Hop total Roman domination in graphs

In this article, we initiate a study of hop total Roman domination defined as follows: a hop total Roman dominating function (HTRDF) on a graph G=(V,E) is a function f:V?{0,1,2} such that for every vertex u with f(u) = 0 there exists a vertex v at distance 2 from u with f(v) = 2 and the subgraph induced by the vertices assigned non-zero values under f has no isolated vertices. The weight of an HTRDF is the sum of its function values over all vertices, and the hop total Roman domination number ?dtR(G) equals the minimum weight of an HTRDF on G. We provide several properties on the hop total Roman domination number. More precisely, we show that the decision problem corresponding to the hop total Roman domination problem is NP-complete for bipartite graphs, and we determine the exact value of ?(htR)(G) for paths and cycles. Moreover, we characterize all connected graphs G of order n with ?(htR)(G?{2,3,4,n}. Finally, we show that for every tree T of diameter at least 3, ?(htR)(T=?(ht)(T)+2, where ?ht(T) is the hop total domination number.