Computability of graphs

We consider topological pairs $(A,B)$, $B\subseteq A$, which have computable type, which means that they have the following property: if X is a computable topological space and $f:A\to X$ a topological imbedding such that $f(A)$ and $f(B)$ are semicomputable sets in X, then $f(A)$ is a computable set in X. It is known, e.g., that $(M,\partial M)$ has computable type if M is a compact manifold with boundary. In this paper we examine topological spaces called graphs and we show that we can in a natural way associate to each graph G a discrete subspace E so that $(G,E)$ has computable type. Furthermore, we use this result to conclude that certain noncompact semicomputable graphs in computable metric spaces are computable.