Hyperspaces C(p,X) of finite graphs

Given a continuum X and p∈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∈X. In this paper we give some conditions on the points p,q∈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 Xn such that K(Xn) 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).