki dosežejo domnevno mejo ▫$3/5$▫. Leplenje dreves iz nekaterih izmed teh družin napoljuben graf nam da konstrukcijo grafov ▫$G$▫, ki imajo igralno dominacijsko število enako ▫$3|V(G)|/5$▫. Z računalnikom smo poiskali vsa ekstremna drevesa znajveč 20 vozlišči. V posebnem, na 20 vozliščih obstaja natanko deset dreves ▫$T$▫, za katere velja ▫$gamma_g(T) = 12$▫, in vsa pripadajo skonstruiranim družinam.
Ključne besede: matematika, teorija grafov, dominacijska igra, igralno dominacijsko številko, 3/5-domneva, računalniško iskanje, mathematics, graph theory, domination game, game domination number, 3/5-conjecture, computer search
Objavljeno: 10.07.2015; Ogledov: 469; Prenosov: 49
URL Povezava na celotno besedilo

82.
Bucolic complexes
Boštjan Brešar, Jérémie Chalopin, Victor Chepoi, Tanja Gologranc, Damian Osajda, 2013, izvirni znanstveni članek

Opis: Vpeljemo in obravnavamo bukolične kompleksne, skupno posplošitev sistoličnih in CAT(0) kubnih kompleksov. Definirani so kot enostavno povezani kompleksi prizem, ki zadoščajo določenim lokalnim kombinatornim pogojem. Raziskujemo različne pristope k bukoličnim kompleksom: gledamo jih iz vidika teorije grafov in topološkega vidika kot tudi iz perspektive geometrijske teorije grup. Tako med drugim okarakteriziramo bukolične komplekse preko nekih lastnosti njihovih 2-skeletov in 1-skeletov (ki jim pravimo bukolični grafi), s čimer posplošimo več prej znanih rezultatov. Prav tako dokažemo, da so lokalno končni bukolični kompleksi kontraktibilni in da zadoščajo nekim lastnostim tipa nepozitivnih ukrivljenosti.
Ključne besede: CAT(0) kubni in sistolični kompleksi, medianski in mostovni grafi, zastražena amalgamacija, kartezični produkt, kompleksi prizem, retrakti, fiksne točke, asferičnost, CAT(0) cubical and systolic complexes, median and bridged graphs, gated amalgamation, Cartesian product, prism complexes, retracts, fixed points, asphericity
Objavljeno: 10.07.2015; Ogledov: 381; Prenosov: 11
URL Povezava na celotno besedilo

83.
On the vertex k-path cover
Boštjan Brešar, Marko Jakovac, Ján Katrenič, Gabriel Semanišin, Andrej Taranenko, 2013, izvirni znanstveni članek

Opis: A subset ▫$S$▫ of vertices of a graph ▫$G$▫ is called a vertex ▫$k$▫-path cover if every path of order ▫$k$▫ in ▫$G$▫ contains at least one vertex from ▫$S$▫. Denote by ▫$psi_k(G)$▫ the minimum cardinality of a vertex ▫$k$▫-path cover in ▫$G$▫. In this paper, an upper bound for ▫$psi_3$▫ in graphs with a given average degree is presented. A lower bound for ▫$psi_k$▫ of regular graphs is also proven. For grids, i.e. the Cartesian products of two paths, we give an asymptotically tight bound for ▫$psi_k$▫ and the exact value for ▫$psi_3$▫.
Ključne besede: matematika, teorija grafov, vozliščno pokritje, regularni grafi, mreže, mathematics, graph theory, vertex cover, grids
Objavljeno: 10.07.2015; Ogledov: 380; Prenosov: 8
URL Povezava na celotno besedilo

84.
Guarded subgraphs and the domination game
Boštjan Brešar, Sandi Klavžar, Gašper Košmrlj, Douglas F. Rall, 2015, izvirni znanstveni članek

Opis: 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.
Ključne besede: dominacijska igra, igralno dominacijsko številko, konveksni podgraf, (2-)izometrični podgraf
Objavljeno: 10.07.2017; Ogledov: 354; Prenosov: 47
.pdf Celotno besedilo (690,85 KB)
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85.
Arboreal structure and regular graphs of median-like classes
Boštjan Brešar, 2003, izvirni znanstveni članek

Opis: We consider classes of graphs that enjoy the following properties: they are closed for gated subgraphs, gated amalgamation and Cartesian products, and for any gated subgraph the inverse of gate function maps vertices to gated subsets. We prove that any graph of such a class contains a peripheral subgraph which is a Cartesian product of two graphs: a gated subgraph and a prime graph minus a vertex. Therefore, these graphs admit a peripheral elimination procedure which is a generalization of analogous procedure in median graphs. We characterize regular graphs of these classes whenever they enjoy an additional properties. As a corollary we derive that regular weakly median graphs are precisely Cartesian products in which each factor is a complete graph or a hyperoctahedron.
Ključne besede: mathematics, graph theory, median graph, tree, gatedness, amalgam, periphery, regular graph
Objavljeno: 31.03.2017; Ogledov: 221; Prenosov: 156
.pdf Celotno besedilo (127,35 KB)
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86.
Edge-connectivity of strong products of graphs
Boštjan Brešar, Simon Špacapan, 2007, izvirni znanstveni članek

Opis: The strong product ▫$G_1 \boxtimes G_2$▫ of graphs ▫$G_1$▫ and ▫$G_2$▫ is the graph with ▫$V(G_1) \times V(G_2)$▫ as the vertex set, and two distinct vertices ▫$(x_1,x_2)$▫ and ▫$(y_1,y_2)$▫ are adjacent whenever for each ▫$i\in \{1,2\}$▫ either ▫$x_i=y_i$▫ or ▫$x_iy_i \in E(G_i)$▫. In this note we show that for two connected graphs ▫$G_1$▫ and ▫$G_2$▫ the edge-connectivity ▫$\lambda(G_1 \boxtimes G_2)$▫ equals ▫$\min\{\delta(G_1\boxtimes G_2), \lambda(G_1)(|V(G_2)|+2|E(G_2)|), \lambda(G_2)(|V(G_1)|+2|E(G_1)|)\}$▫. In addition, we fully describe the structure of possible minimum edge cut sets in strong products of graphs.
Ključne besede: mathematics, graph theory, connectivity, strong product, graph product, separating set
Objavljeno: 31.03.2017; Ogledov: 383; Prenosov: 155
.pdf Celotno besedilo (173,26 KB)
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87.
Median and quasi-median direct products of graphs
Boštjan Brešar, Pranava Jha, Sandi Klavžar, Blaž Zmazek, 2005, izvirni znanstveni članek

Opis: Median graphs are characterized among direct products of graphs on at least three vertices. Beside some trivial cases, it is shown that one component of ▫$G \times P_3$▫ is median if and only if ▫$G$▫ is a tree in that the distance between any two vertices of degree at least 3 is even. In addition, some partial results considering median graphs of the form ▫$G \times K_2$▫ are proved, and it is shown that the only nonbipartite quasi-median direct product is ▫$K_3 \times K_3$▫.
Ključne besede: mathematics, graph theory, median graph, direct product, quasi-median graph, isometric embeddings, convexity
Objavljeno: 31.03.2017; Ogledov: 291; Prenosov: 137
.pdf Celotno besedilo (174,14 KB)
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88.
On Vizing's conjecture
Boštjan Brešar, 2001, izvirni znanstveni članek

Opis: 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 ▫$\gamma(G) = \gamma(H) = 3$▫.
Ključne besede: mathematics, graph theory, graph, Cartesian product, domination number
Objavljeno: 31.03.2017; Ogledov: 434; Prenosov: 47
.pdf Celotno besedilo (126,98 KB)
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89.
The periphery graph of a median graph
Boštjan Brešar, Manoj Changat, Ajitha R. Subhamathi, Aleksandra Tepeh, 2010, izvirni znanstveni članek

Opis: 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.
Ključne besede: mathematics, graph theory, median graph, Cartesian product, geodesic, periphery, peripheral expansion
Objavljeno: 31.03.2017; Ogledov: 496; Prenosov: 185
.pdf Celotno besedilo (145,86 KB)
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90.
Tree-like isometric subgraphs of hypercubes
Boštjan Brešar, Wilfried Imrich, Sandi Klavžar, 2003, izvirni znanstveni članek

Opis: 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.
Ključne besede: mathematics, graph theory, Isometric embeddings, partial cubes, expansion procedures, trees, median graphs, graph automorphisms, automorphism groups, dismantlable graphs
Objavljeno: 31.03.2017; Ogledov: 462; Prenosov: 170
.pdf Celotno besedilo (135,80 KB)
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