We prove the following theorem for a compact, metric θ n {\theta _n} -continuum (i.e., a compact, connected, metric space that is not separated into more than n n components by any subcontinuum). The continuum X X admits a monotone, upper semicontinuous decomposition D \mathfrak {D} such that the elements of D \mathfrak {D} have void interiors and the quotient space X / D X/\mathfrak {D} is a finite graph, if and only if, for each nowhere dense subcontinuum H H of X X , the continuum T ( H ) = { x | T(H) = \{ x| if K K is a subcontinuum of X X and x ∈ K ∘ x \in {K^ \circ } , then K ∩ H ≠ ∅ } K \cap H \ne \emptyset \} is nowhere dense. The elements of the decomposition are characterized in terms of the set function T T . An example is given showing that the condition that requires T ( x ) T(x) to have void interior for all x ∈ X x \in X is not strong enough to guarantee the decomposition.