We investigate the exploration and mapping of anonymous graphs by a mobile agent. It is long known that, without global information about the graph, it is not possible to make the agent halt after the exploration except if the graph is a tree. We therefore endow the agent with binoculars, a sensing device that can show the local structure of the environment at a constant distance of the agent's current location and investigate networks that can be efficiently explored in this setting. In the case of trees, the exploration without binoculars is fast (i.e. using a DFS traversal of the graph, there is a number of moves linear in the number of nodes). We consider here the family of Weetman graphs that is a generalization of the standard family of chordal graphs and present a new deterministic algorithm that realizes Exploration of any Weetman graph, without knowledge of size or diameter and for any port numbering. The number of moves is linear in the number of nodes, despite the fact that Weetman graphs are not sparse, some having a number of edges that is quadratic in the number of nodes. At the end of the Exploration, the agent has also computed a map of the anonymous graph.