مشخصات پژوهش

Abstract A Nordhaus–Gaddum-type result is a lower or an upper bound on the sum or the product of a parameter of a graph and its complement. In this paper we continue the study of Nordhaus–Gaddum bounds for the total Roman domination number γt R. Let G be a graph on n vertices and let G denote the complement of G, and let δ ∗ (G) denote the minimum degree among all vertices in G and G. For δ ∗ (G) ≥ 1, we show that (i) if G and G are connected, then (γt R(G)−4)(γt R(G)−4) ≤ 4δ ∗ (G)−4, (ii) if γt R(G), γt R(G) ≥ 8, then γt R(G)+γt R(G) ≤ 2δ ∗ (G)+5 and (iii) γt R(G)+γt R(G) ≤ n + 5 and γt R(G)γt R(G) ≤ 6n − 5.