We prove some general results on irredundant sets of queens on chessboards, and determine the irredundance numbers of the queens graph \(Q_n\), for \(n = 5, 6\).

Let \(G\) be a graph. The weak domination number of \(G\), \(\gamma_w(G)\), is the minimum cardinality of a set \(D\) of vertices where every vertex \(u \notin D\) is adjacent to a vertex \(v \in D\), where \(\deg(v) \leq \deg(u)\). The strong domination number of \(G\), \(\gamma_s(G)\), is the minimum cardinality of a set  \(D\) of vertices where every vertex \(u \notin D\) is adjacent to a vertex \(v \in D\), where \(\deg(v) \geq \deg(u)\). Similarly, the independent weak domination number, \(i_w(G)\), and the independent strong domination number, \(i_{st}(G)\), are defined with the additional requirement that the set \(D\) is independent. We find upper bounds on the number of edges of a graph in terms of the number of vertices and for each of these four domination parameters. We also characterize all graphs where equality is achieved in each of the four bounds.

For \(k \geq 2\), the \(P_k\)-free domination number \(\gamma(G; -P_k)\) is the minimum cardinality of a dominating set \(S\) in \(G\) such that the subgraph \(\langle S \rangle\) induced by \(S\) contains no path \(P_k\) on \(k\) vertices. The path-free domination number is at least the domination number and at most the independent domination number of the graph. We show that if \(G\) is a connected graph of order \(n \geq 2\), then \(\gamma(G; -P_k) \leq n + 2(k – 1) – 2\sqrt{n(k-1)}\), and this bound is sharp. We also give another bound on \(\gamma(G; -P_k)\) that yields the corollary: if \(G\) is a graph with \(\gamma(G) \geq 2\) that is \(K_{1,t+1}\)-free and \((K_{1,t+1}+e)\)-free (\(t \geq 3\)), then \(\gamma(G; -P_3) \leq (t-2)\gamma(G) – 2(t-3)\), and we characterize the extremal graphs for the corollary’s bound. Every graph \(G\) with maximum degree at most \(3\) is shown to have equal domination number and \(P_3\)-free domination number. We define a graph \(G\) to be \(P_k\)-domination perfect if \(\gamma(H) = \gamma(H; -P_k)\) for every induced subgraph \(H\) of \(G\). We show that a graph \(G\) is \(P_3\)-domination perfect if and only if \(\gamma(H) = \gamma(H; -P_3)\) for every induced subgraph \(H\) of \(G\) with \(\gamma(H) = 3\).