On independent $[1,2]$-sets in trees

TL;DR

This paper characterizes trees with independent [1,2]-sets, provides necessary and sufficient conditions for certain graph classes, and introduces a linear algorithm to determine their existence and minimal size.

Contribution

It offers a complete characterization of trees with independent [1,2]-sets and presents an efficient linear algorithm for their detection and size computation.

Findings

Necessary condition for graphs to have independent [1,2]-sets

Characterization of trees where every vertex belongs to some independent [1,2]-set

Linear algorithm to decide existence and minimal size of such sets

Abstract

An independent $[1,k]$-set $S$ in a graph $G$ is a dominating set which is independent and such that every vertex not in $S$ has at most $k$ neighbors in it. The existence of such sets is not guaranteed in every graph and trees having an independent $[1,k]$-set have been characterized. In this paper we solve some problems previously posed by other authors about independent $[1,2]$-sets. We provide a necessary condition for a graph to have an independent $[1,2]$-set, in terms of spanning trees and we prove that this condition is also sufficient for cactus graphs. We follow the concept of excellent tree and characterize the family of trees such that any vertex belong to some independent $[1,2]$-set. Finally we describe a linear algorithm to decide whether a tree has an independent $[1,2]$-set. Such algorithm can be easily modified to obtain the cardinality of the smallest independent $[1,2]$-set of a tree.