Non-solvable groups whose character degree graph has a cut-vertex. I (Q6050118)

Let \(\mathrm{cd}(G)\) be the set of all irreducible character degrees of a finite group \(G\). A simple graph \(\Delta(G)\) is said to be the \textit{character degree graph} of \(G\) if its vertices are the prime divisors of the numbers in \(\mathrm{cd}(G)\), and two distinct vertices \(p, q\) are adjacent if and only if \(pq\) divides some number in \(\mathrm{cd}(G)\). In this paper, the authors show that \(\Delta(G)\) has at most one cut-vertex, a vertex whose removal increases the number of connected components of the graph. Moreover, they study the structure of the non-solvable groups whose character degree graph has a cut-vertex.