IWOCA 2011: Combinatorial Algorithms pp 19-30

# Algorithmic Aspects of Dominator Colorings in Graphs

• S. Arumugam
• K. Raja Chandrasekar
• Neeldhara Misra
• Geevarghese Philip
• Saket Saurabh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7056)

## Abstract

In this paper we initiate a systematic study of a problem that has the flavor of two classical problems, namely Coloring and Domination, from the perspective of algorithms and complexity. A dominator coloring of a graph G is an assignment of colors to the vertices of G such that it is a proper coloring and every vertex dominates all the vertices of at least one color class. The minimum number of colors required for a dominator coloring of G is called the dominator chromatic number of G and is denoted by χ d (G). In the Dominator Coloring (DC) problem, a graph G and a positive integer k are given as input and the objective is to check whether χ d (G) ≤ k. We first show that unless P=NP, DC cannot be solved in polynomial time on bipartite, planar, or split graphs. This resolves an open problem posed by Chellali and Maffray [Dominator Colorings in Some Classes of Graphs, Graphs and Combinatorics, 2011] about the polynomial time solvability of DC on chordal graphs. We then complement these hardness results by showing that the problem is fixed parameter tractable (FPT) on chordal graphs and in graphs which exclude a fixed apex graph as a minor.

## Keywords

Dominator Coloring Fixed-Parameter Tractability Chordal Graphs Apex-Minor-Free Graphs

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## Authors and Affiliations

• S. Arumugam
• 1
• 2
• K. Raja Chandrasekar
• 1
• Neeldhara Misra
• 3
• Geevarghese Philip
• 3
• Saket Saurabh
• 3
1. 1.National Centre for Advanced Research in Discrete Mathematics (n-CARDMATH)Kalasalingam UniversityKrishnankoilIndia
2. 2.School of Electrical Engineering and Computer ScienceThe University of NewcastleAustralia
3. 3.The Institute of Mathematical SciencesChennaiIndia