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    Wyświetlanie 1-11 z 11
    Tytuł:
    Arboreal structure and regular graphs of median-like classes
    Autorzy:
    Brešar, Bostjan
    Powiązania:
    https://bibliotekanauki.pl/articles/743403.pdf
    Data publikacji:
    2003
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    median graph
    tree
    gatedness
    amalgam
    periphery
    regular graph
    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 the 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 of the graph 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 property. 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.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2003, 23, 2; 215-225
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    On Vizings conjecture
    Autorzy:
    Bresar, Bostjan
    Powiązania:
    https://bibliotekanauki.pl/articles/743820.pdf
    Data publikacji:
    2001
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    graph
    Cartesian product
    domination number
    Opis:
    A dominating set D for a graph G is a subset of V(G) such that any vertex in V(G)-D has a neighbor in D, and a domination number γ(G) is the size of a minimum dominating set for G. For the Cartesian product G ⃞ H Vizing's conjecture [10] states that γ(G ⃞ H) ≥ γ(G)γ(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 γ(G) = γ(H) = 3.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2001, 21, 1; 5-11
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Edge-connectivity of strong products of graphs
    Autorzy:
    Bresar, Bostjan
    Spacapan, Simon
    Powiązania:
    https://bibliotekanauki.pl/articles/743791.pdf
    Data publikacji:
    2007
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    connectivity
    strong product
    graph product
    separating set
    Opis:
    The strong product G₁ ⊠ G₂ of graphs G₁ and G₂ is the graph with V(G₁)×V(G₂) as the vertex set, and two distinct vertices (x₁,x₂) and (y₁,y₂) are adjacent whenever for each i ∈ {1,2} either $x_i = y_i$ or $x_i y_i ∈ E(G_i)$. In this note we show that for two connected graphs G₁ and G₂ the edge-connectivity λ (G₁ ⊠ G₂) equals min{δ(G₁ ⊠ G₂), λ(G₁)(|V(G₂)| + 2|E(G₂)|), λ(G₂)(|V(G₁)| + 2|E(G₁)|)}. In addition, we fully describe the structure of possible minimum edge cut sets in strong products of graphs.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2007, 27, 2; 333-343
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Graphs that are Critical for the Packing Chromatic Number
    Autorzy:
    Brešar, Boštjan
    Ferme, Jasmina
    Powiązania:
    https://bibliotekanauki.pl/articles/32318620.pdf
    Data publikacji:
    2022-05-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    packing coloring
    critical graph
    diameter
    block graph
    tree
    Opis:
    Given a graph G, a coloring c : V (G) → {1, …, k} such that c(u) = c(v) = i implies that vertices u and v are at distance greater than i, is called a packing coloring of G. The minimum number of colors in a packing coloring of G is called the packing chromatic number of G, and is denoted by χρ(G). In this paper, we propose the study of χρ-critical graphs, which are the graphs G such that for any proper subgraph H of G, χρ(H) < χρ(G). We characterize χρ-critical graphs with diameter 2, and χρ-critical block graphs with diameter 3. Furthermore, we characterize χρ-critical graphs with small packing chromatic number, and we also consider χρ-critical trees. In addition, we prove that for any graph G and every edge e ∈ E(G), we have (χρ(G)+1)/2 ≤ χρ(G−e) ≤ χρ(G), and provide a corresponding realization result, which shows that χρ(G − e) can achieve any of the integers between these bounds.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2022, 42, 2; 569-589
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Tree-like isometric subgraphs of hypercubes
    Autorzy:
    Brešar, Bostjan
    Imrich, Wilfried
    Klavžar, Sandi
    Powiązania:
    https://bibliotekanauki.pl/articles/743405.pdf
    Data publikacji:
    2003
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    isometric embeddings
    partial cubes
    expansion procedures
    trees
    median graphs
    graph automorphisms
    automorphism groups
    dismantlable graphs
    Opis:
    Tree-like isometric subgraphs of hypercubes, or tree-like partial cubes as we shall 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 a 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.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2003, 23, 2; 227-240
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Median and quasi-median direct products of graphs
    Autorzy:
    Brešar, Boštjan
    Jha, Pranava
    Klavžar, Sandi
    Zmazek, Blaž
    Powiązania:
    https://bibliotekanauki.pl/articles/744332.pdf
    Data publikacji:
    2005
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    median graph
    direct product
    quasi-median graph
    isometric embeddings
    convexity
    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×P₃ 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×K₂ are proved, and it is shown that the only nonbipartite quasi-median direct product is K₃×K₃.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2005, 25, 1-2; 183-196
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    The periphery graph of a median graph
    Autorzy:
    Brešar, Boštjan
    Changat, Manoj
    Subhamathi, Ajitha
    Tepeh, Aleksandra
    Powiązania:
    https://bibliotekanauki.pl/articles/744496.pdf
    Data publikacji:
    2010
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    median graph
    Cartesian product
    geodesic
    periphery
    peripheral expansion
    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.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2010, 30, 1; 17-32
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    On Total Domination in the Cartesian Product of Graphs
    Autorzy:
    Brešar, Boštjan
    Hartinger, Tatiana Romina
    Kos, Tim
    Milanič, Martin
    Powiązania:
    https://bibliotekanauki.pl/articles/31342240.pdf
    Data publikacji:
    2018-11-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    total domination
    Cartesian product
    total domination quotient
    Opis:
    Ho proved in [A note on the total domination number, Util. Math. 77 (2008) 97–100] 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. We extend a result of Lu and Hou from [Total domination in the Cartesian product of a graph and $ K_2 $ or $ C_n $, Util. Math. 83 (2010) 313–322] by characterizing the pairs of graphs $G$ and $H$ for which $ \gamma_t (G \square H)=1/2 \gamma_t (G) \gamma_t (H) $, whenever $ \gamma_t (H) = 2 $. In addition, we present an infinite family of graphs $ G_n $ with $ \gamma_t (G_n) = 2n $, which asymptotically approximate equality in $ \gamma_t (G_n \square H_n ) \ge 1/2 \gamma_t (G_n)^2 $.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2018, 38, 4; 963-976
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    A Survey on Packing Colorings
    Autorzy:
    Brešar, Boštjan
    Ferme, Jasmina
    Klavžar, Sandi
    Rall, Douglas F.
    Powiązania:
    https://bibliotekanauki.pl/articles/31804166.pdf
    Data publikacji:
    2020-11-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    packing coloring
    packing chromatic number
    subcubic graph
    S -packing chromatic number
    computational complexity
    Opis:
    If S = (a1, a2, . . .) is a non-decreasing sequence of positive integers, then an S-packing coloring of a graph G is a partition of V (G) into sets X1, X2, . . . such that for each pair of distinct vertices in the set Xi, the distance between them is larger than ai. If there exists an integer k such that V (G) = X1 ∪ ∪ Xk, then the partition is called an S-packing k-coloring. The S-packing chromatic number of G is the smallest k such that G admits an S-packing k-coloring. If ai = i for every i, then the terminology reduces to packing colorings and packing chromatic number. Since the introduction of these generalizations of the chromatic number in 2008 more than fifty papers followed. Here we survey the state of the art on the packing coloring, and its generalization, the S-packing coloring. We also list several conjectures and open problems.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2020, 40, 4; 923-970
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    A New Framework to Approach Vizing’s Conjecture
    Autorzy:
    Brešar, Boštjan
    Hartnell, Bert L.
    Henning, Michael A.
    Kuenzel, Kirsti
    Rall, Douglas F.
    Powiązania:
    https://bibliotekanauki.pl/articles/32222699.pdf
    Data publikacji:
    2021-08-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    Cartesian product
    total domination
    Vizing’s conjecture
    Clark and Suen bound
    Opis:
    We introduce a new setting for dealing with the problem of the domination number of the Cartesian product of graphs related to Vizing’s conjecture. The new framework unifies two different approaches to the conjecture. The most common approach restricts one of the factors of the product to some class of graphs and proves the inequality of the conjecture then holds when the other factor is any graph. The other approach utilizes the so-called Clark-Suen partition for proving a weaker inequality that holds for all pairs of graphs. We demonstrate the strength of our framework by improving the bound of Clark and Suen as follows: $ \gamma (X \square Y) \ge \max \{\frac{1}{2} \gamma (X) \gamma_t (Y), \frac{1}{2} \gamma_t (X) \gamma (Y) \} $, where $ \gamma $ stands for the domination number, $ \gamma_t $ is the total domination number, and $ X \square Y $ is the Cartesian product of graphs $X$ and $Y$.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2021, 41, 3; 749-762
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    On Grundy Total Domination Number in Product Graphs
    Autorzy:
    Brešar, Boštjan
    Bujtás, Csilla
    Gologranc, Tanja
    Klavžar, Sandi
    Košmrlj, Gašper
    Marc, Tilen
    Patkós, Balázs
    Tuza, Zsolt
    Vizer, Máté
    Powiązania:
    https://bibliotekanauki.pl/articles/32083828.pdf
    Data publikacji:
    2021-02-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    total domination
    Grundy total domination number
    graph product
    Opis:
    A longest sequence $(v_1, . . ., v_k)$ of vertices of a graph $G$ is a Grundy total dominating sequence of $G$ if for all $i$, \(N(υ_i)\backslash\bigcup_{j=1}^{i-1}N(υ_j)≠∅\). The length $k$ of the sequence is called the Grundy total domination number of $G$ and denoted $\gamma_{gr}^t(G)$. In this paper, the Grundy total domination number is studied on four standard graph products. For the direct product we show that $\gamma_{gr}^t(G×H)≥\gamma_{gr}^t(G)\gamma_{gr}^t(H)$, conjecture that the equality always holds, and prove the conjecture in several special cases. For the lexicographic product we express $\gamma_{gr}^t(G∘H)$ in terms of related invariant of the factors and find some explicit formulas for it. For the strong product, lower bounds on $\gamma_{gr}^t(G⊠H)$ are proved as well as upper bounds for products of paths and cycles. For the Cartesian product we prove lower and upper bounds on the Grundy total domination number when factors are paths or cycles.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2021, 41, 1; 225-247
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
      Wyświetlanie 1-11 z 11

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