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1
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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On k-factor-critical graphs

100%
Favaron O.
Discussiones Mathematicae Graph Theory
|
1996
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tom 16
|
nr 1
41-51
EN
A graph is said to be k-factor-critical if the removal of any set of k vertices results in a graph with a perfect matching. We study some properties of k-factor-critical graphs and show that many results on q-extendable graphs can be improved using this concept.
2
Content available remote

The B-Domatic Number of a Graph

100%
Favaron O.
Discussiones Mathematicae Graph Theory
|
2013
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tom 33
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nr 4
747-757
EN
Besides the classical chromatic and achromatic numbers of a graph related to minimum or minimal vertex partitions into independent sets, the b-chromatic number was introduced in 1998 thanks to an alternative definition of the minimality of such partitions. When independent sets are replaced by dominating sets, the parameters corresponding to the chromatic and achromatic numbers are the domatic and adomatic numbers d(G) and ad(G). We introduce the b-domatic number bd(G) as the counterpart of the b-chromatic number by giving an alternative definition of the maximality of a partition into dominating sets. We initiate the study of bd(G) by giving some properties and examples.
3
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Pancyclism and small cycles in graphs

51%
Faudree R., Favaron O., Flandrin E., Li H.
Discussiones Mathematicae Graph Theory
|
1996
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tom 16
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nr 1
27-40
EN
We first show that if a graph G of order n contains a hamiltonian path connecting two nonadjacent vertices u and v such that d(u)+d(v) ≥ n, then G is pancyclic. By using this result, we prove that if G is hamiltonian with order n ≥ 20 and if G has two nonadjacent vertices u and v such that d(u)+d(v) ≥ n+z, where z = 0 when n is odd and z = 1 otherwise, then G contains a cycle of length m for each 3 ≤ m ≤ max (d_C(u,v)+1, [(n+19)/13]), $d_C(u,v)$ being the distance of u and v on a hamiltonian cycle of G.
4
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Factor-criticality and matching extension in DCT-graphs

51%
Favaron O., Favaron E., Ryjáček Z.
Discussiones Mathematicae Graph Theory
|
1997
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tom 17
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nr 2
271-278
EN
The class of DCT-graphs is a common generalization of the classes of almost claw-free and quasi claw-free graphs. We prove that every even (2p+1)-connected DCT-graph G is p-extendable, i.e., every set of p independent edges of G is contained in a perfect matching of G. This result is obtained as a corollary of a stronger result concerning factor-criticality of DCT-graphs.
5
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Maximal k-independent sets in graphs

51%
Blidia M., Chellali M., Favaron O., Meddah N.
Discussiones Mathematicae Graph Theory
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2008
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tom 28
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nr 1
151-163
EN
A subset of vertices of a graph G is k-independent if it induces in G a subgraph of maximum degree less than k. The minimum and maximum cardinalities of a maximal k-independent set are respectively denoted iₖ(G) and βₖ(G). We give some relations between βₖ(G) and $β_j(G)$ and between iₖ(G) and $i_j(G)$ for j ≠ k. We study two families of extremal graphs for the inequality i₂(G) ≤ i(G) + β(G). Finally we give an upper bound on i₂(G) and a lower bound when G is a cactus.
6
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Matchings and total domination subdivision number in graphs with few induced 4-cycles

51%
Favaron O., Karami H., Khoeilar R., Sheikholeslami S.
Discussiones Mathematicae Graph Theory
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2010
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tom 30
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nr 4
611-618
EN
A set S of vertices of a graph G = (V,E) without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number γₜ(G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number $sd_{γₜ(G)}$ is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. Favaron, Karami, Khoeilar and Sheikholeslami (Journal of Combinatorial Optimization, to appear) conjectured that: For any connected graph G of order n ≥ 3, $sd_{γₜ(G)} ≤ γₜ(G)+1$. In this paper we use matchings to prove this conjecture for graphs with at most three induced 4-cycles through each vertex.
7
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Offensive alliances in graphs

33%
Favaron O., Fricke G., Goddard W., Hedetniemi S., Hedetniemi S., Kristiansen P., Laskar R., Skaggs R.
Discussiones Mathematicae Graph Theory
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2004
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tom 24
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nr 2
263-275
EN
A set S is an offensive alliance if for every vertex v in its boundary N(S)- S it holds that the majority of vertices in v's closed neighbourhood are in S. The offensive alliance number is the minimum cardinality of an offensive alliance. In this paper we explore the bounds on the offensive alliance and the strong offensive alliance numbers (where a strict majority is required). In particular, we show that the offensive alliance number is at most 2/3 the order and the strong offensive alliance number is at most 5/6 the order.
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