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On Unique Minimum Dominating Sets in Some Cartesian Product Graphs

100%
Hedetniemi J. T.
Discussiones Mathematicae Graph Theory
|
2015
|
tom 35
|
nr 4
615-628
EN
Unique minimum vertex dominating sets in the Cartesian product of a graph with a complete graph are considered. We first give properties of such sets when they exist. We then show that when the first factor of the product is a tree, consideration of the tree alone is sufficient to determine if the product has a unique minimum dominating set.
2
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Fractional domination in prisms

100%
Walsh M.
Discussiones Mathematicae Graph Theory
|
2007
|
tom 27
|
nr 3
541-547
EN
Mynhardt has conjectured that if G is a graph such that γ(G) = γ(πG) for all generalized prisms πG then G is edgeless. The fractional analogue of this conjecture is established and proved by showing that, if G is a graph with edges, then $γ_f(G×K₂) > γ_f(G)$.
3
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A cancellation property for the direct product of graphs

100%
Hammack R.
Discussiones Mathematicae Graph Theory
|
2008
|
tom 28
|
nr 1
179-184
EN
Given graphs A, B and C for which A×C ≅ B×C, it is not generally true that A ≅ B. However, it is known that A×C ≅ B×C implies A ≅ B provided that C is non-bipartite, or that there are homomorphisms from A and B to C. This note proves an additional cancellation property. We show that if B and C are bipartite, then A×C ≅ B×C implies A ≅ B if and only if no component of B admits an involution that interchanges its partite sets.
4
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On the domination number of prisms of graphs

100%
Burger A., Mynhardt C., Weakley W.
Discussiones Mathematicae Graph Theory
|
2004
|
tom 24
|
nr 2
303-318
EN
For a permutation π of the vertex set of a graph G, the graph π G is obtained from two disjoint copies G₁ and G₂ of G by joining each v in G₁ to π(v) in G₂. Hence if π = 1, then πG = K₂×G, the prism of G. Clearly, γ(G) ≤ γ(πG) ≤ 2 γ(G). We study graphs for which γ(K₂×G) = 2γ(G), those for which γ(πG) = 2γ(G) for at least one permutation π of V(G) and those for which γ(πG) = 2γ(G) for each permutation π of V(G).
5
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Associative graph products and their independence, domination and coloring numbers

88%
Nowakowski R., Rall D.
Discussiones Mathematicae Graph Theory
|
1996
|
tom 16
|
nr 1
53-79
EN
Associative products are defined using a scheme of Imrich & Izbicki [18]. These include the Cartesian, categorical, strong and lexicographic products, as well as others. We examine which product ⊗ and parameter p pairs are multiplicative, that is, p(G⊗H) ≥ p(G)p(H) for all graphs G and H or p(G⊗H) ≤ p(G)p(H) for all graphs G and H. The parameters are related to independence, domination and irredundance. This includes Vizing's conjecture directly, and indirectly the Shannon capacity of a graph and Hedetniemi's coloring conjecture.
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