Unicyclic 3-colored digraphs with bicyclic inverses

Abstract

The class of unicyclic $3$-colored digraphs with the cycle weight $\pm\mathrm{i}$ and with a unique perfect matching, denoted by $\mathcal{U}_g$, is considered in this article. Kalita \& Sarma [On the inverse of unicyclic 3-coloured digraphs, Linear and Multilinear Algebra, DOI: 10.1080/03081087.2021.1948956] introduced the notion of inverse of $3$-colored digraphs. They characterized the unicyclic $3$-colored digraphs in $\mathcal{U}_g$ possessing unicyclic inverses. This article provides a complete characterization of the unicyclic $3$-colored digraphs in $\mathcal{U}_g$ possessing bicyclic inverses.