Abstract
The connectivity of a network is an important indicator to its reliability and fault tolerability. Since the faulty elements in the network may have some special structures, two new kinds of conditional connectivity, called h-restricted H-structure connectivity and h-restricted H-substructure connectivity, are proposed as a generalization of conditional connectivity, where h & GE;1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\ge 1$$\end{document}, and H is some special structure. In this paper, we establish both h-restricted H-structure connectivity and h-restricted H-substructure connectivity for the hypercube Qn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_n$$\end{document}, where the special structures are K1,K1,1,K1,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_1, K_{1,1}, K_{1,2}$$\end{document}, respectively.