# On the Intersection Number of Transformed Digraphs

## TL;DR

This paper introduces new proofs and formulas for the intersection number of various transformed digraphs, including middle, total, and a new union transformation, extending previous results to cases with loops and specific restrictions.

## Contribution

It provides new proofs and formulas for the intersection number of transformed digraphs, including cases with loops and a novel union transformation.

## Key findings

- Derived the intersection number for the middle digraph with loops.
- Calculated the intersection number for the union of a digraph and its subdivision.
- Extended previous results to the total digraph with certain loop restrictions.

## Abstract

For any simple digraph $D$ we offer a new proof for the intersection number of its middle digraph, $M(D)$; while doing so we also solve for the intersection number when $D$ has loops. In addition, a new transformation, the union of $D$ and its subdivision digraph, is introduced and its intersection number calculated in full generality. For the total digraph, we extend previous arguments letting us solve for the intersection number of $T(D)$ with $D$ possibly having loops, but under the restriction loops of the digraph only touch (are to or from) themselves, sinks, and sources.