In this paper, we extend the concept of k-distance total domination to the concept of L-distance total domination, which is to find a minimum vertex set \(D\subseteq V\) such that each vertex v of the graph G is at a distance \(0<d(v,u)\leqslant a_v\) from some vertex \(u\in D\) , where \(a_v\) is an arbitrary positive integer assigned to v, and \(L=\{a_{v}|~v\in V(G)\}\) . We then compute the exact values of the L-distance total domination numbers of paths and cycles. Finally, by constructing a proper order of the vertices and using a labeling method, we provide an \(O(n^2)\) time algorithm to find a minimum L-distance total dominating set of a block graph, a superclass of trees. Since k-distance total domination is a special form of L-distance total domination, the above algorithm can also determine the k-distance total domination number of a block graph.
