Neighborhood total domination of a graph and its complement

A neighborhood total dominating set in a graph G is a dominating set S of G with the property that the subgraph induced by N(S), the open neighborhood of the set S, has no isolated vertex. The neighborhood total domination number γnt(G) is the minimum cardinality of a neighborhood total dominating set of G. Arumugam and Sivagnanam introduced and studied the concept of neighborhood total domination in graphs [S. Arumugam and C. Sivagnanam, Opuscula Math. 31 (2011) 519–531]. They proved that if G and G are connected, then γnt(G) + γnt(G) ≤ { d 2 e+ 2 if diam(G) ≥ 3. d 2 e+ 3 if diam(G) = 2. , where G is the complement of G. The problem of characterizing graphs attaining equality in the previous bounds was left as an open problem by the authors. In this paper, we address this open problem by studying sharpness and strictness of the above inequalities. ∗ Also at Department of Mathematics, University of Tafresh, Tafresh, Iran. D.A. MOJDEH ET AL. /AUSTRALAS. J. COMBIN. 65 (1) (2016), 37–44 38