Johannes H. Hattingh

Restrained domination in graphs

Abstract In this paper, we initiate the study of a variation of standard domination, namely restrained domination. Let G =( V , E ) be a graph. A restrained dominating set is a set S ⊆ V where every vertex in V − S is adjacent to a vertex in S as well as another vertex in V − S . The restrained domination number of G , denoted by γ r ( G ), is the smallest cardinality of a restrained dominating set of G . We determine best possible upper and lower bounds for γ r ( G ), characterize those graphs achieving these bounds and find best possible upper and lower bounds for γ r (G)+γ r ( G ) where G is a connected graph. Finally, we give a linear algorithm for determining γ r ( T ) for any tree and show that the decision problem for γ r ( G ) is NP-complete even for bipartite and chordal graphs.

Restrained domination in trees

Abstract Let G=(V,E) be a graph. A set S⊆V is a restrained dominating set if every vertex not in S is adjacent to a vertex in S and to a vertex in V−S . The restrained domination number of G , denoted by γ r (G) , is the smallest cardinality of a restrained dominating set of G . We show that if T is a tree of order n , then γ r (T)⩾⌈(n+2)/3⌉ . Moreover, we constructively characterize the extremal trees T of order n achieving this lower bound.

Majority domination in graphs

Abstract A two-valued function f defined on the vertices of a graph G = (V, E), f: V → -1, 1, is a majority dominating function if the sum of its function values over at least half the closed neighborhoods is at least one. That is, for at least half the vertices v ϵ V, f(N[v]) ⩾ 1, where N[v] consists of v and every vertex adjacent to v. The weight of a majority dominating function is f(V) = ∑f(v), over all vertices v ϵ V. The majority domination number of a graph G, denoted γmaj(G), equals the minimum weight of a majority dominating function of G. In this paper we present properties of the majority domination number and establish its value for various classes of graphs. We show that the decision problem corresponding to the problem of computing γmaj(G) is NP-complete.

Maximum sizes of graphs with given domination parameters

We find the maximum number of edges for a graph of given order and value of parameter for several domination parameters. In particular, we consider the total domination and independent domination numbers.

Characterizations of trees with equal domination parameters

Several results concerning existence of k-paths, for which the sum of their vertex degrees is small, are presented.

Trees with Equal Domination and Restrained Domination Numbers

Let G = (V,E) be a graph and let SV. The set S is a packing in G if the vertices of S are pairwise at distance at least three apart in G. The set S is a dominating set (DS) if every vertex in V − S is adjacent to a vertex in S. Further, if every vertex in V − S is also adjacent to a vertex in V − S, then S is a restrained dominating set (RDS). The domination number of G, denoted by γ(G), is the minimum cardinality of a DS of G, while the restrained domination number of G, denoted by γr(G), is the minimum cardinality of a RDS of G. The graph G is γ-excellent if every vertex of G belongs to some minimum DS of G. A constructive characterization of trees with equal domination and restrained domination numbers is presented. As a consequence of this characterization we show that the following statements are equivalent: (i) T is a tree with γ(T)=γr(T); (ii) T is a γ-excellent tree and T ≠ K2; and (iii) T is a tree that has a unique maximum packing and this set is a dominating set of T. We show that if T is a tree of order n with ℓ leaves, then γr(T) ≤ (n + ℓ + 1)/2, and we characterize those trees achieving equality.

Augmenting a graph of minimum degree 2 to have two disjoint total dominating sets

A total dominating set of a graph is a set of vertices such that every vertex is adjacent to a vertex in the set. We show that given a graph of order n with minimum degree at least 2, one can add at most (n-2n)/4+O(logn) edges such that the resulting graph has two disjoint total dominating sets, and this bound is best possible.

Star-path bipartite Ramsey numbers

Abstract For bipartite graphs G 1 , G 2 , …, G k , the bipartite Ramsey number b ( G 1 , G 2 , …, G k ) is the least positive integer b so that any colouring of the edges of K b , b with k colours will result in a copy of G i in the i th colour for some i . In this note, we establish the exact value of the bipartite Ramsey number b ( P m , K l , n ) for all integers m , n ⩾ 2, where P m denotes a path on m vertices.

On irredundant Ramsey numbers for graphs

The irredundant Ramsey number s(m, n) is the smallest p such that in every two-coloring of the edges of Kp using colors red (R) and blue (B), either the blue graph contains an m-element irredundant set or the red graph contains an n-element irredundant set. We develop techniques to obtain upper bounds for irredundant Ramsey numbers of the form s(3, n) and prove that 18 ≤ s(3,7) ≤ 19.

Asymptotic bounds for irredundant and mixed Ramsey numbers

The irredundant Ramsey number s(m, n) is the smallest N such that in every red-blue coloring of the edges of KN, either the blue graph contains an m-element irredundant set or the red graph contains an n-element irredundant set. The definition of the mixed Ramsey number t(m, n) differs from s(m, n) in that the n-element irredundant set is replaced by an n-element independent set. We prove asymptotic lower bounds for s(n, n) and t(m, n) (with m fixed and n large) and a general upper bound for t(3, n).