Bull. Korean Math. Soc. 56 (2019), No. 4, pp. 1059–1075 https://doi.org/10.4134/BKMS.b180891 pISSN: 1015-8634 / eISSN: 2234-3016 TOTAL DOMINATION NUMBER OF CENTRAL GRAPHS Farshad Kazemnejad and Somayeh Moradi Abstract. Let G be a graph with no isolated vertex. A total dominating set, abbreviated TDS of G is a subset S of vertices of G such that every vertex of G is adjacent to a vertex in S. The total domination number of G is the minimum cardinality of a TDS of G. In this paper, we study the total domination number of central graphs. Indeed, we obtain some tight bounds for the total domination number of a central graph C(G) in terms of some invariants of the graph G. Also we characterize the total domination number of the central graph of some families of graphs such as path graphs, cycle graphs, wheel graphs, complete graphs and complete multipartite graphs, explicitly. Moreover, some Nordhaus-Gaddum-like relations are presented for the total domination number of central graphs. Introduction The concept of total domination in graphs was first introduced by Cockayne, Dawes and Hedetniemi [2] and has been studied extensively by many researchers in the last years. The literature on this subject has been surveyed and detailed in the recent book [3]. In this paper, we study the total domination number of central graphs. In the sequel we remind some concepts and terminology which are used in this paper. Let G be a graph with the vertex set V (G) of order n and the edge set E(G) of size m. The open neighborhood and the closed neighborhood of a vertex v ∈ V (G) are N G (v)= {u ∈ V (G): uv ∈ E(G)} and N G [v]= N G (v) ∪{v}, respectively. The degree of a vertex v is defined as deg G (v)= |N G (v)|. The minimum and maximum degree of a vertex in G are denoted by δ = δ(G) and Δ = Δ(G), respectively. We write K n , C n and P n for a complete graph, a cycle graph and a path graph of order n, respectively, while G[S], W n and K n1,n2,...,np denote the subgraph of G induced on the vertex set S,a wheel graph of order n + 1, and a complete p-partite graph, respectively. The complement of a graph G, denoted by G, is a graph with the vertex set V (G) such that for every two vertices v and w, vw ∈ E( G) if and only if vw ∈ E(G). A vertex cover of the graph G is a set D ⊆ V (G) such Received September 19, 2018; Revised January 23, 2019; Accepted March 4, 2019. 2010 Mathematics Subject Classification. Primary 05C76; Secondary 97K30. Key words and phrases. total domination number, central graph, Nordhaus-Gaddum-like relation. c 2019 Korean Mathematical Society 1059