# Parameterized and Exact Algorithms for Class Domination Coloring

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10139)

## Abstract

A class domination coloring (also called as cd-coloring) of a graph is a proper coloring such that for every color class, there is a vertex that dominates it. The minimum number of colors required for a cd-coloring of the graph G, denoted by $$\chi _{cd}(G)$$, is called the class domination chromatic number (cd-chromatic number) of G. In this work, we consider two problems associated with the cd-coloring of a graph in the context of exact exponential-time algorithms and parameterized complexity. (1) Given a graph G on n vertices, find its cd-chromatic number. (2) Given a graph G and integers k and q, can we delete at most k vertices such that the cd-chromatic number of the resulting graph is at most q? For the first problem, we give an exact algorithm with running time $$\mathcal {O}(2^n n^4 \log n)$$. Also, we show that the problem is $$\mathsf {FPT}$$ with respect to the number of colors q as the parameter on chordal graphs. On graphs of girth at least 5, we show that the problem also admits a kernel with $$\mathcal {O}(q^3)$$ vertices. For the second (deletion) problem, we show $$\mathsf {NP}$$-hardness for each $$q \ge 2$$. Further, on split graphs, we show that the problem is $$\mathsf {NP}$$-hard if q is a part of the input and $$\mathsf {FPT}$$ with respect to k and q. As recognizing graphs with cd-chromatic number at most q is $$\mathsf {NP}$$-hard in general for $$q \ge 4$$, the deletion problem is unlikely to be $$\mathsf {FPT}$$ when parameterized by the size of deletion set on general graphs. We show fixed parameter tractability for $$q \in \{2,3\}$$ using the known algorithms for finding a vertex cover and an odd cycle transversal as subroutines.

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© Springer International Publishing AG 2017

## Authors and Affiliations

• R. Krithika
• 1
• Ashutosh Rai
• 1
• Saket Saurabh
• 1
• 2
• Prafullkumar Tale
• 1
Email author
1. 1.The Institute of Mathematical SciencesHBNIChennaiIndia
2. 2.University of BergenBergenNorway