Note on the connectivity of line graphs

doi:10.1016/j.ipl.2004.03.013

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Volume 91, Issue 1, 16 July 2004, Pages 7-10

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Abstract

Let G be a connected graph with vertex set V(G), edge set E(G), vertex-connectivity κ(G) and edge-connectivity λ(G).

A subset S of E(G) is called a restricted edge-cut if G−S is disconnected and each component contains at least two vertices. The restricted edge-connectivity λ₂(G) is the minimum cardinality over all restricted edge-cuts. Clearly λ₂(G)⩾λ(G)⩾κ(G).

In 1969, Chartrand and Stewart have shown that $κ L(G) ⩾λ(G),$ if λ(G)⩾2, where L(G) denotes the line graph of G.

In the present paper we show that $κ L(G) =λ_{2} (G),$ if |V(G)|⩾4 and G is not a star, which improves the result of Chartrand and Stewart. As a direct consequence of this identity, we obtain the known inequality λ₂(G)⩽ξ(G) by Esfahanian and Hakimi, where ξ(G) is the minimum edge degree.

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A connected graph G is super-connected (resp. super edge-connected) if every minimum vertex-cut (resp. edge-cut) isolates a vertex of G. In [Super connectivity of line graphs, Inform. Process. Lett. 94 (2005) 191–195], Xu et al. shows that a super-connected graph with minimum degree at least 4 is also super edge-connected. In this paper, a characterization of all super-connected but not super edge-connected graphs is given. It follows from this result that there is a unique super-connected but not super edge-connected graph with minimum degree 3, that is, the Ladder graph $L_{3}$ of order 6, and that there are infinitely many super-connected but not super edge-connected graphs with minimum degree 1 or 2.
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Citation Excerpt :
For the study of type I restricted vertex cuts, we refer the reader to the papers [1,9]. For the study of type II restricted vertex cuts, we refer the reader to [11,12]. Definitions and terminology not given here follow [6].
In this paper, we study two types of restricted connectivity: $κ_{k} (G)$ is the cardinality of a minimum vertex cut $S$ such that every component of $G - S$ has at least $k$ vertices; $κ_{k}^{'} (G)$ is the cardinality of a minimum vertex cut $S$ such that there are at least two components in $G - S$ of order at least $k$ . In this paper, we give some sufficient conditions for the existence and upper bound of $κ_{k} (G)$ and/or $κ_{k}^{'} (G)$ , and study some properties of these two parameters.
Maximally edge-connected and vertex-connected graphs and digraphs: A survey
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Let D be a graph or a digraph. If $δ (D)$ is the minimum degree, $λ (D)$ the edge-connectivity and $κ (D)$ the vertex-connectivity, then $κ (D) ⩽ λ (D) ⩽ δ (D)$ is a well-known basic relationship between these parameters. The graph or digraph D is called maximally edge-connected if $λ (D) = δ (D)$ and maximally vertex-connected if $κ (D) = δ (D)$ . In this survey we mainly present sufficient conditions for graphs and digraphs to be maximally edge-connected as well as maximally vertex-connected. We also discuss the concept of conditional or restricted edge-connectivity and vertex-connectivity, respectively.
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In this paper, we explore the 2-extraconnectivity of a special class of graphs $G (G_{0}, G_{1}; M)$ proposed by Chen et al. [Y.-C. Chen, J.J.M. Tan, L.-H. Hsu, S.-S. Kao, Super-connectivity and super edge-connectivity for some interconnection networks, Applied Mathematics and Computation 140 (2003) 245–254]. As applications of the results, we obtain that the 2-extraconnectivities of several well-known interconnection networks, such as hypercubes, twisted cubes, crossed cubes, Möbius cubes and locally twisted cubes, are all equal to $3 n - 5$ when their dimension n is not less than 8. That is, when $n ⩾ 8$ , at least $3 n - 5$ vertices must be removed to disconnect any one of these n-dimensional networks provided that the removal of these vertices does not isolate a vertex or an edge.
On reliability of the folded hypercubes
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In this paper, we explore the 2-extra connectivity and 2-extra-edge-connectivity of the folded hypercube FQ_n. We show that κ₂(FQ_n) = 3n − 2 for n ⩾ 8; and λ₂(FQ_n) = 3n − 1 for n ⩾ 5. That is, for n ⩾ 8 (resp. n ⩾ 5), at least 3n − 2 vertices (resp. 3n − 1 edges) of FQ_n are removed to get a disconnected graph that contains no isolated vertices (resp. edges). When the folded hypercube is used to model the topological structure of a large-scale parallel processing system, these results can provide more accurate measurements for reliability and fault tolerance of the system.
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The super connectivity $κ^{'}$ and the super edge-connectivity $λ^{'}$ are more refined network reliability indices than connectivity κ and edge-connectivity λ. This paper shows that for a connected graph G with order at least four rather than a star and its line graph $L (G)$ , $κ^{'} (L (G)) = λ^{'} (G)$ if and only if G is not super- $λ^{'}$ . As a consequence, we obtain the result of Hellwig et al. [Note on the connectivity of line graphs, Inform. Process. Lett. 91 (2004) 7] that $κ (L (G)) = λ^{'} (G)$ . Furthermore, the authors show that the line graph of a super- $λ^{'}$ graph is super-λ if the minimum degree is at least three.

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Note on the connectivity of line graphs

Abstract

Discrete Math.

Inform. Process. Lett.

Discrete Appl. Math.

Graphs and Digraphs

The connectivity of line-graphs

Math. Ann.