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