Decomposing semi-complete multigraphs and directed graphs into paths of length two

Abstract

A P3-decomposition of a graph is a partition of the edges of the graph into paths of length two. We give a simple necessary and sufficient condition for a semi-complete multigraph, that is a multigraph with at least one edge between each pair of vertices, to have a P3- decomposition. We show that this condition can be tested in strongly polynomial-time, and that the same condition applies to a larger class of multigraphs. We give a similar condition for a P3-decomposition of a semi-complete directed graph. In particular, we show that a tournament admits a P3-decomposition iff its outdegree sequence is the degree sequence of a simple undirected graph.