Arc reversals of cycles in orientations of $G$ vertex-multiplications

TL;DR

This paper extends known equivalence results for tournaments and bipartite tournaments to orientations of vertex-multiplied graphs, using advanced techniques to analyze score list and parity properties.

Contribution

It introduces new methods to analyze cycle reversals in orientations of vertex-multiplied graphs, generalizing previous results to broader graph classes.

Findings

Established $C_3$-equivalence for orientations with same score list.

Proved $C_4$-equivalence for bipartite orientations with same score lists.

Developed new extension and reduction techniques for cycle analysis.

Abstract

Ryser proved that any two tournaments with the same score sequence are $C_3$-equivalent while Beineke and Moon proved the $C_4$-equivalence for any two bipartite tournaments with the same score lists. In this paper, we extend these results to orientations of $G$ vertex-multiplications. We focus on two main areas, namely orientations with the same score list and with score-list parity. Our main tools are extensions of the refinement technique, directed difference graph and a reduction lemma.