Exponential Independence

TL;DR

This paper introduces the concept of exponential independence in graphs, explores its properties, and characterizes graphs where exponential independence equals traditional independence, providing exact values, bounds, and extremal graphs.

Contribution

It defines exponential independence, analyzes its properties, and characterizes graphs where exponential and traditional independence numbers coincide, including trees.

Findings

Exact values for specific graphs

Tight bounds and extremal graphs

Characterization of graphs with equal exponential and traditional independence

Abstract

For a set $S$ of vertices of a graph $G$, a vertex $u$ in $V(G)\setminus S$, and a vertex $v$ in $S$, let ${\rm dist}_{(G,S)}(u,v)$ be the distance of $u$ and $v$ in the graph $G-(S\setminus \{ v\})$. Dankelmann et al. (Domination with exponential decay, Discrete Math. 309 (2009) 5877-5883) define $S$ to be an exponential dominating set of $G$ if $w_{(G,S)}(u)\geq 1$ for every vertex $u$ in $V(G)\setminus S$, where $w_{(G,S)}(u)=\sum\limits_{v\in S}\left(\frac{1}{2}\right)^{{\rm dist}_{(G,S)}(u,v)-1}$. Inspired by this notion, we define $S$ to be an exponential independent set of $G$ if $w_{(G,S\setminus \{ u\})}(u)<1$ for every vertex $u$ in $S$, and the exponential independence number $\alpha_e(G)$ of $G$ as the maximum order of an exponential independent set of $G$. Similarly as for exponential domination, the non-local nature of exponential independence leads to many interesting effects and challenges. Our results comprise exact values for special graphs as well as tight bounds and the corresponding extremal graphs. Furthermore, we characterize all graphs $G$ for which $\alpha_e(H)$ equals the independence number $\alpha(H)$ for every induced subgraph $H$ of $G$, and we give an explicit characterization of all trees $T$ with $\alpha_e(T)=\alpha(T)$.