A $\overrightarrow{P_{3}}$-decomposition of tournaments and bipartite digraphs

TL;DR

This paper characterizes when tournaments and bipartite digraphs can be decomposed into directed paths of length two, solving a previously open problem in directed graph decompositions.

Contribution

It provides a complete characterization for the existence of $ ightarrow P_3$-decompositions in tournaments and bipartite digraphs, addressing a problem posed by Diwan.

Findings

Characterization for $ ightarrow P_3$-decomposition in tournaments

Characterization for $ ightarrow P_3$-decomposition in bipartite digraphs

Solves an open problem in directed graph decompositions

Abstract

A $\overrightarrow{P_{3}}$-decomposition of a directed graph $D$ is a partition of the arcs of $D$ into directed paths of length $2$. In this paper, we give a characterization for a tournament and a bipartite digraph admitting a $\overrightarrow{P_{3}}$-decomposition. This solves a problem posed by Diwan ($\overrightarrow{P_{3}}$-decomposition of directed graphs, Discrete Appl. Math., http:// dx.doi.org/10.1016/j.dam.2016.01.039.).