On the number of maximal independent sets in minimum colorings of split graphs

Abstract

The largest number of color classes that are dominating sets (or, equivalently, maximal independent sets) among all $\chi \left(G\right)$-colorings of a graph $G$ is called the dominating-$\chi$-color number of $G$, and denoted $d_{\chi }\left(G\right)$. It was shown in Arumugam et al. (2011) that determining whether $d_{\chi }\left(G\right)\ge 2$ is NP-complete even in 3-chromatic graphs $G$. In this note, we prove that in split graphs the dominating-$\chi$-color number equals to 1 if and only if its domination number is greater than 2. While such graphs can be efficiently recognized, we prove that for split graphs with domination number at most 2 the decision version of the problem of determining dominating-$\chi$-color number is NP-complete. We also present existence results for split graphs attaining prescribed values of dominating-$\chi$-color numbers and some other relevant parameters.