Structural results and bounds on the $k$-isolate domination number in classical graphs

Abstract

In this paper, the concept of domination plays a critical role in analyzing structural properties of graphs. A recent extension, known as the $k$-isolate domination number ($\gamma_{k,0}(G)$), focuses on dominating sets that isolate exactly $k$ vertices while ensuring the rest of the dominating subgraph is connected or structured. This paper investigates the $k$-isolate domination number for fundamental graph families such as paths ($P_n$), cycles ($C_n$), and a few other special graphs. We establish exact values for $\gamma_{k,0}(G)$ for these graphs and illustrate the patterns and bounds that emerge, contributing to the broader understanding of domination-related parameters in combinatorial structures.