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.