Abstract
For a graph \(G = (V, E)\) with vertex set V and edge set E, a subset F of E is called an edge dominating set (resp. a total edge dominating set) if every edge in \(E\backslash F\) (resp. in E) is adjacent to at least one edge in F, the minimum cardinality of an edge dominating set (resp. a total edge dominating set) of G is the edge domination number (resp. total edge domination number) of G, denoted by \(\gamma ^{\prime }(G)\) (resp. \(\gamma _t^{\prime }(G)\)). In the present paper, we first prove that the total edge domination problem is NP-complete for bipartite graphs with maximum degree 3. Then, for a graph G, we give the inequality \(\gamma ^{\prime }(G)\leqslant \gamma ^{\prime }_{t}(G)\leqslant 2\gamma ^{\prime }(G)\) and characterize the trees T which obtain the upper or lower bounds in the inequality.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cockayne EJ, Hedetniemi ST (1977) Towards a theory of domination in graphs. Networks 7(3):247–261
Haynes TW, Hedetniemi ST, Slater PJ (1998) Fundermentals of domination in graphs. Marcel Dekker, New York
Horton JD, Kilakos K (1993) Minimum edge dominating sets. SIAM J Discret Math 6(3):375–387
Karp R (1972) Reducibility among combinatorial problems. In: Miller RE, Thatcher JW (eds) Complexity of computer computations. Plenum Press, New York, pp 85–104
Kalbflisch JG, Stanton R, Horton JD (1971) On covering sets and error correcting codes. J Comb 11A:233–250
Kilakos K (1998) On the complexity of edge domination. Master’s Thesis, University of New Brunswick, New Brunswick, Canada
Kulli VR, Patwari DK (1991) On the edge domination number of a graph. In: Proceedings of the symposium on graph theory and combinatorics, Publication, vol 21, pp 75–81. Centre Math Sci Trivandrum, Cochin
Lru CL (1968) Introduction to combinatorial mathematics. McGraw-Hill, New York
Mitchell S, Hedetniemi ST (1977) Edge domination in trees. Congr Numer 19:489–509
Muddebihal MH, Sedamkar AR (2013) Characterization of trees with equal edge domination and end edge domination numbers. Math Theory Model 5:33–42
Paspasan MNS, Canoy SR (2016) Edge domination and total edge domination in the join of graphs. Appl Math Sci 10:1077–1086
Velammal S (2014) Equality of connected edge domination and total edge domaination in graphs. Int J Enhanc Res Sci Technol Eng 5:198–201
Xu B (2006) Two classes of edge domination in graphs. Discret Appl Math 154:1541–1546
Yannakakis M (1981) Edge-deletion problems. SIAM J Comput 10:297–309
Yannakakis M, Gavril F (1980) Edge dominating sets in graphs. SIAM J Appl Math 38:364–372
Zhao YC, Liao ZH, Miao LY (2014) On the algorithmic complexity of edge total domination. Theor Comput Sci 6:28–33
Acknowledgements
The authors would like to thank anonymous reviewers for their helpful comments and suggestions which lead to a considerably improved presentation. This work is partially supported by National Natural Science Foundation of China (Grant Nos. 11571155, 11761070, 11201208).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Pan, Z., Yang, Y., Li, X. et al. The complexity of total edge domination and some related results on trees. J Comb Optim 40, 571–589 (2020). https://doi.org/10.1007/s10878-020-00596-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-020-00596-y