Double Domination in Graphs

Each vertex of a graph \(G = (V, E)\) is said to dominate every vertex in its closed neighborhood. A set \(S \subseteq V\) is a double dominating set for \(G\) if each vertex in \(V\) is dominated by at least two vertices in \(S\). The smallest cardinality of a double dominating set is called the double domination number \(dd(G)\). We initiate the study of double domination in graphs and present bounds and some exact values for \(dd(G)\). Also, relationships between \(dd(G)\) and other domination parameters are explored. Then we extend many results of double domination to multiple domination.