  Given a continuum $X$ and $p\in X$, we will consider the hyperspace $C(p,X)$
of all subcontinua of $X$ containing $p$ and the family $K(X)$ of all
hyperspaces $C(q,X)$, where $q\in X$. In this paper we give some conditions on
the points $p,q\in X$ to guarantee that $C(p,X)$ and $C(q,X)$ are homeomorphic,
for finite graphs $X$. Also, we study the relationship between the homogeneity
degree of a finite graph $X$ and the number of topologically distinct spaces in
$K(X)$, called the size of $K(X)$. In addition, we construct for each positive
integer $n$, a finite graph $X_n$ such that $K(X_n)$ has size $n$, and we
present a theorem that allows to construct finite graphs $X$ with a degree of
homogeneity different from the size of the family $K(X)$.
