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An interesting class of problems in character theory arises from considering how the structure of a group $G$ and the set of its irreducible character degrees, are related. One of the techniques to answer these questions, is to associate an appropriate graph to this set. An example of this, is the prime graph. For a finite group $G$, the prime graph of $G$, denoted by $\Delta(G)$, is an undirected graph whose vertex set is $\rho(G)$ which is the set of prime divisors of irreducible character degrees, and there is an edge between two distinct primes $p$ and $q$ if and only if $pq$ divides some irreducible character degree of $G$. On the other hand, a cut-vertex of a graph, is a vertex whose deletion increases the number of connected components. Since cut-vertices play an important role in a graph, in this talk, we consider the influence of a cut-vertex $v$ of $\Delta(G)$ on the structure of the subgroup $O^{v}(G)$, which is the unique minimal normal subgroup of $G$ such that $\frac{G}{O^{v}(G)}$ is a $v$-group.