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1.
2.
Cage-amalgamation graphs, a common generalization of chordal and median graphs
Boštjan Brešar, Aleksandra Tepeh, 2009, original scientific article

Abstract: V članku je vpeljan in na različne načine okarakteriziran nov razred grafov, imenovan grafi amalgamov kletk, ki je vsebovan v šibko modularnih grafih in grafih zastraženih inverzov in ki vsebuje tako medianske kot tetivne grafe. Vpeljemo tudi variacijo Hammingovega polinoma in jo uporabimo pri izpeljavi dveh enakosti drevesnega tipa za ta razred grafov, ki sta bili prej znani za tetivne in medianske grafe. Prva enakost je ▫$sum_{ige 0}, (-1)^{i}, rho_i(G)=1$▫, kjer je ▫$rho_i(G)$▫ število ▫$i$▫-regularnih Hammingovih podgrafov v grafu amalgamov kletk ▫$G$▫.
Keywords: matematika, teorija grafov, medianski grafi, tetivni grafi, konveksnost, amalgamacija, enakosti drevesnega tipa, mathematics, graph theory, median graphs, chordal graphs, convexity, amalgamation, tree-like equalities
Published: 10.07.2015; Views: 271; Downloads: 50
URL Link to full text

3.
Cube intersection concepts in median graphs
Boštjan Brešar, Tadeja Kraner Šumenjak, 2009, original scientific article

Abstract: Obravnavamo različne razrede presečnih grafov maksimalnih hiperkock medianskih grafov. Za medianski graf ▫$G$▫ in celo število ▫$k ge 0$▫ je presečni graf ▫${mathcal{Q}}_k(G)$▫ definiran kot tisti graf, katerega vozlišča so maksimalne hiperkocke (z ozirom na inkluzijo) grafa ▫$G$▫ in sta dve vozlišči ▫$H_x$▫ in ▫$H_y$▫ v njem sosednji tedaj, ko presek ▫$H_x cap H_y$▫ vsebuje podgraf izomorfen ▫$Q_k$▫. V članku predstavimo karakterizacije kličnih grafov z uporabo omenjenih presečnih konceptov, ko je ▫$k>0$▫. Vpeljemo tudi t.i. maksimalno 2-presečni graf maksimalnih hiperkock medianskega grafa ▫$G$▫, ki ga označimo z ▫${mathcal{Q}}_{m2}(G)$▫ in predstavlja tisti graf, katerega vozlišča somaksimalne hiperkocke grafa ▫$G$▫, dve vozlišči v njem pa sta sosednji, če presek pripadajočih hiperkock ni strogo vsebovan v kakem preseku dveh maksimalnih hiperkock. Dokažemo, da je graf ▫$H$▫ brez induciranih diamantov, če in samo če obstaja takšen medianski graf ▫$G$▫, da je ▫$H$▫ izomorfen ▫${mathcal{Q}}_{m2}(G)$▫. Obravnavamo tudi konvergenco medianskega grafa h grafu na enem vozlišču glede na vse vpeljane operacije.
Keywords: matematika, teorija grafov, kartezični produkt, medianski graf, graf kock, presečni graf, konveksnost, mathematics, graph theory, Cartesian product, median graph, cube graph, intersection graph, convexity
Published: 10.07.2015; Views: 381; Downloads: 65
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4.
On some polynomially convex maximal real submanifolds in C [sup] {2n} and a related Riemann-Hilbert problem
Matej Zajec, 2009, original scientific article

Abstract: Dokažemo, da so določeni maksimalno realni grafi preslikav iz ▫${mathbb{C}}^n$▫ v ▫${mathbb{C}}^n$▫ polinomsko konveksni. Vpeljemo mnogoterosti, ki so fibrirane nad ▫$partialDelta$▫ in katerih vlakna so taki grafi, in rešujemo pripadajoči Riemann-Hilbertov problem. Dokažemo obstoj rešitve in opišemo njeno lokalno strukturo.
Keywords: matematika, kompleksna analiza, polinomska konveksnost, Riemann-Hilbertov problem, analitični disk, mathematics, complex analysis, polynomial convexity, Riemann-Hilbert problem, analytic disc
Published: 10.07.2015; Views: 329; Downloads: 11
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5.
The pre-hull number and lexicographic product
Iztok Peterin, 2012, published scientific conference contribution

Abstract: Nedavno sta Polat in Sabidussi v [On the geodesic pre-hull number of a graph, Europ. J. Combin. 30 (2009), 1205--1220] vpeljala invarianto ko-točkovno pred-ovojnično število ▫$mathrm{ph}(G)$▫ grafa ▫$G$▫, ki meri nekonveksnost konveksnega prostora. Vpeljemo podobno invarianto imenovano konveksno pred-ovojnično število, ki je naravna zgornja meja za ko-točkovno pred-ovojnično število. Obe invarianti študiramo na leksikografskem produktu in podamo natančne vrednosti za obe invarianti glede na lastnosti faktorjev.
Keywords: matematika, teorija grafov, pred-ovojnično število, geodetska konveksnost, leksikografski produkt, mathematics, graph theory, pre-hull number, geodesic convexity, lexicographic product
Published: 10.07.2015; Views: 423; Downloads: 53
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6.
Some Steiner concepts on lexicographic products of graphs
Bijo S. Anand, Manoj Changat, Iztok Peterin, Prasanth G. Narasimha-Shenoi, 2012, original scientific article

Abstract: The smallest tree that contains all vertices of a subset ▫$W$▫ of ▫$V(G)$▫ is called a Steiner tree. The number of edges of such a tree is the Steiner distance of ▫$W$▫ and union of all Steiner trees of ▫$W$▫ form a Steiner interval. Both of them are described for the lexicographic product in the present work. We also give a complete answer for the following invariants with respect to the Steiner convexity: the Steiner number, the rank, the hull number, and the Carathéodory number, and a partial answer for the Radon number. At the end we locate and repair a small mistake from [J. Cáceres, C. Hernando, M. Mora, I. M. Pelayo, M. L. Puertas, On the geodetic and the hull numbers in strong product graphs, Comput. Math. Appl. 60 (2010) 3020--3031].
Keywords: teorija grafov, leksikografski produkt, Steinerjeva konveksnost, Steinerjeva množica, Steinerjeva razdalja, graph theory, lexicographic product, Steiner convexity, Steiner set, Steiner distance
Published: 10.07.2015; Views: 406; Downloads: 59
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7.
Intervals and convex sets in strong product of graphs
Iztok Peterin, 2013, original scientific article

Abstract: Obravnavamo intervale in konveksne množice krepkega produkta. Vozlišča poljubnega intervala iz ▫$G boxtimes H$▫ so klasificirana z najkrajšimi potmi v enem faktorju in s sprehodi v dugem rahlo modificiranem faktorju. Konveksne množice krepkega produkta so karakterizirane s konveksnostjo obeh projekcij in še tremi lokalnimi lastnostmi, med katerimi je tudi 2-konveksnost.
Keywords: teorija grafov, krepki produkt, geodetska konveksnost, interval, graph theory, strong product, geodesic convexity, interval
Published: 10.07.2015; Views: 321; Downloads: 64
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8.
Median and quasi-median direct products of graphs
Boštjan Brešar, Pranava Jha, Sandi Klavžar, Blaž Zmazek, 2005, original scientific article

Abstract: 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 ▫$K_3 \times K_3$▫.
Keywords: mathematics, graph theory, median graph, direct product, quasi-median graph, isometric embeddings, convexity
Published: 31.03.2017; Views: 335; Downloads: 171
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9.
n-ary transit functions in graphs
Manoj Changat, Joseph Mathews, Iztok Peterin, Prasanth G. Narasimha-Shenoi, 2010, original scientific article

Abstract: ▫$n$▫-ary transit functions are introduced as a generalization of binary (2-ary) transit functions. We show that they can be associated with convexities in natural way and discuss the Steiner convexity as a natural ▫$n$▫-ary generalization of geodesicaly convexity. Furthermore, we generalize the betweenness axioms to ▫$n$▫-ary transit functions and discuss the connectivity conditions for underlying hypergraph. Also ▫$n$▫-ary all paths transit function is considered.
Keywords: mathematics, graph theory, n-arity, transit function, betweenness, Steiner convexity
Published: 31.03.2017; Views: 386; Downloads: 181
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10.
Recognizing weighted directed Cartesian graph bundles
Blaž Zmazek, Janez Žerovnik, 2000, original scientific article

Abstract: In this paper we show that methods for recognizing Cartesian graph bundles can be generalized to weighted digraphs. The main result is an algorithm which lists the sets of degenerate arcs for all representations of digraph as a weighted directed Cartesian graph bundle over simple base digraphs not containing transitive tournament on three vertices. Two main notions are used.The first one is the new relation ▫$\vec{\delta}^\ast$▫ defined among the arcs of a digraph as a weighted directed analogue of the well-known relation ▫$\delta^\ast$▫. The second one is the concept of half-convex subgraphs. A subgraph ▫$H$▫ is half-convex in ▫$G$▫ if any vertex ▫$x \in G \setminus H$▫ has at most one predecessor and at most one successor
Keywords: mathematics, graph theory, graph bundles, Cartesian graph product, weighted digraphs, half-convexity
Published: 31.03.2017; Views: 377; Downloads: 200
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