On Upper Total Domination Versus Upper Domination in Graphs

A total dominating set of a graph G is a dominating set S of G such that the subgraph induced by S contains no isolated vertex, where a dominating set of G is a set of vertices of G such that each vertex in $$V(G){\setminus } S$$V(G)\S has a neighbor in S. A (total) dominating set S is said to be minimal if $$S{\setminus } \{v\}$$S\{v} is not a (total) dominating set for every $$v\in S$$v∈S. The upper total domination number $$\varGamma _t(G)$$Γt(G) and the upper domination number $$\varGamma (G)$$Γ(G) are the maximum cardinalities of a minimal total dominating set and a minimal dominating set of G, respectively. For every graph G without isolated vertices, it is known that $$\varGamma _t(G)\le 2\varGamma (G)$$Γt(G)≤2Γ(G). The case in which $$\frac{\varGamma _t(G)}{\varGamma (G)}=2$$Γt(G)Γ(G)=2 has been studied in Cyman et al. (Graphs Comb 34:261–276, 2018), which focused on the characterization of the connected cubic graphs and proposed one problem to be solved and two questions to be answered in terms of the value of $$\frac{\varGamma _t(G)}{\varGamma (G)}$$Γt(G)Γ(G). In this paper, we solve this problem, i.e., the characterization of the subcubic graphs G that satisfy $$\frac{\varGamma _t(G)}{\varGamma (G)}=2$$Γt(G)Γ(G)=2, by constructing a class of subcubic graphs, which we call triangle-trees. Moreover, we show that the answers to the two questions are negative by constructing connected cubic graphs G that satisfy $$\frac{\varGamma _t(G)}{\varGamma (G)}>\frac{3}{2}$$Γt(G)Γ(G)>32 and a class of regular non-complete graphs G that satisfy $$\frac{\varGamma _t(G)}{\varGamma (G)}=2$$Γt(G)Γ(G)=2.