covering total double roman domination in graphs

For a graph g with no isolated vertex, a covering total double roman dominating function (ctdrd function) f of g is a total double roman dominating function (tdrd function) of g for which the set {v∈v(g)|f(v)≠0} is a vertex cover set. the covering total double roman domination number γctdr(g) equals the minimum weight of an ctdrd function on g. an ctdrd function on g with weight γctdr(g) is called a γctdr(g)-function. in this paper, the graphs g with small γctdr(g) are characterised. we show that the decision problem associated with ctdrd is np-complete even when restricted to planer graphs with maximum degree at most four. we then show that for every graph g without isolated vertices, γoitr(g)<γctdr(g)<2γoitr(g) and for every tree t, 2β(t)+1≤γctdr(t)≤4β(t), where γoitr(g) and β(t) are the outer independent total roman domination number of g, and the minimum vertex cover number of t respectively. moreover we investigate the γctdr of corona of two graphs.