On the pseudo-achromatic number problem

Authors:
Jianer Chen;Iyad A. Kanj;Jie Meng;Ge Xia;Fenghui Zhang
Affiliations:
Department of Computer Science, Texas A&M University, College Station, TX 77843-3112, USA;School of Computing, DePaul University, 243 S. Wabash Avenue, Chicago, IL 60604-2301, USA;Department of Computer Science, Texas A&M University, College Station, TX 77843-3112, USA;Department of Computer Science, Lafayette College, Easton, PA 18042, USA;Department of Computer Science, Texas A&M University, College Station, TX 77843-3112, USA
Venue:
Theoretical Computer Science
Year:
2009

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Abstract

We study the parameterized complexity of the pseudo-achromatic number problem: Given an undirected graph and a parameter k, determine if the graph can be partitioned into k groups such that every two groups are connected by at least one edge. This problem has been extensively studied in graph theory and combinatorial optimization. We show that the problem has a kernel of at most (k-2)(k+1) vertices that is constructable in time O(mn), where n and m are the number of vertices and edges, respectively, in the graph, and k is the parameter. This directly implies that the problem is fixed-parameter tractable. We also study generalizations of the problem and show that they are parameterized intractable.