Transitivity on subclasses of chordal graphs
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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 A dominates B if every vertex of B is adjacent to at least one vertex of A in G. A vertex partition π = {V_1, V_2, …, 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 partitions. It was known that the decision version of Maximum Transitivity Problem is NP-complete for chordal graphs [Iterated colorings of graphs, Discrete Mathematics, 278, 2004]. In this paper, we first prove that this problem can be solved in linear time for split graphs and for the complement of bipartite chain graphs, two subclasses of chordal graphs. We also discuss Nordhaus-Gaddum type relations for transitivity and provide counterexamples for an open problem posed by J. T. Hedetniemi and S. T. Hedetniemi [The transitivity of a graph, J. Combin. Math. Combin. Comput, 104, 2018]. Finally, we characterize transitively critical graphs having fixed transitivity.
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