  Let $G$ be a finite group. Denoting by ${\rm{cd}}(G)$ the set of the degrees
of the irreducible complex characters of $G$, we consider the {\it character
degree graph} of $G$: this is the (simple, undirected) graph whose vertices are
the prime divisors of the numbers in ${\rm{cd}}(G)$, and two distinct vertices
$p$, $q$ are adjacent if and only if $pq$ divides some number in
${\rm{cd}}(G)$. This paper completes the classification, started in [5] and
[6], of the finite non-solvable groups whose character degree graph has a {\it
cut-vertex}, i.e. a vertex whose removal increases the number of connected
components of the graph. More specifically, it was proved in [6] that these
groups have a unique non-solvable composition factor $S$, and that $S$ is
isomorphic to a group belonging to a restricted list of non-abelian simple
groups. In [5] and [6] all isomorphism types for $S$ were treated, except the
case \(S\cong{\rm{PSL}}_2(2^a)\) for some integer $a\geq 2$; the remaining case
is addressed in the present paper.
