Distance Independent Domination in Iterated Line Graphs

Abstract

Let \(k \geq 1\) be an integer and let \(G = (V, E)\) be a graph. A set \(S\) of vertices of \(G\) is \(k\)-independent if the distance between any two vertices of \(S\) is at least \(k+1\). We denote by \(\rho_k(G)\) the maximum cardinality among all \(k\)-independent sets of \(G\). Number \(\rho_k(G)\) is called the \(k\)-packing number of \(G\). Furthermore, \(S\) is defined to be \(k\)-dominating set in \(G\) if every vertex in \(V(G) – S\) is at distance at most \(k\) from some vertex in \(S\). A set \(S\) is \(k\)-independent dominating if it is both \(k\)-independent and \(k\)-dominating. The \(k\)-independent dominating number, \(i_k(G)\), is the minimum cardinality among all \(k\)-independent dominating sets of \(G\). We find the values \(i_k(G)\) and \(\rho_k(G)\) for iterated line graphs.