Henning, Michael A.; Oellermann, Ortrud R.; Swart, Henda C. Relationships between distance domination parameters. (English) Zbl 0801.05038 Math. Pannonica 5, No. 1, 67-77 (1994). Some numerical invariants of graphs are generalized. Let \(n\) be a positive integer. A subset \(D\) of the vertex set \(V(G)\) of a graph \(G\) is called \(P_{\leq n}\)-dominating (or total \(P_{\leq n}\)-dominating) if each vertex of \(V(G)- D\) (or of \(V(G)\) respectively) has distance at most \(n\) from some vertex of \(D\). A subset \(I\) of \(V(G)\) is called \(P_{\leq n}\)- independent, if any two distinct vertices of \(I\) have distance at least \(n\). Using these definitions, the \(P_{\leq n}\)-domination number \(\gamma_ n(G)\), the total \(P_{\leq n}\)-domination number \(\gamma^ t_ n(G)\), the \(P_{\leq n}\)-independence number \(\beta_ n(G)\) and the \(P_{\leq n}\)-independent domination number \(i_ n(G)\) are introduced. Also the \(P_{\leq n}\)-covering number is defined. Some inequalities in terms of these graph invariants are proved. Reviewer: B.Zelinka (Liberec) Cited in 1 ReviewCited in 6 Documents MSC: 05C35 Extremal problems in graph theory 05C99 Graph theory 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) Keywords:distance domination; independence number; domination number; graph invariants PDFBibTeX XMLCite \textit{M. A. Henning} et al., Math. Pannonica 5, No. 1, 67--77 (1994; Zbl 0801.05038) Full Text: EuDML