Algorithmic study on 2-transitivity of 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 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 V...

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Bibliographic Details
Published in:Discrete Applied Mathematics Vol. 358; pp. 57 - 75
Main Authors: Paul, Subhabrata, Santra, Kamal
Format: Journal Article
Language:English
Published: Elsevier B.V 15.12.2024
Subjects:
2-transitivity
Bipartite graphs
Chordal graphs
Linear-time algorithm
NP-completeness
Split graphs
Split graphs
2-transitivity
Chordal graphs
Bipartite graphs
NP-completeness
Linear-time algorithm
ISSN:0166-218X
Online Access:Get full text
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Summary: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.
ISSN:0166-218X
DOI:10.1016/j.dam.2024.06.030