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A characterization of locating-total domination edge critical graphs

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Blidia M., Dali W.

Discussiones Mathematicae Graph Theory

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2011

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tom 31

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nr 1

197-202

EN

For a graph G = (V,E) without isolated vertices, a subset D of vertices of V is a total dominating set (TDS) of G if every vertex in V is adjacent to a vertex in D. The total domination number γₜ(G) is the minimum cardinality of a TDS of G. A subset D of V which is a total dominating set, is a locating-total dominating set, or just a LTDS of G, if for any two distinct vertices u and v of V(G)∖D, $N_G(u) ∩ D ≠ N_G(v) ∩ D$. The locating-total domination number $γ_L^t(G)$ is the minimum cardinality of a locating-total dominating set of G. A graph G is said to be a locating-total domination edge removal critical graph, or just a $γ_L^{t+}$-ER-critical graph, if $γ_L^t(G-e) > γ_L^t(G)$ for all e non-pendant edge of E. The purpose of this paper is to characterize the class of $γ_L^{t+}$-ER-critical graphs.

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Gallai's innequality for critical graphs of reducible hereditary properties

80%

Mihók P., Skrekovski R.

Discussiones Mathematicae Graph Theory

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2001

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tom 21

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nr 2

167-177

EN

In this paper Gallai's inequality on the number of edges in critical graphs is generalized for reducible additive induced-hereditary properties of graphs in the following way. Let $𝓟₁,𝓟₂,...,𝓟ₖ$ (k ≥ 2) be additive induced-hereditary properties, $𝓡 = 𝓟₁ ∘ 𝓟₂ ∘ ... ∘𝓟ₖ$ and $δ = ∑_{i=1}^k δ(𝓟_i)$. Suppose that G is an 𝓡 -critical graph with n vertices and m edges. Then 2m ≥ δn + (δ-2)/(δ²+2δ-2)*n + (2δ)/(δ²+2δ-2) unless 𝓡 = 𝓞² or $G = K_{δ+1}$. The generalization of Gallai's inequality for 𝓟-choice critical graphs is also presented.

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Critical Graphs for R(Pn, Pm) and the Star-Critical Ramsey Number for Paths

80%

Hook J.

Discussiones Mathematicae Graph Theory

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2015

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tom 35

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nr 4

689-701

EN

The graph Ramsey number R(G,H) is the smallest integer r such that every 2-coloring of the edges of Kr contains either a red copy of G or a blue copy of H. The star-critical Ramsey number r∗(G,H) is the smallest integer k such that every 2-coloring of the edges of Kr − K1,r−1−k contains either a red copy of G or a blue copy of H. We will classify the critical graphs, 2-colorings of the complete graph on R(G,H) − 1 vertices with no red G or blue H, for the path-path Ramsey number. This classification will be used in the proof of r∗(Pn, Pm).

Ograniczanie wyników

3 Discussiones Mathematicae Graph Theory

1 Blidia M.

1 Dali W.

1 Hook J.

1 Mihók P.

1 Skrekovski R.

1 2015

1 2011

1 2001

A characterization of locating-total domination edge critical graphs

Gallai's innequality for critical graphs of reducible hereditary properties

Critical Graphs for R(Pn, Pm) and the Star-Critical Ramsey Number for Paths