An Analysis on the Reliability of the Alternating Group Graph

For interconnection network losing processors, usually, when the surviving network has a large connected component, it can be used as a functional subsystem without leading to severe performance degradation. Consequently, it is crucial to characterize the interprocessor communication ability and efficiency of the surviving structure. In this article, we prove that when a subset <formula><tex>$D$</tex></formula> of at most <formula><tex>$6n-17$</tex></formula> processors is deleted from an <formula><tex>$n$</tex></formula>-dimensional alternating group graph <formula><tex>$\text{AG}_n$</tex></formula>, there exists a largest component with cardinality greater or equal to <formula><tex>$|V(\text{AG}_n)|-|D|-3$</tex></formula> for <formula><tex>$n\geq 6$</tex></formula> in the remaining network, and the union of small components is, first, an empty graph; or, second, a 3-cycle, or an edge, or a 2-path, or a singleton; or, third, an edge and a singleton, or two singletons. Then, we prove that when a subset <formula><tex>$D$</tex></formula> of at most <formula><tex>$8n-25$</tex></formula> processors is deleted from <formula><tex>$\text{AG}_n$</tex></formula>, there exists a largest component with cardinality greater or equal to <formula><tex>$|V(\text{AG}_n)|-|D|-5$</tex></formula> for <formula><tex>$n\geq 6$</tex></formula> in the remaining network, and the union of small components is, first, an empty graph; or, second, a 5-cycle, or a 4-path, or a 4-claw, or a 4-cycle, or a 3-path, or a 3-claw, or a 3-cycle, or a 2-path, or an edge, or a singleton; or, third, a 4-cycle and a singleton, or a 3-path and a singleton, or a 3-claw and a singleton, or a 2-path and a singleton, two edges, an edge and a singleton, or two singletons; or, fourth, two edges and a singleton, or a 2-path and two singletons, or an edge and two singletons, or three singletons.