A 2n-component non-splittable spatial graph G = G1 ⋯ G2n in S3 possibly admits a smooth involution on its exterior whose fixed point set is a closed surface F of genus g ≥ 1. It is known that n ≥ g holds if G is a graph link in S3. In this paper, we consider G to be a hyperbolic spatial graph in a closed orientable 3-manifold M. We first study the case where M is the 3-sphere, and provide a method for constructing G and F with a1 + ⋯ + an &gt
g &gt
1, where 1 - ai is the Euler characteristic of Gi for 1 ≤ i ≤ n. Next, we study the case where M is a closed orientable 3-manifold in relation to Heegaard splittings and Dehn surgeries.