Transitivity on subclasses of bipartite graphs

Let G=(V,E) 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 pi={V-1,V-2, horizontexpressionl ellipsis ,V-k} of G is called a transitive k-partition if V-i dominates V-j for all i, j where 1 <= i < j <= k. 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).