Total restrained domination in trees

Let G=(V,E) be a graph. A set S⊆V is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of V-S is adjacent to a vertex in V-S. The total restrained domination number of G, denoted by γ_tr(G), is the smallest cardinality of a total restrained dominating set of G. We show that if T is a tree of order n, then . Moreover, we show that if T is a tree of order , then . We then constructively characterize the extremal trees T of order n achieving these lower bounds.     

An upper bound for the restrained domination number of a graph with minimum degree at least two in terms of order and minimum degree

Let G=(V,E) be a graph. A set S⊆V is a restrained dominating set if every vertex in V−S is adjacent to a vertex in S and to a vertex in V−S. The restrained domination number of G, denoted γ_r(G), is the smallest cardinality of a restrained dominating set of G. We will show that if G is a connected graph of order n and minimum degree δ and not isomorphic to one of nine exceptional graphs, then .     

Nordhaus-Gaddum results for restrained domination and total restrained domination in graphs

Let G=(V,E) be a graph. A set S⊆V is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of V-S is adjacent to a vertex in V-S. A set S⊆V is a restrained dominating set if every vertex in V-S is adjacent to a vertex in S and to a vertex in V-S. The total restrained domination number of G (restrained domination number of G, respectively), denoted by γ_tr(G) (γ_r(G), respectively), is the smallest cardinality of a total restrained dominating set (restrained dominating set, respectively) of G. We bound the sum of the total restrained domination numbers of a graph and its complement, and provide characterizations of the extremal graphs achieving these bounds. It is known (see [G.S. Domke, J.H. Hattingh, S.T. Hedetniemi, R.C. Laskar, L.R. Markus, Restrained domination in graphs, Discrete Math. 203 (1999) 61-69.]) that if G is a graph of order n?2 such that both G and are not isomorphic to P₃, then . We also provide characterizations of the extremal graphs G of order n achieving these bounds.     

Total restrained domination in graphs with minimum degree two

In this paper, we continue the study of total restrained domination in graphs, a concept introduced by Telle and Proskurowksi (Algorithms for vertex partitioning problems on partial k-trees, SIAM J. Discrete Math. 10 (1997) 529-550) as a vertex partitioning problem. A set S of vertices in a graph G=(V,E) is a total restrained dominating set of G if every vertex is adjacent to a vertex in S and every vertex of V?S is adjacent to a vertex in V?S. The minimum cardinality of a total restrained dominating set of G is the total restrained domination number of G, denoted by γ_tr(G). Let G be a connected graph of order n with minimum degree at least 2 and with maximum degree Δ where Δ?n-2. We prove that if n?4, then and this bound is sharp. If we restrict G to a bipartite graph with Δ?3, then we improve this bound by showing that and that this bound is sharp.     

Total domination and total domination subdivision number of a graph and its complement

A set S of vertices of a graph G=(V,E) with no isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination numberγ_t(G) is the minimum cardinality of a total dominating set of G. The total domination subdivision numbersd_γt(G) is the minimum number of edges that must be subdivided in order to increase the total domination number. We consider graphs of order n?4, minimum degree δ and maximum degree Δ. We prove that if each component of G and has order at least 3 and , then and if each component of G and has order at least 2 and at least one component of G and has order at least 3, then . We also give a result on stronger than a conjecture by Harary and Haynes.     

On the degree distance of a graph

If G is a connected graph with vertex set V, then the degree distance of G, D^′(G), is defined as , where degw is the degree of vertex w, and d(u,v) denotes the distance between u and v. We prove the asymptotically sharp upper bound for graphs of order n and diameter d. As a corollary we obtain the bound for graphs of order n. This essentially proves a conjecture by Tomescu [I. Tomescu, Some extremal properties of the degree distance of a graph, Discrete Appl. Math. (98) (1999) 159-163].