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If \(n\) is an integer, \(n \geq 2\), and \(u\) and \(v\) are vertices of a graph \(G\), then \(u\) and \(v\) are said to be \(K_n\)-adjacent vertices of \(G\) if there is a subgraph of \(G\), isomorphic to \(K_n\), containing \(u\) and \(v\). A total \(K_n\)-dominating set of \(G\) is a set \(D\) of vertices such that every vertex of \(G\) is \(K_n\)-adjacent to a vertex of \(D\). The total \(K_n\)-domination number \(\gamma_{K_n}^t(G)\) of \(G\) is the minimum cardinality among the total \(K_n\)-dominating sets of vertices of \(G\). It is shown that, for \(n \in \{3, 4, 5\}\), if \(G\) is a graph with no \(K_n\)-isolated vertex, then \(\gamma_{K_n}^t(G) \leq (2p)/{n}\). Further, \(K_n\)-connectivity is defined and it is shown that, for \(n \in \{3, 4\}\), if \(G\) is a \(K_n\)-connected graph of order \(\geq n + 1\), then \(\gamma_{K_n}^t(G) \leq (2p)/(n + 1)\). We establish that the upper bounds obtained are best possible.