Strong Hotspot Domination Integrity of Corona Products, Comb Products and Shadow Graphs with Applications

Abstract

The study of graph vulnerability parameters provides valuable insights into the structural stability and resilience of networks when there is a disruption. The strong hotspot domination integrity (DISH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$DI_{SH}$$\end{document}) is one such measure, combining domination, integrity, and heatmap centrality into a single context that captures both control and resilience of a network. It is defined as DISH(G)=min{|SSH|+m(G-SSH):SSH}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$DI_{SH}(G)=\min \{|S_{SH}|+m(G-S_{SH}): S_{SH}\}$$\end{document} and the parameter is taken over all strong hotspot dominating sets SSH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{SH}$$\end{document}, where domination is governed by the heatmap centrality condition ensuring that for every vertex u∈V(G)-SSH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in V(G)-S_{SH}$$\end{document}, there exists v∈SSH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v\in S_{SH}$$\end{document} such that HC(v)≥HC(u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$HC(v)\ge HC(u)$$\end{document}. In this paper, we investigate the strong hotspot domination integrity for three important graph operations: the corona product, the shadow graph, and the comb product of certain graphs and understand how the parameter behaves under structural transformations. Moreover, we present few applications of strong hotspot domination integrity in acupuncture, electric power networks, and medicine.