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On the generalized (edge-)connectivity of graphs
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It is obtained that $\lambda(G)-1\leq \lambda_3(G)$ if $G$ is a connected planar graph, and the relation between the generalized3-connectivity and generalized 3-edge-Connectivity of a graph and its line graph is studied.Abstract:
The generalized $k$-connectivity $\kappa_k(G)$ of a graph $G$ was introduced by Chartrand et al. in 1984. It is natural to introduce the concept of generalized $k$-edge-connectivity $\lambda_k(G)$. For general $k$, the generalized $k$-edge-connectivity of a complete graph is obtained. For $k\geq 3$, tight upper and lower bounds of $\kappa_k(G)$ and $\lambda_k(G)$ are given for a connected graph $G$ of order $n$, that is, $1\leq \kappa_k(G)\leq n-\lceil\frac{k}{2}\rceil$ and $1\leq \lambda_k(G)\leq n-\lceil\frac{k}{2}\rceil$. Graphs of order $n$ such that $\kappa_k(G)=n-\lceil\frac{k}{2}\rceil$ and $\lambda_k(G)=n-\lceil\frac{k}{2}\rceil$ are characterized, respectively. Nordhaus-Gaddum-type results for the generalized $k$-connectivity are also obtained. For $k=3$, we study the relation between the edge-connectivity and the generalized 3-edge-connectivity of a graph. Upper and lower bounds of $\lambda_3(G)$ for a graph $G$ in terms of the edge-connectivity $\lambda$ of $G$ are obtained, that is, $\frac{3\lambda-2}{4}\leq \lambda_3(G)\leq \lambda$, and two graph classes are given showing that the upper and lower bounds are tight. From these bounds, we obtain that $\lambda(G)-1\leq \lambda_3(G)\leq \lambda(G)$ if $G$ is a connected planar graph, and the relation between the generalized 3-connectivity and generalized 3-edge-connectivity of a graph and its line graph.read more
Citations
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Journal ArticleDOI
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The generalized 3-connectivity of graph products ☆
TL;DR: The generalized k-connectivity κk(G) of a graph G, which was introduced by Chartrand et al. (1984), is a generalization of the concept of vertex connectivity and, for this generalization, the generalized 2-connectivities κ2(G), which is exactly the connectivity of G.
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Nordhaus-Gaddum-type results for the generalized edge-connectivity of graphs
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TL;DR: In this paper, the Nordhaus-Gaddum-type results for the parameter λ-k(G) were considered and sharp upper and lower bounds of λ k(G), λ 2 (G) and λ 3 (G 2 ) were derived for a graph of order n and size m.
Posted Content
A survey on the generalized connectivity of graphs
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TL;DR: This paper summarizes the known results on the generalized connectivity and generalized edge-connectivity of graphs with large generalized (edge-)connectivity, Nordhaus-Gaddum-type results, graph operations, extremal problems, and some results for random graphs and multigraphs.
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The generalized 3-connectivity of lexicographic product graphs
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TL;DR: It is proved that for any two connected graphs G and H, κ3(G^H)≥ κ 3(G)|V(H)|, and the upper bounds are sharp.
References
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Algorithms for VLSI Physical Design Automation
TL;DR: This book is a core reference for graduate students and CAD professionals and presents a balance of theory and practice in a intuitive manner.
Proceedings ArticleDOI
Packing Steiner trees
TL;DR: An algorithm with an asymptotic approximation factor of |S|/4 gives a sufficient condition for the existence of k edge-disjoint Steiner trees in a graph in terms of the edge-connectivity of the graph.
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