Algorithmic Aspects of Dominator Colorings in Graphs

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.