On the number of k-dominating independent sets

TL;DR

This paper investigates the maximum number of k-dominating independent sets in graphs, providing bounds for trees and general graphs, and exploring constructions from product graphs, bipartite graphs, and finite geometries.

Contribution

It establishes bounds on the number of k-dominating independent sets in graphs, especially trees, and links graph constructions to MDS codes and finite geometries.

Findings

Maximum number of k-dominating independent sets in n-vertex graphs is bounded between exponential functions.

For k=2, the number is between 1.22^n and 1.246^n.

Product graphs and finite geometries are key constructions for many such 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 Erd\H{o}s 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 $c_k'\cdot\sqrt[k+1]{2}^n$ if $k\geq 2$, moreover the maximum number of $2$-dominating independent sets in $n$-vertex graphs is between $c\cdot 1.22^n$ and $c'\cdot1.246^n$. 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.