# The Component Connectivity of Alternating Group Graphs and Split-Stars

## TL;DR

This paper determines the exact values of the $oldsymbol{ extit{ ext{ell}}}$-component connectivity for alternating group graphs and split-stars, providing insights into their fault-tolerance and robustness as interconnection network models.

## Contribution

It derives exact $oldsymbol{ extit{ ext{ell}}}$-connectivity values for $AG_n$ and $S_n^2$, advancing understanding of their structural robustness.

## Key findings

- $oxed{ ext{For } AG_n, ext{ } oldsymbol{ ext{kappa}_3=4n-10, ext{kappa}_4=6n-16, ext{kappa}_5=8n-24}}$
- $oxed{ ext{For } S_n^2, ext{ } oldsymbol{ ext{kappa}_3=4n-8, ext{kappa}_4=6n-14, ext{kappa}_5=8n-20}}$
- These results quantify the fault-tolerance of the networks for various component thresholds.

## Abstract

For an integer $\ell\geqslant 2$, the $\ell$-component connectivity of a graph $G$, denoted by $\kappa_{\ell}(G)$, is the minimum number of vertices whose removal from $G$ results in a disconnected graph with at least $\ell$ components or a graph with fewer than $\ell$ vertices. This is a natural generalization of the classical connectivity of graphs defined in term of the minimum vertex-cut and is a good measure of robustness for the graph corresponding to a network. So far, the exact values of $\ell$-connectivity are known only for a few classes of networks and small $\ell$'s. It has been pointed out in~[Component connectivity of the hypercubes, Int. J. Comput. Math. 89 (2012) 137--145] that determining $\ell$-connectivity is still unsolved for most interconnection networks, such as alternating group graphs and star graphs. In this paper, by exploring the combinatorial properties and fault-tolerance of the alternating group graphs $AG_n$ and a variation of the star graphs called split-stars $S_n^2$, we study their $\ell$-component connectivities. We obtain the following results: (i) $\kappa_3(AG_n)=4n-10$ and $\kappa_4(AG_n)=6n-16$ for $n\geqslant 4$, and $\kappa_5(AG_n)=8n-24$ for $n\geqslant 5$; (ii) $\kappa_3(S_n^2)=4n-8$, $\kappa_4(S_n^2)=6n-14$, and $\kappa_5(S_n^2)=8n-20$ for $n\geqslant 4$.