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    Wyświetlanie 1-3 z 3
    Tytuł:
    The well-covered dimension of products of graphs
    Autorzy:
    Birnbaum, Isaac
    Kuneli, Megan
    McDonald, Robyn
    Urabe, Katherine
    Vega, Oscar
    Powiązania:
    https://bibliotekanauki.pl/articles/30148720.pdf
    Data publikacji:
    2014-11-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    well-covered dimension
    maximal independent sets
    Opis:
    We discuss how to find the well-covered dimension of a graph that is the Cartesian product of paths, cycles, complete graphs, and other simple graphs. Also, a bound for the well-covered dimension of $K_n × G$ is found, provided that $G$ has a largest greedy independent decomposition of length $c < n$. Formulae to find the well-covered dimension of graphs obtained by vertex blowups on a known graph, and to the lexicographic product of two known graphs are also given.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2014, 34, 4; 811-827
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    On well-covered graphs of odd girth 7 or greater
    Autorzy:
    Randerath, Bert
    Vestergaard, Preben
    Powiązania:
    https://bibliotekanauki.pl/articles/743557.pdf
    Data publikacji:
    2002
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    well-covered
    independence number
    domination number
    odd girth
    Opis:
    A maximum independent set of vertices in a graph is a set of pairwise nonadjacent vertices of largest cardinality α. Plummer [14] defined a graph to be well-covered, if every independent set is contained in a maximum independent set of G. One of the most challenging problems in this area, posed in the survey of Plummer [15], is to find a good characterization of well-covered graphs of girth 4. We examine several subclasses of well-covered graphs of girth ≥ 4 with respect to the odd girth of the graph. We prove that every isolate-vertex-free well-covered graph G containing neither C₃, C₅ nor C₇ as a subgraph is even very well-covered. Here, a isolate-vertex-free well-covered graph G is called very well-covered, if G satisfies α(G) = n/2. A vertex set D of G is dominating if every vertex not in D is adjacent to some vertex in D. The domination number γ(G) is the minimum order of a dominating set of G. Obviously, the inequality γ(G) ≤ α(G) holds. The family $_{γ=α}$ of graphs G with γ(G) = α(G) forms a subclass of well-covered graphs. We prove that every connected member G of $_{γ=α}$ containing neither C₃ nor C₅ as a subgraph is a K₁, C₄,C₇ or a corona graph.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2002, 22, 1; 159-172
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    On Well-Covered Direct Products
    Autorzy:
    Kuenzel, Kirsti
    Rall, Douglas F.
    Powiązania:
    https://bibliotekanauki.pl/articles/32315158.pdf
    Data publikacji:
    2022-05-01
    Wydawca:
    Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
    Tematy:
    well-covered graph
    direct product of graphs
    isolatable vertex
    Opis:
    A graph G is well-covered if all maximal independent sets of G have the same cardinality. In 1992 Topp and Volkmann investigated the structure of well-covered graphs that have nontrivial factorizations with respect to some of the standard graph products. In particular, they showed that both factors of a well-covered direct product are also well-covered and proved that the direct product of two complete graphs (respectively, two cycles) is well-covered precisely when they have the same order (respectively, both have order 3 or 4). Furthermore, they proved that the direct product of two well-covered graphs with independence number one-half their order is well-covered. We initiate a characterization of nontrivial connected well-covered graphs G and H, whose independence numbers are strictly less than one-half their orders, such that their direct product G × H is well-covered. In particular, we show that in this case both G and H have girth 3 and we present several infinite families of such well-covered direct products. Moreover, we show that if G is a factor of any well-covered direct product, then G is a complete graph unless it is possible to create an isolated vertex by removing the closed neighborhood of some independent set of vertices in G.
    Źródło:
    Discussiones Mathematicae Graph Theory; 2022, 42, 2; 627-640
    2083-5892
    Pojawia się w:
    Discussiones Mathematicae Graph Theory
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
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