The Star-Structure Connectivity and Star-Substructure Connectivity of Hypercubes and Folded Hypercubes. (15th September 2021) - Ba, Lina; Zhang, Heping
Abstract: 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. 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 $2≤ r≤ 3$ and $n≥ 3$ (Lin, C.-K., Zhang, L.-L., Fan, J.-X. and Wang, D.-J. (2016) Structure connectivity and substructure connectivity of hypercubes. Theor. Comput. Sci., 634, 97–107). Sabir et al. obtained that $κ (Q_{n};K_{1, 4})=κ ^{s}(Q_{n};K_{1, 4})= \lceil \frac{n}{2}\rceil $ for $n≥ 6$ 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 $2≤ r≤ 3$ and $n≥ 7$ (Sabir, E. and Meng, J.(2018) Structure fault tolerance of hypercubes and folded hypercubes. Theor. Comput. Sci., 711, 44–55). 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 $r≥ 2$, $κ (Q_{n};K_{1, r})$ $=κ ^{s}(Q_{n};K_{1, r})$ $=\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 $4≤ r≤ 6$, we separately confirm the above result holds for $Q_n$ in the remaining cases.