Hypercube is one of the most important networks to interconnect processors in multiprocessor computer systems. Different kinds of connectivities are important parameters to measure the fault tolerability of networks. Lin et al.\cite{LinStructure} introduced the concept of $H$-structure connectivity $κ(Q_n;H)$ (resp. $H$-substructure connectivity $κ^s(Q_n;H)$) as the minimum cardinality of $F=\{H_1,\dots,H_m\}$ such that $H_i (i=1,\dots,m)$ is isomorphic to $H$ (resp. $F=\{H'_1,\dots,H'_m\}$ such that $H'_i (i=1,\dots,m)$ is isomorphic to connected subgraphs of $H$) such that $Q_n-V(F)$ is disconnected or trivial. In this paper, we discuss $κ(Q_n;H)$ and $κ^s(Q_n;H)$ for hypercubes $Q_n$ with $n\geq 3$ and $H\in \{P_k,C_k|3\leq k\leq 2^{n-1}\}$. As a by-product, we solve the problem mentioned in \cite{ManeStructure}.
Let $G$ be a graph and $T$ a certain connected subgraph of $G$. The $T$-structure connectivity $κ(G; T)$ (or resp., $T$-substructure connectivity $κ^{s}(G; T)$) of $G$ is the minimum number of a set of subgraphs $\mathcal{F}=\{T_{1}, T_{2}, \ldots, T_{m}\}$ (or resp., $\mathcal{F}=\{T^{'}_{1}, T^{'}_{2}, \ldots, T^{'}_{m}\}$) such that $T_{i}$ is isomorphic to $T$ (or resp., $T^{'}_{i}$ is a connected subgraph of $T$) for every $1\leq i \leq m$, and $\mathcal{F}$'s removal will disconnect $G$. The twisted hypercube $H_{n}$ is a new variant of hypercubes with asymptotically optimal diameter int…
The component connectivity is the generalization of connectivity which is an parameter for the reliability evaluation of interconnection networks. The $g$-component connectivity $cκ_{g}(G)$ of a non-complete connected graph $G$ is the minimum number of vertices whose deletion results in a graph with at least $g$ components. The results in [Component connectivity of the hypercubes, International Journal of Computer Mathematics 89 (2012) 137-145] by Hsu et al. determines the component connectivity of the hypercubes. As an invariant of the hypercube, we determine the $(g+1)$-component connectivit…
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…
The restricted $h$-connectivity of a graph $G$, denoted by $κ^h(G)$, is defined as the minimum cardinality of a set of vertices $F$ in $G$, if exists, whose removal disconnects $G$ and the minimum degree of each component of $G-F$ is at least $h$. In this paper, we study the restricted $h$-connectivity of the balanced hypercube $BH_n$ and determine that $κ^1(BH_n)=κ^2(BH_n)=4n-4$ for $n\geq2$. We also obtain a sharp upper bound of $κ^3(BH_n)$ and $κ^4(BH_n)$ of $n$-dimension balanced hypercube for $n\geq3$ ($n\neq4$). In particular, we show that $κ^3(BH_3)=κ^4(BH_3)=12$.
A connected graph $G$ is called strongly Menger edge connected if $G$ has min\{deg$_G(x)$, deg$_G(y)$\} edge-disjoint paths between any two distinct vertices $x$ and $y$ in $G$. In this paper, we consider two types of strongly Menger edge connectivity of the line graphs of $n$-dimensional hypercube-like networks with faulty edges, namely the $m$-edge-fault-tolerant and $m$-conditional edge-fault-tolerant strongly Menger edge connectivity. We show that the line graph of any $n$-dimensional hypercube-like network is $(2n-4)$-edge-fault-tolerant strongly Menger edge connected for $n\geq 3$ and $(…
The connectivity is an important parameter to evaluate the robustness of a network. As a generalization, structure connectivity and substructure connectivity of graphs were proposed. For connected graphs $G$ and $H$, the $H$-structure connectivity $κ(G; H)$ (resp. $H$-substructure connectivity $κ^{s}(G; H)$) of $G$ is the minimum cardinality of a set of subgraphs $F$ of $G$ that each is isomorphic to $H$ (resp. to a connected subgraph of $H$) so that $G-F$ is disconnected or the singleton. As popular variants of hypercubes, the $n$-dimensional folded hypercubes $FQ_{n}$ and augmented cubes $AQ…
The connectivity of a network directly signifies its reliability and fault-tolerance. Structure and substructure connectivity are two novel generalizations of the connectivity. Let $H$ be a subgraph of a connected graph $G$. The structure connectivity (resp. substructure connectivity) of $G$, denoted by $κ(G;H)$ (resp. $κ^s(G;H)$), is defined to be the minimum cardinality of a set $F$ of connected subgraphs in $G$, if exists, whose removal disconnects $G$ and each element of $F$ is isomorphic to $H$ (resp. a subgraph of $H$). In this paper, we shall establish both $κ(BH_n;H)$ and $κ^s(BH_n;H)$…
The $g$-extra edge-connectivity is an important measure for the reliability of interconnection networks. Recently, Yang et al. [Appl. Math. Comput. 320 (2018) 464--473] determined the $3$-extra edge-connectivity of balanced hypercubes $BH_n$ and conjectured that the $g$-extra edge-connectivity of $BH_n$ is $λ_g(BH_n)=2(g+1)n-4g+4$ for $2\leq g\leq 2n-1$. In this paper, we confirm their conjecture for $n\geq 6-\dfrac{12}{g+1}$ and $2\leq g\leq 8$, and disprove their conjecture for $n\geq \dfrac{3e_g(BH_n)}{g+1}$ and $9\leq g\leq 2n-1$, where $e_g(BH_n)=\max\{|E(BH_n[U])|\mid U\subseteq V(BH_n),…
It is well known that \textit{every} Eulerian orientation of an Eulerian $2k$-edge connected (undirected) graph is strongly $k$-edge connected. An important goal in the area is to obtain analogous results for other types of connectivity, such as node connectivity and element connectivity. We show that \textit{every} Eulerian orientation of the hypercube of degree $2k$ is strongly $k$-node connected.
College of Mathematics and System Sciences, Xinjiang University（新疆大学数学与系统科学学院）；Department of Computer Science, National Yang Ming Chiao Tung University（国立阳明交通大学计算机科学系）
The $g$-component edge connectivity $cλ_g(G)$ of a non-complete graph $G$ is the minimum number of edges whose deletion results in a graph with at least $g$ components. In this paper, we determine the component edge connectivity of the folded hypercube $cλ_{g+1}(FQ_{n})=(n+1)g-(\sum\limits_{i=0}^{s}t_i2^{t_i-1}+\sum\limits_{i=0}^{s} i\cdot 2^{t_i})$ for $g\leq 2^{[\frac{n+1}2]}$ and $n\geq 5$, where $g$ be a positive integer and $g=\sum\limits_{i=0}^{s}2^{t_i}$ be the decomposition of $g$ such that $t_0=[\log_{2}{g}],$ and $t_i=[\log_2({g-\sum\limits_{r=0}^{i-1}2^{t_r}})]$ for $i\geq 1$.
As a generalization of the traditional connectivity, the g-component edge connectivity cλg(G) of a non-complete graph G is the minimum number of edges to be deleted from the graph G such that the resulting graph has at least g components. Hypercube-like networks (HL-networks for short) are obtained by manipulating some pairs of edges in hypercubes, which contain several famous interconnection networks such as twisted cubes, Mobius cubes, crossed cubes, locally twisted cubes. In this paper, we determine the (g + 1)-component edge connectivity of the n-dimensional HL-networks.
Given a connected graph $G$ and a non-negative integer $g$, the {\em $g$-extra connectivity} $\k_g(G)$ of $G$ is the minimum cardinality of a set of vertices in $G$, if it exists, whose deletion disconnects $G$ and leaves each remaining component with more than $g$ vertices. This paper focuses on the $g$-extra connectivity of hypercube-like networks (HL-networks for short) which includes numerous well-known topologies, such as hypercubes, twisted cubes, crossed cubes and Möbius cubes. All the known results suggest the equality $\k_g(X_n)=f_n(g)$ holds, where $X_n$ is an $n$-dimensional HL-netw…
In this paper we study the connectivity properties of the random subgraph of the $n$-cube generated by the $k$-out model and denoted by $Q^n(k)$. Let $k$ be an integer, $1\leq k \leq n-1$. We let $Q^n(k)$ be the graph that is generated by independently including for every $v\in V(Q^n)$ a set of $k$ distinct edges chosen uniformly from all the $\binom{n}{k}$ sets of distinct edges that are incident to $v$. We study connectivity the properties of $Q^n(k)$ as $k$ varies. We show that w.h.p. $Q^n(1)$ does not contain a giant component i.e. a component that spans $Ω(2^n)$ vertices. Thereafter we sh…
As a generalization of vertex connectivity, for connected graphs $G$ and $T$, the $T$-structure connectivity $κ(G, T)$ (resp. $T$-substructure connectivity $κ^{s}(G, T)$) of $G$ is the minimum cardinality of a set of subgraphs $F$ of $G$ that each is isomorphic to $T$ (resp. to a connected subgraph of $T$) so that $G-F$ is disconnected. For $n$-dimensional hypercube $Q_{n}$, Lin et al. [6] showed $κ(Q_{n},K_{1,1})=κ^{s}(Q_{n},K_{1,1})=n-1$ and $κ(Q_{n},K_{1,r})=κ^{s}(Q_{n},K_{1,r})=\lceil\frac{n}{2}\rceil$ for $2\leq r\leq 3$ and $n\geq 3$. Sabir et al. [11] obtained that $κ(Q_{n},K_{1,4})=κ^{…
The status of a vertex $u$ in a connected graph $G$, denoted by $σ_G(u)$, is defined as the sum of the distances between $u$ and all other vertices of a graph $G$. The first and second status connectivity indices of a graph $G$ are defined as $S_{1}(G) = \sum_{uv \in E(G)}[σ_G(u)+ σ_G(v)]$ and $S_{2}(G) = \sum_{uv \in E(G)}σ_G(u)σ_G(v)$ respectively, where $E(G)$ denotes the edge set of $G$. In this paper we have defined the first and second status co-indices of a graph $G$ as $\overline{S_{1}}(G) = \sum_{uv \notin E(G)}[σ_G(u)+ σ_G(v)]$ and $\overline{S_{2}}(G) = \sum_{uv \notin E(G)}σ_G(u)σ_…
Let $G_n$ be an $n$-dimensional recursive network. The $h$-embedded connectivity $ζ_h(G_n)$ (resp. edge-connectivity $η_h(G_n)$) of $G_n$ is the minimum number of vertices (resp. edges) whose removal results in disconnected and each vertex is contained in an $h$-dimensional subnetwork $G_h$. This paper determines $ζ_h$ and $η_h$ for the hypercube $Q_n$ and the star graph $S_n$, and $η_3$ for the bubble-sort network $B_n$.
The Steiner tree problem in graphs has applications in network design or circuit layout. Given a set $S$ of vertices, $|S| \geq 2,$ a tree connecting all vertices of $S$ is called an $S$-Steiner tree (tree connecting $S$). The reliability of a network $G$ to connect any $S$ vertices ($|S|$ number of vertices) in $G$ can be measure by this parameter. For an $S$-Steiner tree, if the degree of each vertex in $S$ is equal to one, then that tree is called a pendant S-Steiner tree. Two pendant $S$-Steiner trees $T$ and $T'$ are said to be internally disjoint if $E(T) \cap E(T') = \emptyset$ and $V(T…
Reliability evaluation and fault tolerance of an interconnection network of some parallel and distributed systems are discussed separately under various link-faulty hypotheses in terms of different $\mathcal{P}$-conditional edge-connectivity. With the help of edge isoperimetric problem's method in combinatorics, this paper mainly offers a novel and unified view to investigate the $\mathcal{P}$-conditional edge-connectivities of hamming graph $K_{L}^{n}$ with satisfying the property that each minimum $\mathcal{P}$-conditional edge-cut separates the $K_{L}^{n}$ just into two components, such as …
Last decade, numerous giant data center networks are built to provide increasingly fashionable web applications. For two integers $m\geq 0$ and $n\geq 2$, the $m$-dimensional DCell network with $n$-port switches $D_{m,n}$ and $n$-dimensional BCDC network $B_{n}$ have been proposed. Connectivity is a basic parameter to measure fault-tolerance of networks. As generalizations of connectivity, structure (substructure) connectivity was recently proposed. Let $G$ and $H$ be two connected graphs. Let $\mathcal{F}$ be a set whose elements are subgraphs of $G$, and every member of $\mathcal{F}$ is isom…
This paper considers the conditional fault tolerance, $h$-super connectivity $κ^{h}$ and $h$-super edge-connectivity $λ^{h}$ of the hierarchical cubic network $HCN_n$, an attractive alternative network to the hypercube, and shows $κ^h(HCN_n)=λ^h(HCN_n)=2^h(n+1-h)$ for any $h$ with $0\leq h\leq n-1$. The results imply that at least $2^h(n+1-h)$ vertices or edges have to be removed from $HCN_n$ to make it disconnected with no vertices of degree less than $h$, and generalize some known results.
One of the fundamental problems in distributed computing is how to efficiently perform routing in a faulty network in which each link fails with some probability. This paper investigates how big the failure probability can be, before the capability to efficiently find a path in the network is lost. Our main results show tight upper and lower bounds for the failure probability which permits routing, both for the hypercube and for the $d-$dimensional mesh. We use tools from percolation theory to show that in the $d-$dimensional mesh, once a giant component appears -- efficient routing is possibl…
For an integer $\ell\geqslant 2$, the $\ell$-component connectivity of a graph $G$, denoted by $κ_{\ell}(G)$, is the minimum number of vertices whose removal from $G$ results in a disconnected graph with at least $\ell$ components or a graph with fewer than $\ell$ vertices. This is a natural generalization of the classical connectivity of graphs defined in term of the minimum vertex-cut and is a good measure of robustness for the graph corresponding to a network. So far, the exact values of $\ell$-connectivity are known only for a few classes of networks and small $\ell$'s. It has been pointed…
