Graphs and Combinatorics

, Volume 35, Issue 3, pp 767–778 | Cite as

On Upper Total Domination Versus Upper Domination in Graphs

  • Enqiang Zhu
  • Chanjuan Liu
  • Fei DengEmail author
  • Yongsheng Rao
Original Paper


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 \(V(G){\setminus } S\) has a neighbor in S. A (total) dominating set S is said to be minimal if \(S{\setminus } \{v\}\) is not a (total) dominating set for every \(v\in S\). The upper total domination number \(\varGamma _t(G)\) and the upper domination number \(\varGamma (G)\) 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 \(\varGamma _t(G)\le 2\varGamma (G)\). The case in which \(\frac{\varGamma _t(G)}{\varGamma (G)}=2\) 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 \(\frac{\varGamma _t(G)}{\varGamma (G)}\). In this paper, we solve this problem, i.e., the characterization of the subcubic graphs G that satisfy \(\frac{\varGamma _t(G)}{\varGamma (G)}=2\), 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 \(\frac{\varGamma _t(G)}{\varGamma (G)}>\frac{3}{2}\) and a class of regular non-complete graphs G that satisfy \(\frac{\varGamma _t(G)}{\varGamma (G)}=2\).


Upper domination number Upper total domination number Subcubic graphs 



This work was supported by the National Natural Science Foundation of China (61872101, 61672051, 61702075), the China Postdoctoral Science Foundation under Grant (2017M611223).


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Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Computing Science and TechnologyGuangzhou UniversityGuangzhouChina
  2. 2.School of Computer Science and TechnologyDalian University of TechnologyDalianChina
  3. 3.College of Network SecurityChengdu University of TechnologyChengduChina

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