Sur le probléme des courbes gauches en Topologie

Let G be a nontrivial connected graph of order n and let k be an integer with 2?k?n. For a set S of k vertices of G, let ?(S) denote the maximum number ? of edge-disjoint trees T 1,T 2,?,T ? in G such that V(T i )?V(T j )=S for every pair i,j of distinct integers with 1?i,j??. Chartrand et al. generalized the concept of connectivity as follows: The k-connectivity, denoted by ? k (G), of G is defined by ? k (G)=min{?(S)}, where the minimum is taken over all k-subsets S of V(G). Thus ? 2(G)=?(G), where ?(G) is the connectivity of G, for which there are polynomial-time algorithms to solve it.

This paper mainly focus on the complexity of determining the generalized connectivity of a graph. 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.

Note on the hardness of generalized connectivity

The generalized 3-connectivity of star graphs and bubble-sort graphs

Let G be a nontrivial connected graph of order n, and k an integer with 2 � kn. For a set S of k vertices of G, let �(S) denote the maximum numberof edge-disjoint trees T1;T2;:::;Tin G such that V (Ti) \ V (Tj) = S for every pair i;j of distinct integers with 1 � i;j� `. Chartrand et al. generalized the concept of connectivity as follows: The k-c潮nectivity , denoted byk(G), of G is defined byk(G) =minf�(S)g, where the minimum is taken over all k-subsets S of V (G). Thus �2(G) = �(G), where �(G) is the connectivity of G. Moreover, �n(G) is the maximum number of edge-disjoint spanning trees of G. This paper mainly focus on the k-connectivity of complete bipartite graphs Ka;b. First, we obtain the number of edge-disjoint spanning trees of Ka;b, which is b ab a+b−1 c, and specifically give the b ab a+b−1 c edge-disjoint spanning trees. Then based on this result, we get the k-connectivity of Ka;bfor all 2 � ka+b. Namely, if k > b−a+2

The generalized connectivity of complete bipartite graphs

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.

On the generalized (edge-)connectivity of graphs

Sharp bounds for the generalized connectivity κ 3 (G)

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 $G$. 
In general, the investigation of $\kappa_k(G)$ is very difficult. We therefore focus on the investigation on $\kappa_3(G)$ in this paper. We study the relation between the connectivity and the 3-connectivity of a graph. First we give sharp upper and lower bounds of $\kappa_3(G)$ for general graphs $G$, and construct two kinds of graphs which attain the upper and lower bound, respectively. We then show that if $G$ is a connected planar graph, then $\kappa(G)-1 \leq \kappa_3(G)\leq \kappa(G)$, and give some classes of graphs which attain the bounds. In the end we show that the problem whether $\kappa(G)=\kappa_3(G)$ for a planar graph $G$ can be solved in polynomial time.

Shasha Li

Papers

Sharp bounds for the generalized connectivity κ 3 (G)

Sharp bounds for the generalized connectivity $\kappa_3(G)$