Directed Intersection Representations and the Information Content of Digraphs

Consider a directed graph (digraph) in which vertices are assigned color sets, and two connected if only they share at least one the tail vertex has strictly smaller set than head. We seek to determine smallest possible size of union sets that allows for such digraph representation. To address this problem, we introduce new notion intersection representation digraph, show it is well-defined all acyclic graphs (DAGs). then proceed number (DIN), colors needed represent DAG. Our main results upper bounds on DIN DAGs based what call longest terminal path decomposition set, constructive lower bounds.