168 research outputs found
A survey on the generalized connectivity of graphs
The generalized -connectivity of a graph was introduced
by Hager before 1985. As its a natural counterpart, we introduced the concept
of generalized edge-connectivity , 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 and
, 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 -connectivity of Cartesian product graphs
The generalized -connectivity of a graph , which was
introduced by Chartrand et al.(1984) is a generalization of the concept of
vertex connectivity. Let and be nontrivial connected graphs. Recently,
Li et al. gave a lower bound for the generalized -connectivity of the
Cartesian product graph and proposed a conjecture for the case
that is -connected. In this paper, we give two different forms of lower
bounds for the generalized -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
For and , is the maximum number of
edge-disjoint trees connecting in . For an integer with , the \emph{generalized -edge-connectivity} of is then
defined as .
It is also clear that when , is nothing new but the
standard edge-connectivity of . In this paper, graphs of order
such that is characterized. Furthermore, we determine
the minimal number of edges of a graph of order with
and give a sharp lower bound for .Comment: 10 page
Pendant-tree connectivity of line graphs
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
The concept of generalized -connectivity of a graph
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 with , i.e., for a graph
of order and size with , we proved that
, and the lower bound is sharp by constructing a class of
graphs, only for and . In this paper, we
improve the lower bound to . Moreover, we show that for all
but , there always exists a graph of order with
whose size attains the lower bound .
Whereas for we give examples to show that is
the best possible lower bound. This gives a clear picture on the minimal size
of a graph of order with generalized connectivity .Comment: 7 page
Constructing Internally Disjoint Pendant Steiner Trees in Cartesian Product Networks
The concept of pedant tree-connectivity was introduced by Hager in 1985. For
a graph and a set of at least two vertices,
\emph{an -Steiner tree} or \emph{a Steiner tree connecting } (or simply,
\emph{an -tree}) is a such subgraph of that is a tree with
. For an -Steiner tree, if the degree of each vertex in
is equal to one, then this tree is called a \emph{pedant -Steiner tree}. Two
pedant -Steiner trees and are said to be \emph{internally disjoint}
if and . For
and , the \emph{local pedant tree-connectivity} is the
maximum number of internally disjoint pedant -Steiner trees in . For an
integer with , \emph{pedant tree -connectivity} is
defined as . In this
paper, we prove that for any two connected graphs and , .
Moreover, the bound is sharp.Comment: 22 page
Path connectivity of line graphs
Dirac showed that in a -connected graph there is a path through each
vertices. The path -connectivity of a graph , 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
Let be a nontrivial connected graph of order and an integer with
. For a set of vertices of , let denote
the maximum number of edge-disjoint trees in
such that for every pair of distinct integers with
. Chartrand et al. generalized the concept of connectivity
as follows: The -, denoted by , of is defined
by min, where the minimum is taken over all
-subsets of . Thus , where is
the connectivity of .
This paper mainly focuses on the minimal number of edges of a graph with
. For a graph of order and size with
, we obtain that , 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
The concept of pendant-tree -connectivity of a graph ,
introduced by Hager in 1985, is a generalization of classical
vertex-connectivity. Let be the minimal number of edges of a
graph of order with . In this
paper, we give some exact value or sharp bounds of the parameter .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
The concept of pedant tree-connectivity was introduced by Hager in 1985. For
a graph and a set of at least two vertices,
\emph{an -Steiner tree} or \emph{a Steiner tree connecting } (or simply,
\emph{an -tree}) is a such subgraph of that is a tree with
. For an -Steiner tree, if the degree of each vertex in
is equal to one, then this tree is called a \emph{pedant -Steiner tree}. Two
pedant -Steiner trees and are said to be \emph{internally disjoint}
if and . For
and , the \emph{local pedant-tree connectivity} is the
maximum number of internally disjoint pedant -Steiner trees in . For an
integer with , \emph{-pedant tree-connectivity} is
defined as . 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 -connectivity of a complete multipartite graph. For a connected
graph , we show that , and graphs with
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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