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 $κ(S)$ 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 $κ_k(G)$, of $G$ is defined by $κ_k(G)=$min$\{κ(S)\}$, where the minimum is taken over all $k$-subsets $S$ of $V(G)$. Thus $κ_2(G)=κ(G)$, where $κ(G)$ 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.
We perform a systematic study in the computational complexity of the connected variant of three related transversal problems: Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal. Just like their original counterparts, these variants are NP-complete for general graphs. A graph $G$ is $H$-free for some graph $H$ if $G$ contains no induced subgraph isomorphic to $H$. It is known that Connected Vertex Cover is NP-complete even for $H$-free graphs if $H$ contains a claw or a cycle. We show that the two other connected variants also remain NP-complete if $H$ contains a cycle or claw. In the…
The generalized $k$-connectivity $κ_k(G)$ of a graph $G$ was introduced by Hager before 1985. As its a natural counterpart, we introduced the concept of generalized edge-connectivity $λ_k(G)$, 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 $κ_k(G)$ and $λ_k(G)$, graphs with large generalized (edge-)connectivity, Nordhaus-Gaddum-ty…
Generalized connectivity introduced by Hager (1985) has been studied extensively in undirected graphs and become an established area in undirected graph theory. For connectivity problems, directed graphs can be considered as generalizations of undirected graphs. In this paper, we introduce a natural extension of generalized $k$-connectivity of undirected graphs to directed graphs (we call it strong subgraph $k$-connectivity) by replacing connectivity with strong connectivity. We prove NP-completeness results and the existence of polynomial algorithms. We show that strong subgraph $k$--connecti…
The generalized $k$-connectivity $κ_k(G)$ of a graph $G$, introduced by Hager in 1985, is a nice generalization of the classical connectivity. Recently, as a natural counterpart, we proposed the concept of generalized $k$-edge-connectivity $λ_k(G)$. In this paper, graphs of order $n$ such that $κ_k(G)=n-\frac{k}{2}-1$ and $λ_k(G)=n-\frac{k}{2}-1$ for even $k$ are characterized.
The generalized $k$-connectivity $κ_{k}(G)$ of a graph $G$, which was introduced by Chartrand et al.(1984) is a generalization of the concept of vertex connectivity. Let $G$ and $H$ be nontrivial connected graphs. Recently, Li et al. gave a lower bound for the generalized $3$-connectivity of the Cartesian product graph $G \square H$ and proposed a conjecture for the case that $H$ is $3$-connected. In this paper, we give two different forms of lower bounds for the generalized $3$-connectivity of Cartesian product graphs. The first lower bound is stronger than theirs, and the second confirms the…
The generalized $k$-connectivity $κ_k(G)$ of a graph $G$, introduced by Hager in 1985, is a natural generalization of the concept of connectivity $κ(G)$, which is just for $k=2$. Total graph is generalized line graph and a large graph which obtained by incidence relation between vertices and edges of original graph. T. Hamada and T. Nonaka et al., in \cite{Hamada} determined the connectivity of the total graph $T(G)$ for a graph $G$. In this paper we determine the generalized $k$-(edge)-connectivity of total graph $T(G)$ for $k=3$.
The generalized $k$-connectivity of a graph $G$, denoted by $κ_k(G)$, is the minimum number of internally edge disjoint $S$-trees for any $S\subseteq V(G)$ and $|S|=k$. The generalized $k$-connectivity is a natural extension of the classical connectivity and plays a key role in applications related to the modern interconnection networks. The burnt pancake graph $BP_n$ and the godan graph $EA_n$ are two kinds of Cayley graphs which posses many desirable properties. In this paper, we investigate the generalized 3-connectivity of $BP_n$ and $EA_n$. We show that $κ_3(BP_n)=n-1$ and $κ_3(EA_n)=n-1$…
The generalized $k$-connectivity of a graph $G$, denoted by $κ_k(G)$, is a generalization of the traditional connectivity. It is well known that the generalized $k$-connectivity is an important indicator for measuring the fault tolerance and reliability of interconnection networks. The $n$-dimensional folded hypercube $FQ_n$ is obtained from the $n$-dimensional hypercube $Q_n$ by adding an edge between any pair of vertices with complementary addresses. In this paper, we show that $κ_3(FQ_n)=n$ for $n\ge 2$, that is, for any three vertices in $FQ_n$, there exist $n$ internally disjoint trees co…
In this paper, we present some new results describing connections between the spectrum of a regular graph and its generalized connectivity, toughness, and the existence of spanning trees with bounded degree.
For $S\subseteq V(G)$ with $|S|\ge 2$, let $κ_G (S)$ denote the maximum number of internally disjoint trees connecting $S$ in $G$. For $2\le k\le n$, the generalized $k$-connectivity $κ_k(G)$ of an $n$-vertex connected graph $G$ is defined to be $κ_k(G)=\min \{κ_G(S): S\in V(G) \mbox{ and } |S|=k\}$. The generalized $k$-connectivity can serve for measuring the fault tolerance of an interconnection network. The bubble-sort graph $B_n$ for $n\ge 2$ is a Cayley graph over the symmetric group of permutations on $[n]$ generated by transpositions from the set $\{[1,2],[2,3],\dots, [n-1,n]\}$. In thi…
本文证明了：对于任意整数$n$和$k$，若有$0\leqslant k \leqslant n-2$，则必须从一个$n$维星图中去掉至少$(k+1)!(n-k-1)$点或边，使其不连通且没有度小于$k$的点.这一结果肯定地回答了Wan和Zhang提出的猜想[应用数学快报，22（2009），264-267].
Boesch and Chen (SIAM J. Appl. Math., 1978) introduced the cut-version of the generalized edge-connectivity, named $k$-edge-connectivity. For any integer $k$ with $2\leq k\leq n$, the {\em $k$-edge-connectivity} of a graph $G$, denoted by $λ_k(G)$, is defined as the smallest number of edges whose removal from $G$ produces a graph with at least $k$ components. In this paper, we first compute some exact values and sharp bounds for $λ_k(G)$ in terms of $n$ and $k$. We then discuss the relationships between $λ_k(G)$ and other generalized connectivities. An algorithm in $\mathcal{O}(n^2)$ time will…
Let $S\subseteq V(G)$ and $κ_{G}(S)$ denote the maximum number $r$ of edge-disjoint trees $T_1, T_2, \cdots, T_r$ in $G$ such that $V(T_i)\bigcap V(T_{j})=S$ for any $i, j \in \{1, 2, \cdots, r\}$ and $i\neq j$. For an integer $k$ with $2\leq k\leq n$, the {\em generalized $k$-connectivity} of a graph $G$ is defined as $κ_{k}(G)= min\{κ_{G}(S)|S\subseteq V(G)$ and $|S|=k\}$. The generalized $k$-connectivity is a generalization of the traditional connectivity. In this paper, the generalized $3$-connectivity of the $(n,k)$-bubble-sort graph $B_{n,k}$ is studied for $2\leq k\leq n-1$. By proposin…
The generalized $k$-connectivity $κ_{k}(G)$ of a graph $G$ is a parameter that can measure the reliability of a network $G$ to connect any $k$ vertices in $G$, which is proved to be NP-complete for a general graph $G$. Let $S\subseteq V(G)$ and $κ_{G}(S)$ denote the maximum number $r$ of edge-disjoint trees $T_{1}, T_{2}, \cdots, T_{r}$ in $G$ such that $V(T_{i})\bigcap V(T_{j})=S$ for any $i, j \in \{1, 2, \cdots, r\}$ and $i\neq j$. For an integer $k$ with $2\leq k\leq n$, the {\em generalized $k$-connectivity} of a graph $G$ is defined as $κ_{k}(G)= min\{κ_{G}(S)|S\subseteq V(G)$ and $|S|=k…
Extremal properties and bounds for the generalized algebraic connectivity of graphs in Euclidean spaces
Universidad de Buenos Aires（布宜诺斯艾利斯大学）；CONICET-Universidad de Buenos Aires, INTECIN（阿根廷国家科学研究委员会-布宜诺斯艾利斯大学，智能通信研究所）；Universidad de San Andrés（圣安德烈斯大学）；CONICET（阿根廷国家科学研究委员会）
For a digraph $D=(V(D), A(D))$, and a set $S\subseteq V(D)$ with $r\in S$ and $|S|\geq 2$, an $(S, r)$-tree is an out-tree $T$ rooted at $r$ with $S\subseteq V(T)$. Two $(S, r)$-trees $T_1$ and $T_2$ are said to be arc-disjoint if $A(T_1)\cap A(T_2)=\emptyset$. Two arc-disjoint $(S, r)$-trees $T_1$ and $T_2$ are said to be internally disjoint if $V(T_1)\cap V(T_2)=S$. Let $κ_{S,r}(D)$ and $λ_{S,r}(D)$ be the maximum number of internally disjoint and arc-disjoint $(S, r)$-trees in $D$, respectively. The generalized $k$-vertex-strong connectivity of $D$ is defined as $$κ_k(D)= \min \{κ_{S,r}(D)\…
Let $S\subseteq V(G)$ and $κ_{G}(S)$ denote the maximum number $k$ of edge-disjoint trees $T_{1}, T_{2}, \cdots, T_{k}$ in $G$ such that $V(T_{i})\bigcap V(T_{j})=S$ for any $i, j \in \{1, 2, \cdots, k\}$ and $i\neq j$. For an integer $r$ with $2\leq r\leq n$, the {\em generalized $r$-connectivity} of a graph $G$ is defined as $κ_{r}(G)= min\{κ_{G}(S)|S\subseteq V(G)$ and $|S|=r\}$. The $r$-component connectivity $cκ_{r}(G)$ of a non-complete graph $G$ is the minimum number of vertices whose deletion results in a graph with at least $r$ components. These two parameters are both generalizations…
For a digraph $D=(V(D), A(D))$, and a set $S\subseteq V(D)$ with $r\in S$ and $|S|\geq 2$, an $(S, r)$-tree is an out-tree $T$ rooted at $r$ with $S\subseteq V(T)$. Two $(S, r)$-trees $T_1$ and $T_2$ are said to be arc-disjoint if $A(T_1)\cap A(T_2)=\emptyset$. Two arc-disjoint $(S, r)$-trees $T_1$ and $T_2$ are said to be internally disjoint if $V(T_1)\cap V(T_2)=S$. Let $κ_{S,r}(D)$ and $λ_{S,r}(D)$ be the maximum number of internally disjoint and arc-disjoint $(S, r)$-trees in $D$, respectively. The generalized $k$-vertex-strong connectivity of $D$ is defined as $$κ_k(D)= \min \{κ_{S,r}(D)\…
We study the problem #IndSub(P) of counting all induced subgraphs of size k in a graph G that satisfy the property P. This problem was introduced by Jerrum and Meeks and shown to be #W[1]-hard when parameterized by k for some families of properties P including, among others, connectivity [JCSS 15] and even- or oddness of the number of edges [Combinatorica 17]. Very recently [IPEC 18], two of the authors introduced a novel technique for the complexity analysis of #IndSub(P), inspired by the "topological approach to evasiveness" of Kahn, Saks and Sturtevant [FOCS 83] and the framework of graph m…
NP-hardness of additive approximation for generalized Turán problems
University of Science and Technology of China（中国科学技术大学）；East China Normal University（华东师范大学）；Shaanxi Normal University（陕西师范大学）；Tsinghua University（清华大学）；Fuzhou University（福州大学）
Minimum Label Cut (or Hedge Connectivity) problem is defined as follows: given an undirected graph $G=(V, E)$ with $n$ vertices and $m$ edges, in which, each edge is labeled (with one or multiple labels) from a label set $L=\{\ell_1,\ell_2, ..., \ell_{|L|}\}$, the edges may be weighted with weight set $W =\{w_1, w_2, ..., w_m\}$, the label cut problem(hedge connectivity) problem asks for the minimum number of edge sets(each edge set (or hedge) is the edges with the same label) whose removal disconnects the source-sink pair of vertices or the whole graph with minimum total weights(minimum cardi…
In a standard $f$-connectivity network design problem, we are given an undirected graph $G=(V,E)$, a cut-requirement function $f:2^V \rightarrow {\mathbb{N}}$, and non-negative costs $c(e)$ for all $e \in E$. We are then asked to find a minimum-cost vector $x \in {\mathbb{N}}^E$ such that $x(δ(S)) \geq f(S)$ for all $S \subseteq V$. We focus on the class of such problems where $f$ is a proper function. This encodes many well-studied NP-hard problems such as the generalized survivable network design problem. In this paper we present the first strongly polynomial time FPTAS for solving the LP re…
