2 research outputs found
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
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