  Let $G$ be a finite group, and let ${\rm{cd}}(G)$ 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 ${\rm{cd}}(G)$, and two distinct vertices $p$, $q$
are adjacent if and only if $pq$ divides some number in ${\rm{cd}}(G)$. This
paper continues the work, started in [7], 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 [7], groups with no composition factors isomorphic to
${\rm{PSL}}_2(t^a)$ (for any prime power $t^a\geq 4$) 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 ([8]), that deals with the case $t=2$.
