Mutual d-visibility in Graphs

Identifier:  imarina:9462783

Handle: https://hdl.handle.net/20.500.11797/imarina9462783

Authors:  Kuziak, Dorota; Montejano, Luis P; Rodriguez-Velazquez, Juan A

Abstract:
Let d be a positive integer. The mutual d-visibility number mu d(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu d}(G)$$\end{document} of a graph G is introduced as the cardinality of the largest mutual d-visibility set. That is, X subset of V(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X\subseteq V(G)$$\end{document} is a mutual d-visibility set if for any pair of vertices x,y is an element of X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x,y\in X$$\end{document}, the distance between them is larger than d, or there exists a shortest x, y-path in G whose internal vertices are not in X. Several combinatorial and computational aspects of mu d(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu d}(G)$$\end{document} are given in this work. Finally, the NP-completeness of the decision problem concerning finding mu d(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu d}(G)$$\end{document} is proved.