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Total domination in cubic Knodel graphs
A subset D of vertices of a graph G is a dominating set if for each u ∈ V (G) \ D, u is adjacent to somevertex v ∈ D. The domination number, γ(G) ofG, is the minimum cardinality of a dominating set of G. A setD ⊆ V (G) is a total dominating set if for eachu ∈ V (G), u is adjacent to some vertex v ∈ D. Thetotal domination number, γt (G) of G, is theminimum cardinality of a total dominating set o...
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Let v(G) and γ(G) denote the number of vertices and the domination number of a graph G, respectively, and let ρ(G) = γ(G)/v(G). In 1996 B. Reed conjectured that if G is a cubic graph, then γ(G) ≤ dv(G)/3e. In 2005 A. Kostochka and B. Stodolsky disproved this conjecture for cubic graphs of connectivity one and maintained that the conjecture may still be true for cubic 2-connected graphs. Their m...
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The HAPPYNET problem is defined as follows : Given a undirected simple graph G with integer weights wvu on its edges vu ∈ E(G), find a function s : V (G) −→ {−1, 1} such that ∀v ∈ V (G), v is happy in G, i.e. such that u∈Γ(v) s(v)s(u)wuv ≥ 0. It is easy to see that HAPPYNET has always a solution, no matter what the input is. However, no polynomial algorithm is known for this problem, which is c...
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ژورنال
عنوان ژورنال: Časopis pro pěstování matematiky
سال: 1982
ISSN: 0528-2195
DOI: 10.21136/cpm.1982.118123