In the field of autonomous navigation for robotics, one of the most challenging issues is to locate the Next-Best-View and to guide robotics through a previously unknown environment. Existing methods based on generalized Voronoi graphs (GVGs) have presented feasible solutions but require excessive computation to construct GVGs from metric maps, and the GVGs are usually redundant. This paper proposes a reduced approximated GVG (RAGVG), which provides a topological representation of the explored space with a smaller graph. To be specific, a fast and practical algorithm for constructing RAGVGs from metric maps is presented, and a RAGVG-based autonomous robotic exploration framework is designed and implemented. The proposed method for constructing RAGVGs is validated with two known common maps, while the RAGVG-based autonomous exploration framework is tested on two simulation and one real-world museum. The experimental results show that the proposed algorithm is efficient in constructing RAGVGs, and indicate that the mobile robot controlled by the RAGVG-based autonomous exploration framework, compared with famous frontiers-based method, reduced the total time by approximately 20% for the given tasks.