论文标题
星际结构连通性和超振管的星际结构连通性和折叠式高管
The star-structure connectivity and star-substructure connectivity of hypercubes and folded hypercubes
论文作者
论文摘要
作为顶点连接性的概括，对于连接的图形$ g $和$ t $，$ t $结构连接性$κ（g，t）$（$ t $ $ t $ - substructure $κ^{s}（s} $ g $）的$ g $是$ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f ins $ f ins $ f ins $ f of Ass $属于$ f its $属于$ f its us at iS is ats。 $ t $）的子图是$ g-f $断开连接的。对于$ n $ dimensional hypercube $ q_ {n} $，lin等。 [6]显示$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} {2} \ rceil $ for $ 2 \ leq r \ leq r \ leq 3 $ and $ n \ geq 3 $。 Sabir等。 [11]获得了$κ（q_ {n}，k_ {1,4}）=κ^{s}（q_ {n}，k_ {n}，k_ {1,4}）= \ lceil \ lceil \ lceil \ frac {n} {n} {2} {2} {2} {2} \ rceil $ for $ n \ geq 6 $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ - foff。 $κ（fq_ {n}，k_ {1,1}）=κ^{s}（fq_ {n}，k_ {1,1}）= n $， $κ（fq_ {n}，k_ {1，r}）=κ^{s}（fq_ {n}，k_ {1，r}）= \ lceil \ frac {n+1} {n+1} {2} \ rceil $，带有$ 2 \ leq r \ leq r \ leq r \ leq 3 $和$ n $ g geq 7 $。他们提出了一个开放的问题，即确定$ k_ {1，r} $ - $ q_n $的结构连接和一般$ r $的$ fq_n $。在本文中，我们为每个整数$ r \ geq 2 $，$κ（q_ {n}; k_ {1，r}）=κ^{s}（q_ {n}; k_ {n}; k_ {1，r}） $κ（fq_ {n}; k_ {1，r}）=κ^{s}（fq_ {n}; k_ {1，r}）= \ lceil \ lceil \ frac {n+1} {n+1} {2} \ rceil $用于所有整数$ n $ n $ n $ n $ n $ n $ n $ r $ in quare spare。对于$ 4 \ leq r \ leq 6 $，我们单独确认上述结果在其余案例中以$ q_n $的价格保留。
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 $κ(Q_{n},K_{1,4})=κ^{s}(Q_{n},K_{1,4})=\lceil\frac{n}{2}\rceil$ for $n\geq 6$, and for $n$-dimensional folded hypercube $FQ_{n}$, $κ(FQ_{n},K_{1,1})=κ^{s}(FQ_{n},K_{1,1})=n$, $κ(FQ_{n},K_{1,r})=κ^{s}(FQ_{n},K_{1,r})=\lceil\frac{n+1}{2}\rceil$ with $2\leq r\leq 3$ and $n\geq 7$. They proposed an open problem of determining $K_{1,r}$-structure connectivity of $Q_n$ and $FQ_n$ for general $r$. In this paper, we obtain that for each integer $r\geq 2$, $κ(Q_{n};K_{1,r})=κ^{s}(Q_{n};K_{1,r})=\lceil\frac{n}{2}\rceil$ and $κ(FQ_{n};K_{1,r})=κ^{s}(FQ_{n};K_{1,r})= \lceil\frac{n+1}{2}\rceil$ for all integers $n$ larger than $r$ in quare scale. For $4\leq r\leq 6$, we separately confirm the above result holds for $Q_n$ in the remaining cases.
