The complexity of total edge domination and some related results on trees

Zhuo Pan, Yu Yang, Xianyue Li and Shou-Jun Xu ()
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Zhuo Pan: Lanzhou University
Yu Yang: Lanzhou University
Xianyue Li: Lanzhou University
Shou-Jun Xu: Lanzhou University

Journal of Combinatorial Optimization, No 0, 19 pages

Abstract: Abstract For a graph $$G = (V, E)$$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 $$E\backslash F$$E\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 $$\gamma ^{\prime }(G)$$γ′(G) (resp. $$\gamma _t^{\prime }(G)$$γt′(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 $$\gamma ^{\prime }(G)\leqslant \gamma ^{\prime }_{t}(G)\leqslant 2\gamma ^{\prime }(G)$$γ′(G)⩽γt′(G)⩽2γ′(G) and characterize the trees T which obtain the upper or lower bounds in the inequality.

Keywords: Edge domination; Total edge domination; NP-completeness; Trees (search for similar items in EconPapers)
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DOI: 10.1007/s10878-020-00596-y

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