Distinguishing and Reconstructing Directed Graphs by their \(\pmb {B}\)-Polynomials

The B-polynomial and quasisymmetric B-function, introduced by Awan and Bernardi, extends the widely studied Tutte polynomial and Tutte symmetric function to digraphs. In this article, we address one of the fundamental questions concerning these digraph invariants, which is, the determination of the classes of digraphs uniquely characterized by them. We solve an open question originally posed by Awan and Bernardi, regarding the identification of digraphs that result from replacing every edge of a graph with a pair of opposite arcs. Further, we address the more challenging problem of reconstructing digraphs using their quasisymmetric functions. In particular, we show that the quasisymmetric B-function reconstructs partially symmetric orientations of proper caterpillars. As a consequence, we establish that all orientations of paths and asymmetric proper caterpillars can be reconstructed from their quasisymmetric B-functions. These results enhance the pool of oriented trees distinguishable through quasisymmetric functions.