Strong transitivity of a graph

Abstract

A vertex partition [Formula: see text] of [Formula: see text] is called a transitive partition of size [Formula: see text] if [Formula: see text] dominates [Formula: see text] for all [Formula: see text]. For two disjoint subsets [Formula: see text] and [Formula: see text] of [Formula: see text], we say [Formula: see text] strongly dominates [Formula: see text] if for every vertex [Formula: see text], there exists a vertex [Formula: see text], such that [Formula: see text] and [Formula: see text]. A vertex partition [Formula: see text] of [Formula: see text] is called a strong transitive partition of size [Formula: see text] if [Formula: see text] strongly dominates [Formula: see text] for all [Formula: see text]. The maximum strong transitivity problem involves finding a strong transitive partition of a given graph with the maximum number of parts. In this paper, we initiate the study of this variation of transitive partition from an algorithmic point of view. We show that the decision version of this problem is NP-complete for chordal graphs. On the positive side, we prove that this problem can be solved in linear time for trees and split graphs.