168 research outputs found

    A survey on the generalized connectivity of graphs

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    The generalized kk-connectivity ΞΊk(G)\kappa_k(G) of a graph GG was introduced by Hager before 1985. As its a natural counterpart, we introduced the concept of generalized edge-connectivity Ξ»k(G)\lambda_k(G), recently. In this paper we summarize the known results on the generalized connectivity and generalized edge-connectivity. After an introductory section, the paper is then divided into nine sections: the generalized (edge-)connectivity of some graph classes, algorithms and computational complexity, sharp bounds of ΞΊk(G)\kappa_k(G) and Ξ»k(G)\lambda_k(G), graphs with large generalized (edge-)connectivity, Nordhaus-Gaddum-type results, graph operations, extremal problems, and some results for random graphs and multigraphs. It also contains some conjectures and open problems for further studies.Comment: 51 pages. arXiv admin note: text overlap with arXiv:1303.3881 by other author

    Two bounds for generalized 33-connectivity of Cartesian product graphs

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    The generalized kk-connectivity ΞΊk(G)\kappa_{k}(G) of a graph GG, which was introduced by Chartrand et al.(1984) is a generalization of the concept of vertex connectivity. Let GG and HH be nontrivial connected graphs. Recently, Li et al. gave a lower bound for the generalized 33-connectivity of the Cartesian product graph Gβ–‘HG \square H and proposed a conjecture for the case that HH is 33-connected. In this paper, we give two different forms of lower bounds for the generalized 33-connectivity of Cartesian product graphs. The first lower bound is stronger than theirs, and the second confirms their conjecture

    The minimal size of a graph with given generalized 3-edge-connectivity

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    For SβŠ†V(G)S\subseteq V(G) and ∣S∣β‰₯2|S|\geq 2, Ξ»(S)\lambda(S) is the maximum number of edge-disjoint trees connecting SS in GG. For an integer kk with 2≀k≀n2\leq k\leq n, the \emph{generalized kk-edge-connectivity} Ξ»k(G)\lambda_k(G) of GG is then defined as Ξ»k(G)=min{Ξ»(S):SβŠ†V(G)Β and ∣S∣=k}\lambda_k(G)= min\{\lambda(S) : S\subseteq V(G) \ and \ |S|=k\}. It is also clear that when ∣S∣=2|S|=2, Ξ»2(G)\lambda_2(G) is nothing new but the standard edge-connectivity Ξ»(G)\lambda(G) of GG. In this paper, graphs of order nn such that Ξ»3(G)=nβˆ’3\lambda_3(G)=n-3 is characterized. Furthermore, we determine the minimal number of edges of a graph of order nn with Ξ»3=1,nβˆ’3,nβˆ’2\lambda_3=1,n-3,n-2 and give a sharp lower bound for 2≀λ3≀nβˆ’42\leq \lambda_3\leq n-4.Comment: 10 page

    Pendant-tree connectivity of line graphs

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    The concept of pendant-tree connectivity, introduced by Hager in 1985, is a generalization of classical vertex-connectivity. In this paper, we study pendant-tree connectivity of line graphs.Comment: 19 pagers, 2 figures. arXiv admin note: substantial text overlap with arXiv:1603.03995, arXiv:1508.07202, arXiv:1508.07149. text overlap with arXiv:1103.6095 by other author

    Note on the minimal size of a graph with generalized connectivity kappa_3= 2

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    The concept of generalized kk-connectivity ΞΊk(G)\kappa_{k}(G) of a graph GG was introduced by Chartrand et al. in recent years. In our early paper, extremal theory for this graph parameter was started. We determined the minimal number of edges of a graph of order nn with ΞΊ3=2\kappa_{3}= 2, i.e., for a graph GG of order nn and size e(G)e(G) with ΞΊ3(G)=2\kappa_{3}(G)= 2, we proved that e(G)β‰₯(6/5)ne(G)\geq (6/5)n, and the lower bound is sharp by constructing a class of graphs, only for n≑0Β (modΒ 5)n\equiv 0 \ (mod \ 5) and nβ‰ 10n\neq 10. In this paper, we improve the lower bound to ⌈(6/5)nβŒ‰\lceil(6/5)n\rceil. Moreover, we show that for all nβ‰₯4n\geq 4 but n=9,10n= 9, 10, there always exists a graph of order nn with ΞΊ3=2\kappa_{3}= 2 whose size attains the lower bound ⌈(6/5)nβŒ‰\lceil(6/5)n\rceil. Whereas for n=9,10n= 9, 10 we give examples to show that ⌈(6/5)nβŒ‰+1\lceil(6/5)n\rceil+1 is the best possible lower bound. This gives a clear picture on the minimal size of a graph of order nn with generalized connectivity ΞΊ3=2\kappa_{3}= 2.Comment: 7 page

    Constructing Internally Disjoint Pendant Steiner Trees in Cartesian Product Networks

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    The concept of pedant tree-connectivity was introduced by Hager in 1985. For a graph G=(V,E)G=(V,E) and a set SβŠ†V(G)S\subseteq V(G) of at least two vertices, \emph{an SS-Steiner tree} or \emph{a Steiner tree connecting SS} (or simply, \emph{an SS-tree}) is a such subgraph T=(Vβ€²,Eβ€²)T=(V',E') of GG that is a tree with SβŠ†Vβ€²S\subseteq V'. For an SS-Steiner tree, if the degree of each vertex in SS is equal to one, then this tree is called a \emph{pedant SS-Steiner tree}. Two pedant SS-Steiner trees TT and Tβ€²T' are said to be \emph{internally disjoint} if E(T)∩E(Tβ€²)=βˆ…E(T)\cap E(T')=\varnothing and V(T)∩V(Tβ€²)=SV(T)\cap V(T')=S. For SβŠ†V(G)S\subseteq V(G) and ∣S∣β‰₯2|S|\geq 2, the \emph{local pedant tree-connectivity} Ο„G(S)\tau_G(S) is the maximum number of internally disjoint pedant SS-Steiner trees in GG. For an integer kk with 2≀k≀n2\leq k\leq n, \emph{pedant tree kk-connectivity} is defined as Ο„k(G)=min⁑{Ο„G(S)β€‰βˆ£β€‰SβŠ†V(G),∣S∣=k}\tau_k(G)=\min\{\tau_G(S)\,|\,S\subseteq V(G),|S|=k\}. In this paper, we prove that for any two connected graphs GG and HH, Ο„3(Gβ–‘H)β‰₯min⁑{3βŒŠΟ„3(G)2βŒ‹,3βŒŠΟ„3(H)2βŒ‹}\tau_3(G\Box H)\geq \min\{3\lfloor\frac{\tau_3(G)}{2}\rfloor,3\lfloor\frac{\tau_3(H)}{2}\rfloor\}. Moreover, the bound is sharp.Comment: 22 page

    Path connectivity of line graphs

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    Dirac showed that in a (kβˆ’1)(k-1)-connected graph there is a path through each kk vertices. The path kk-connectivity Ο€k(G)\pi_k(G) of a graph GG, which is a generalization of Dirac's notion, was introduced by Hager in 1986. In this paper, we study path connectivity of line graphs.Comment: 15 pages. arXiv admin note: substantial text overlap with arXiv:1508.07202, arXiv:1207.1838; text overlap with arXiv:1103.6095 by other author

    The minimal size of a graph with generalized connectivity ΞΊ3=2\kappa_3 = 2

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    Let GG be a nontrivial connected graph of order nn and kk an integer with 2≀k≀n2\leq k\leq n. For a set SS of kk vertices of GG, let ΞΊ(S)\kappa (S) denote the maximum number β„“\ell of edge-disjoint trees T1,T2,...,Tβ„“T_1,T_2,...,T_\ell in GG such that V(Ti)∩V(Tj)=SV(T_i)\cap V(T_j)=S for every pair i,ji,j of distinct integers with 1≀i,j≀ℓ1\leq i,j\leq \ell. Chartrand et al. generalized the concept of connectivity as follows: The kk-connectivityconnectivity, denoted by ΞΊk(G)\kappa_k(G), of GG is defined by ΞΊk(G)=\kappa_k(G)=min{ΞΊ(S)}\{\kappa(S)\}, where the minimum is taken over all kk-subsets SS of V(G)V(G). Thus ΞΊ2(G)=ΞΊ(G)\kappa_2(G)=\kappa(G), where ΞΊ(G)\kappa(G) is the connectivity of GG. This paper mainly focuses on the minimal number of edges of a graph GG with ΞΊ3(G)=2\kappa_{3}(G)= 2. For a graph GG of order v(G)v(G) and size e(G)e(G) with ΞΊ3(G)=2\kappa_{3}(G)= 2, we obtain that e(G)β‰₯6/5v(G)e(G)\geq 6/5v(G), and the lower bound is sharp by showing a class of examples attaining the lower bound.Comment: 9 page

    The minimal size of graphs with given pendant-tree connectivity

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    The concept of pendant-tree kk-connectivity Ο„k(G)\tau_k(G) of a graph GG, introduced by Hager in 1985, is a generalization of classical vertex-connectivity. Let f(n,k,β„“)f(n,k,\ell) be the minimal number of edges of a graph GG of order nn with Ο„k(G)=β„“Β (1≀ℓ≀nβˆ’k)\tau_k(G)=\ell \ (1\leq \ell\leq n-k). In this paper, we give some exact value or sharp bounds of the parameter f(n,k,β„“)f(n,k,\ell).Comment: 19 pages. arXiv admin note: substantial text overlap with arXiv:1508.07202, arXiv:1508.07149, arXiv:1603.03995, arXiv:1604.01887; text overlap with arXiv:1103.6095 by other author

    On the pedant tree-connectivity of graphs

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    The concept of pedant tree-connectivity was introduced by Hager in 1985. For a graph G=(V,E)G=(V,E) and a set SβŠ†V(G)S\subseteq V(G) of at least two vertices, \emph{an SS-Steiner tree} or \emph{a Steiner tree connecting SS} (or simply, \emph{an SS-tree}) is a such subgraph T=(Vβ€²,Eβ€²)T=(V',E') of GG that is a tree with SβŠ†Vβ€²S\subseteq V'. For an SS-Steiner tree, if the degree of each vertex in SS is equal to one, then this tree is called a \emph{pedant SS-Steiner tree}. Two pedant SS-Steiner trees TT and Tβ€²T' are said to be \emph{internally disjoint} if E(T)∩E(Tβ€²)=βˆ…E(T)\cap E(T')=\varnothing and V(T)∩V(Tβ€²)=SV(T)\cap V(T')=S. For SβŠ†V(G)S\subseteq V(G) and ∣S∣β‰₯2|S|\geq 2, the \emph{local pedant-tree connectivity} Ο„G(S)\tau_G(S) is the maximum number of internally disjoint pedant SS-Steiner trees in GG. For an integer kk with 2≀k≀n2\leq k\leq n, \emph{kk-pedant tree-connectivity} is defined as Ο„k(G)=min⁑{Ο„G(S)β€‰βˆ£β€‰SβŠ†V(G),∣S∣=k}\tau_k(G)=\min\{\tau_G(S)\,|\,S\subseteq V(G),|S|=k\}. In this paper, we first study the sharp bounds of pedant tree-connectivity. Next, we obtain the exact value of a threshold graph, and give an upper bound of the pedant-tree kk-connectivity of a complete multipartite graph. For a connected graph GG, we show that 0≀τk(G)≀nβˆ’k0\leq \tau_k(G)\leq n-k, and graphs with Ο„k(G)=nβˆ’k,nβˆ’kβˆ’1,nβˆ’kβˆ’2,0\tau_k(G)=n-k,n-k-1,n-k-2,0 are characterized in this paper. In the end, we obtain the Nordhaus-Guddum type results for pedant tree-connectivity.Comment: 25 page
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