The star-structure connectivity and star-substructure connectivity of hypercubes and folded hypercubes

TL;DR

This paper determines the star-structure and star-substructure connectivity of hypercubes and folded hypercubes for all relevant parameters, extending previous results and solving an open problem in graph connectivity theory.

Contribution

It provides exact formulas for the star-structure connectivity of hypercubes and folded hypercubes for all r ≥ 2, resolving an open problem for general r.

Findings

For hypercubes, a7(Q_n;K_{1,r})=a7^s(Q_n;K_{1,r})=\u23a1(n/2) for all r  and n > r.

For folded hypercubes, a7(FQ_n;K_{1,r})=a7^s(FQ_n;K_{1,r})=((n+1)/2) for all r  and n > r.

The results confirm the conjecture for specific ranges of r and n, completing the characterization of these connectivities.

Abstract

As a generalization of vertex connectivity, for connected graphs $G$ and $T$, the $T$-structure connectivity $\kappa(G, T)$ (resp. $T$-substructure connectivity $\kappa^{s}(G, T)$) 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 $\kappa(Q_{n},K_{1,1})=\kappa^{s}(Q_{n},K_{1,1})=n-1$ and $\kappa(Q_{n},K_{1,r})=\kappa^{s}(Q_{n},K_{1,r})=\lceil\frac{n}{2}\rceil$ for $2\leq r\leq 3$ and $n\geq 3$. Sabir et al. [11] obtained that $\kappa(Q_{n},K_{1,4})=\kappa^{s}(Q_{n},K_{1,4})=\lceil\frac{n}{2}\rceil$ for $n\geq 6$, and for $n$-dimensional folded hypercube $FQ_{n}$, $\kappa(FQ_{n},K_{1,1})=\kappa^{s}(FQ_{n},K_{1,1})=n$, $\kappa(FQ_{n},K_{1,r})=\kappa^{s}(FQ_{n},K_{1,r})=\lceil\frac{n+1}{2}\rceil$ 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 $r\geq 2$, $\kappa(Q_{n};K_{1,r})=\kappa^{s}(Q_{n};K_{1,r})=\lceil\frac{n}{2}\rceil$ and $\kappa(FQ_{n};K_{1,r})=\kappa^{s}(FQ_{n};K_{1,r})= \lceil\frac{n+1}{2}\rceil$ 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.