Algorithmic study on 2-transitivity of graphs - Paul, Subhabrata; Santra, Kamal (Discrete Applied Mathematics, 2024)
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 of V, we say AdominatesB if every vertex of B is adjacent to at least one vertex of A. A vertex partition π={V1,V2,…,Vk} of G is called a transitive partition of size k if Vi dominates Vj for all 1≤i<j≤k. In this article, we study a variation of transitive partition, namely 2-transitive partition. For two disjoint subsets A and B of V, we say A 2-dominatesB if every vertex of B is adjacent to at least two vertices of A. A vertex partition π={V1,V2,…,Vk} of G is called a 2-transitive partition of size k if Vi2-dominates Vj for all 1≤i<j≤k. The Maximum 2-Transitivity Problem is to find a 2-transitive partition of a given graph with the maximum number of parts. We show that the decision version of this problem is NP-complete for chordal and bipartite graphs. On the positive side, we design three linear-time algorithms for solving Maximum 2-Transitivity Problem in trees, split, and bipartite chain graphs.