3-extra connectivity of 3-ary n-cube networks

Let G be a connected graph and S be a set of vertices. The h-extra connectivity of G is the cardinality of a minimum set S such that G-S is disconnected and each component of G-S has at least h+1 vertices. The h-extra connectivity is an important parameter to measure the reliability and fault tolerance ability of large interconnection networks. The h-extra connectivity for h=1,2 of k-ary n-cube are gotten by Hsieh et al. in [Theoretical Computer Science, 443 (2012) 63-69] for k>=4 and Zhu et al. in [Theory of Computing Systems, arxiv.org/pdf/1105.0991v1 [cs.DM] 5 May 2011] for k=3. In this paper, we show that the h-extra connectivity of the 3-ary n-cube networks for h=3 is equal to 8n-12, where n>=3.Comment: 20 pages,1 figures. arXiv admin note: substantial text overlap with arXiv:1309.496

The tightly super 3-extra connectivity and 3-extra diagnosability of crossed cubes

Many multiprocessor systems have interconnection networks as underlying topologies and an interconnection network is usually represented by a graph where nodes represent processors and links represent communication links between processors. In 2016, Zhang et al. proposed the $g$-extra diagnosability of $G$, which restrains that every component of $G-S$ has at least $(g +1)$ vertices. As an important variant of the hypercube, the $n$-dimensional crossed cube $CQ_{n}$ has many good properties. In this paper, we prove that $CQ_{n}$ is tightly $(4n-9)$ super 3-extra connected for $n\geq 7$ and the 3-extra diagnosability of $CQ_{n}$ is $4n-6$ under the PMC model $(n\geq5)$ and MM$^*$ model $(n\geq7)$

On the gg-extra connectivity of graphs

Connectivity and diagnosability are two important parameters for the fault tolerant of an interconnection network $G$. In 1996, F\`{a}brega and Fiol proposed the $g$-extra connectivity of $G$. A subset of vertices $S$ is said to be a \emph{cutset} if $G-S$ is not connected. A cutset $S$ is called an \emph{$R_g$-cutset}, where $g$ is a non-negative integer, if every component of $G-S$ has at least $g+1$ vertices. If $G$ has at least one $R_g$-cutset, the \emph{$g$-extra connectivity} of $G$, denoted by $\kappa_g(G)$, is then defined as the minimum cardinality over all $R_g$-cutsets of $G$. In this paper, we first obtain the exact values of $g$-extra connectivity of some special graphs. Next, we show that $1\leq \kappa_g(G)\leq n-2g-2$ for $0\leq g\leq \left\lfloor \frac{n-3}{2}\right\rfloor$, and graphs with $\kappa_g(G)=1,2,3$ and trees with $\kappa_g(T_n)=n-2g-2$ are characterized, respectively. In the end, we get the three extremal results for the $g$-extra connectivity.Comment: 20 pages; 2 figure

Relationship between Conditional Diagnosability and 2-extra Connectivity of Symmetric Graphs

The conditional diagnosability and the 2-extra connectivity are two important parameters to measure ability of diagnosing faulty processors and fault-tolerance in a multiprocessor system. The conditional diagnosability $t_c(G)$ of $G$ is the maximum number $t$ for which $G$ is conditionally $t$-diagnosable under the comparison model, while the 2-extra connectivity $\kappa_2(G)$ of a graph $G$ is the minimum number $k$ for which there is a vertex-cut $F$ with $|F|=k$ such that every component of $G-F$ has at least $3$ vertices. A quite natural problem is what is the relationship between the maximum and the minimum problem? This paper partially answer this problem by proving $t_c(G)=\kappa_2(G)$ for a regular graph $G$ with some acceptable conditions. As applications, the conditional diagnosability and the 2-extra connectivity are determined for some well-known classes of vertex-transitive graphs, including, star graphs, $(n,k)$-star graphs, alternating group networks, $(n,k)$-arrangement graphs, alternating group graphs, Cayley graphs obtained from transposition generating trees, bubble-sort graphs, $k$-ary $n$-cube networks and dual-cubes. Furthermore, many known results about these networks are obtained directly

Neighbor connectivity of kk-ary nn-cubes

The neighbor connectivity of a graph $G$ is the least number of vertices such that removing their closed neighborhoods from $G$ results in a graph that is disconnected, complete or empty. If a~graph is used to model the topology of an interconnection network, this means that the failure of a network node causes failures of all its neighbors. We completely determine the neighbor connectivity of $k$-ary $n$-cubes for all $n\ge1$ and $k\ge2$.Comment: 11 pages, 1 figur