Chain--collider--fork Decompositions of Transitive Tournament

Abstract: A transitive tournament is an acyclic orientation of a complete graph. We study decompositions and packings of the transitive tournament (TT_n) into connected two-arc motifs. The three motifs considered are chains, colliders, and forks, which are also fundamental local configurations in directed acyclic graphs. We first construct decompositions of (TT_n) into mixtures of these motifs whenever such decompositions exist. We then consider the corresponding pure packing problem for each individual motif. For (H) equal to a chain, a collider, or a fork, we determine the maximum number of arc-disjoint copies of (H) in (TT_n). These results give a precise extremal description of two-arc motif packings in transitive tournaments and suggest further questions on motif decompositions in broader classes of directed acyclic graphs.