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Some Toughness Results in Independent Domination Critical Graphs

100%
Ananchuen N., Ananchuen W.
Discussiones Mathematicae Graph Theory
|
2015
|
tom 35
|
nr 4
703-713
EN
A subset S of V (G) is an independent dominating set of G if S is independent and each vertex of G is either in S or adjacent to some vertex of S. Let i(G) denote the minimum cardinality of an independent dominating set of G. A graph G is k-i-critical if i(G) = k, but i(G+uv) < k for any pair of non-adjacent vertices u and v of G. In this paper, we establish that if G is a connected 3-i-critical graph and S is a vertex cutset of G with |S| ≥ 3, then [...] improving a result proved by Ao [3], where ω(G−S) denotes the number of components of G−S. We also provide a characteriza- tion of the connected 3-i-critical graphs G attaining the maximum number of ω(G − S) when S is a minimum cutset of size 2 or 3.
2
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Long cycles and neighborhood union in 1-tough graphs with large degree sums

80%
Hoa V.
Discussiones Mathematicae Graph Theory
|
1998
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tom 18
|
nr 1
5-13
EN
For a 1-tough graph G we define σ₃(G) = min{d(u) + d(v) + d(w):{u,v,w} is an independent set of vertices} and $NC_{σ₃-n+5}(G)$ = $max{⋃_{i = 1}^{σ₃-n+5}$ $N(v_i) : {v₁, ..., v_{σ₃-n+5}}$ is an independent set of vertices}. We show that every 1-tough graph with σ₃(G) ≥ n contains a cycle of length at least $min{n,2NC_{σ₃-n+5}(G)+2}$. This result implies some well-known results of Faßbender [2] and of Flandrin, Jung & Li [6]. The main result of this paper also implies that c(G) ≥ min{n,2NC₂(G)+2} where NC₂(G) = min{|N(u) ∪ N(v)|:d(u,v) = 2}. This strengthens a result that c(G) ≥ min{n, 2NC₂(G)} of Bauer, Fan and Veldman [3].
3
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The Existence Of P≥3-Factor Covered Graphs

70%
Zhou S., Wu J., Zhang T.
Discussiones Mathematicae Graph Theory
|
2017
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tom 37
|
nr 4
1055-1065
EN
A spanning subgraph F of a graph G is called a P≥3-factor of G if every component of F is a path of order at least 3. A graph G is called a P≥3-factor covered graph if G has a P≥3-factor including e for any e ∈ E(G). In this paper, we obtain three sufficient conditions for graphs to be P≥3-factor covered graphs. Furthermore, it is shown that the results are sharp.
4
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Some recent results on domination in graphs

61%
Plummer M.
Discussiones Mathematicae Graph Theory
|
2006
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tom 26
|
nr 3
457-474
EN
In this paper, we survey some new results in four areas of domination in graphs, namely: (1) the toughness and matching structure of graphs having domination number 3 and which are "critical" in the sense that if one adds any missing edge, the domination number falls to 2; (2) the matching structure of graphs having domination number 3 and which are "critical" in the sense that if one deletes any vertex, the domination number falls to 2; (3) upper bounds on the domination number of cubic graphs; and (4) upper bounds on the domination number of graphs embedded in surfaces.
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