On the restricted connectivity and superconnectivity in graphs with given girth

Balbuena, C.; Cera, M.; Diánez, A.; García–Vázquez, P.; Marcote, X.

doi:10.1016/j.disc.2006.07.016

Cited by 22 publications

(9 citation statements)

References 7 publications

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“…In fact, (G) > (G) is equivalent to saying that the graph G is edge-superconnected, i.e., to saying that every minimum edge-cut is equal to the edge-neighborhood of some vertex of minimum degree, see [6,7]. Some sufficient conditions for a graph to be -optimal have been given in terms of the girth in [1,2].…”

Section: Terminology and Notationmentioning

confidence: 97%

On the edge-connectivity and restricted edge-connectivity of a product of graphs

Balbuena

Cera

Diánez

et al. 2007

Discrete Applied Mathematics

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The product graph G m * G p of two given graphs G m and G p was defined by Bermond et al. [Large graphs with given degree and diameter II, J. Combin. Theory Ser. B 36 (1984) 32-48]. For this kind of graphs we provide bounds for two connectivity parameters ( and , edge-connectivity and restricted edge-connectivity, respectively), and state sufficient conditions to guarantee optimal values of these parameters. Moreover, we compare our results with other previous related ones for permutation graphs and cartesian product graphs, obtaining several extensions and improvements. In this regard, for any two connected graphs G m ,

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Section: Terminology and Notationmentioning

confidence: 97%

On the edge-connectivity and restricted edge-connectivity of a product of graphs

Balbuena

Cera

Diánez

et al. 2007

Discrete Applied Mathematics

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“…The restricted connectivity κ (G) and super connectivity κ 1 (G) are related [2,4] and have received the attention of many researchers (see, for example, [1,9,14,15,19]). Obviously, κ 1 (G) ≤ κ (G) for any restricted vertex-cut S is a super vertex-cut.…”

Section: It Is Well Known That κ(G) ≤ δ(G) a Graph G Is Said To Be Mmentioning

confidence: 99%

“…Suppose v = (x, y ) ∈ X j is one of the vertices that have a neighbor in G 2x 2 . Take x 1 ∈ W arbitrarily.…”

mentioning

confidence: 99%

On super connectivity of Cartesian product graphs

Lü

Chen

et al. 2008

Networks

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The super connectivity κ 1 of a connected graph G is the minimum number of vertices whose deletion results in a disconnected graph without isolated vertices; this is a more refined index than the connectivity parameter κ. This article provides bounds for the super connectivity κ 1 of the Cartesian product of two connected graphs, and thus generalizes the main result of Shieh on the super connectedness of the Cartesian product of two regular graphs with maximum connectivity. Particularly, we determine that κ 1 (K m × K n ) = min{m + 2n − 4, 2m + n − 4} for m + n ≥ 6 and state sufficient conditions to guarantee κ 1 (K 2 × G) = 2κ(G). As a consequence, we immediately obtain the super connectivity of the n-cube for n ≥ 3.

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“…In [1], the authors have studied the κ 2 -connected graphs. We will study the relation between κ 3 (G) and ξ 3 (G) of κ 3 -connected graphs with given girth in the next part.…”

Section: §1 Introductionmentioning

confidence: 99%

3-restricted connectivity of graphs with given girth

Guo

Meng

2008

Appl. Math. J. Chin. Univ.

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Let G = (V, E) be a connected graph. X ⊂ V (G) is a vertex set. X is a 3-restricted cut of G, if G−X is not connected and every component of G−X has at least three vertices. The 3-restricted connectivity κ3(G) (in short κ3) of G is the cardinality of a minimum 3-restrictedA network is often modelled by a graph G = (V, E) with the vertices representing nodes such as processors or stations, and the edges representing links between the nodes. Throughout this paper, we assume the graphs considered are simple.Let d(u, v) denotes the length of a shortest (u, v)-path. If X, Y ⊂ V , d(X, Y ) = min{d(x, y) : for any x ∈ X and any y ∈ Y } denotes the distance between X and Y . v ∈ V, r ≥ 0 is an integer,is the subgraph induced by X. We denote the diameter and girth by D and g, respectively, and write G − v for G − {v}. A path is called k-path, if its length of edges is k.An edge set S is called a k-restricted edge cut of G, if G − S is not connected and every connected component of G−S has at least k vertices. The cardinality of a minimum k-restricted edge cut is the k-restricted edge connectivity of G, denoted by λ k (G). As for the recent studies in this aspect, we can see [2][3][4][5][6][7][8][9].

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On the restricted connectivity and superconnectivity in graphs with given girth

Cited by 22 publications

References 7 publications

On the edge-connectivity and restricted edge-connectivity of a product of graphs

On the edge-connectivity and restricted edge-connectivity of a product of graphs

On super connectivity of Cartesian product graphs

3-restricted connectivity of graphs with given girth

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