@article{Xu2024, 
author = {Xiaoyan Xu and Xiaohua Xu and Jin Chen and Shixun Lin},
title = {On forbidden subgraphs of main supergraphs of groups},
year = {2024},
journal = {Electronic Research Archive},
volume = {32},
number = {8},
pages = {4845-4857},
keywords = {main supergraph, finite group, split graph, threshold graph, cograph},
url = {https://www.sciopen.com/article/10.3934/era.2024222},
doi = {10.3934/era.2024222},
abstract = {In this study, we explore the main supergraph  S(G) of a finite group  G, defined as an undirected, simple graph with a vertex set  G in which two distinct vertices,  a and  b, are adjacent in  S(G) if the order of one is a divisor of the order of the other. This is denoted as either  o(a)∣o(b) or  o(b)∣o(a), where  o(⋅) is the order of an element. We classify finite groups for which the main supergraph is either a split graph or a threshold graph. Additionally, we characterize finite groups whose main supergraph is a cograph. Our classification extends to finite groups  G with  S(G), a cograph that includes when  G is a direct product of two non-trivial groups, as well as when  G is either a dihedral group, a generalized quaternion group, a symmetric group, an alternating group, or a sporadic simple group.}
}