Transitivity on subclasses of bipartite graphs

Let G=(V,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=(V, E)$$\end{document} be a graph where V and E are the vertex and edge sets, respectively. For two disjoint subsets A and B, we say A dominates B if every vertex of B is adjacent to at least one vertex of A. A vertex partition π={V1,V2,…,Vk}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi = \{V_1, V_2, \ldots , V_k\}$$\end{document} of G is called a transitive k-partition if Vi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_i$$\end{document} dominates Vj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_j$$\end{document} for all i, j where 1≤i<j≤k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le i<j\le k$$\end{document}. The maximum integer k for which the above partition exists is called transitivity of G and it is denoted by Tr(G). The Maximum Transitivity Problem is to find a transitive partition of a given graph with the maximum number of parts. It was known that the decision version of Maximum Transitivity Problem is NP-complete for chordal graphs, which was proved by Hedetniemi et al. (Discret Math 278:81–108, 2004)]. This paper first strengthens the NP-completeness result by showing that this problem remains NP-complete for perfect elimination bipartite graphs. On the other hand, we propose a linear-time algorithm for finding the transitivity of a given bipartite chain graph. We then characterize graphs with transitivity at least t for any integer t. This result answers two open questions posed by J. T. Hedetniemi and S. T. Hedetniemi (J Combin Math Combin Comput 104:75–91, 2018).