Hardness results on generalized connectivity

Details Hardness results on generalized connectivity Hardness results on generalized connectivity

2010-05-04

arxiv.org/abs/1005.0488v1

Mathematics

Combinatorics

10 pages

Scientific paper

Let $G$ be a nontrivial connected graph of order $n$ and let $k$ be an integer with $2\leq k\leq n$. For a set $S$ of $k$ vertices of $G$, let $\kappa (S)$ denote the maximum number $\ell$ of edge-disjoint trees $T_1,T_2,...,T_\ell$ in $G$ such that $V(T_i)\cap V(T_j)=S$ for every pair $i,j$ of distinct integers with $1\leq i,j\leq \ell$. A collection $\{T_1,T_2,...,T_\ell\}$ of trees in $G$ with this property is called an internally disjoint set of trees connecting $S$. Chartrand et al. generalized the concept of connectivity as follows: The $k$-$connectivity$, denoted by $\kappa_k(G)$, of $G$ is defined by $\kappa_k(G)=$min$\{\kappa(S)\}$, where the minimum is taken over all $k$-subsets $S$ of $V(G)$. Thus $\kappa_2(G)=\kappa(G)$, where $\kappa(G)$ is the connectivity of $G$, for which there are polynomial-time algorithms to solve it. This paper mainly focus on the complexity of the generalized connectivity. At first, we obtain that for two fixed positive integers $k_1$ and $k_2$, given a graph $G$ and a $k_1$-subset $S$ of $V(G)$, the problem of deciding whether $G$ contains $k_2$ internally disjoint trees connecting $S$ can be solved by a polynomial-time algorithm. Then, we show that when $k_1$ is a fixed integer of at least 4, but $k_2$ is not a fixed integer, the problem turns out to be NP-complete. On the other hand, when $k_2$ is a fixed integer of at least 2, but $k_1$ is not a fixed integer, we show that the problem also becomes NP-complete. Finally we give some open problems.

Affiliated with

Also associated with

No associations

Rating

Hardness results on generalized connectivity does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Hardness results on generalized connectivity, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Hardness results on generalized connectivity will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-531962