Some Results on [1, k]-sets of Lexicographic Products of Graphs

TL;DR

This paper studies special vertex subsets called $[1,k]$-sets in lexicographic product graphs, characterizes their existence, computes their minimum sizes, and proves that finding the smallest total $[1,k]$-set is NP-complete.

Contribution

It provides a complete characterization of $[1,k]$-sets in lexicographic product graphs and determines their minimum sizes, introducing new insights into their structure and computational complexity.

Findings

Characterization of graphs with total $[1,k]$-sets in lexicographic products

Exact formulas for $oldsymbol{eta_{[1,k]}(G owtie H)}$, $oldsymbol{eta_{t[1,k]}(G owtie H)}$, and $oldsymbol{eta_{i[1,k]}(G owtie H)}$

NP-completeness of finding the smallest total $[1,k]$-set

Abstract

A subset $S \subseteq V$ in a graph $G = (V,E)$ is called a $[1, k]$-set, if for every vertex $v \in V \setminus S$, $1 \leq | N_G(v) \cap S | \leq k$. The $[1,k]$-domination number of $G$, denoted by $\gamma_{[1, k]}(G)$ is the size of the smallest $[1,k]$-sets of $G$. A set $S'\subseteq V(G)$ is called a total $[1,k]$-set, if for every vertex $v \in V$, $1 \leq | N_G(v) \cap S | \leq k$. If a graph $G$ has at least one total $[1, k]$-set then the cardinality of the smallest such set is denoted by $\gamma_{t[1, k]}(G)$. We consider $[1, k]$-sets that are also independent. Note that not every graph has an independent $[1, k]$-set. For graphs having an independent $[1, k]$-set, we define $[1, k]$-independence numbers which is denoted by $\gamma_{i[1, k]}(G)$. In this paper, we investigate the existence of $[1,k]$-sets in lexicographic products $G\circ H$. Furthermore, we completely characterize graphs which their lexicographic product has at least one total $[1,k]$-set. Also, we determine $\gamma_{[1, k]}(G\circ H)$, $\gamma_{t[1, k]}(G\circ H)$ and $\gamma_{i[1, k]}(G\circ H)$. Finally, we show that finding smallest total $[1, k]$-set is $NP$-complete.