LAUSR

Abstract 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 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 ) .