Non-solvable groups whose character degree graph has a cut-vertex. II

Let G be a finite group, and let cd(G) denote the set of degrees of the irreducible complex characters of G. Define then the character degree graph delta(G) as the (simple undirected) graph whose vertices are the prime divisors of the numbers in cd(G), and two distinct vertices p, q are adjacent if and only if pq divides some number in cd(G). This paper continues the work, started in Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. I, Vietnam J. Math., to appear), toward the classification of the finite non-solvable groups whose degree graph possesses a cut-vertex, i.e. a vertex whose removal increases the number of connected components of the graph. While, in Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. I, Vietnam J. Math., to appear), groups with no composition factors isomorphic to PSL2(t(a)) (for any prime power ta >= 4) were treated, here we consider the complementary situation in the case when t is odd and t(a) > 5. The proof of this classification will be then completed in the third and last paper of this series (Dolfi et al., Non-solvable groups whose character degree graph has a cut-vertex. III, preprint 2022, submitted. https://doi.org/10.48550/arXiv.2209.07161) that deals with the case t = 2.