We consider "ordinary" graphs: that is, finite undirected graphs with no loops or multiple edges. An intersection representation of a graph G is a function r from V(G), the set of vertices of G, into a family of sets S such that distinct points χ_α and χ_β of V(G) are neighbors in G precisely when γ(χ_α)∩γ(χ_β)≠Φ, A graph G is a rigid circuit grouph if every cycle in G has at least one triangular chord in G. In this paper we consider the main theorem; A graph G has an intersection representation by arcs on an acyclic graph if and only if is a normal rigid circuit graph.