On the number of k-dominating independent sets

Abstract

We study the existence and the number of $k$-dominating independent sets in certain graph families. While the case $k=1$ namely the case of maximal independent sets - which is originated from Erdős and Moser - is widely investigated, much less is known in general. In this paper we settle the question for trees and prove that the maximum number of $k$-dominating independent sets in $n$-vertex graphs is between $c_k\cdot {\sqrt[2k]{2}}^n$ and $c_k^{\prime }\cdot {\sqrt[k+1]{2}}^n$ if $k\ge 2$, moreover the maximum number of $2$-dominating independent sets in $n$-vertex graphs is between $c\cdot 1.22^n$ and $c^{\prime }\cdot 1.246^n$. Graph constructions containing a large number of $k$-dominating independent sets are coming from product graphs, complete bipartite graphs and with finite geometries. The product graph construction is associated with the number of certain MDS codes.

13 pages

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