# Directed Intersection Representations and the Information Content of Digraphs

## TL;DR

This paper introduces the concept of directed intersection representations for DAGs, establishing bounds on the minimum number of colors needed to encode their structure through shared color sets.

## Contribution

It defines the directed intersection number (DIN) for DAGs and provides new upper and lower bounds based on graph decompositions, advancing understanding of digraph representations.

## Key findings

- Established the directed intersection number (DIN) for DAGs.
- Provided upper bounds on DIN using longest terminal path decomposition.
- Derived constructive lower bounds for DIN.

## Abstract

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