Ernest J. Cockayne

Total domination in graphs

A set D of vertices of a finite, undirected graph G = (V, E) is a total dominating set if every vertex of V is adjacent to some vertex of D. In this paper we initiate the study of total dominating sets in graphs and, in particular, obtain results concerning the total domination number of G (the smallest number of vertices in a total dominating set) and the total domatic number of G (the largest order of a partition of G into total dominating sets).

Towards a theory of domination in graphs

This paper presents a quick review of results and applications concerning dominating sets in graphs. The domatic number of a graph is defined and studied. It is seen that the theory of domination resembles the well known theory of colorings of graphs.

Graph‐theoretic parameters concerning domination, independence, and irredundance

A vertex x in a subset X of vertices of an undericted graph is redundant if its closed neighbourhood is contained in the union of closed neighborhoods of vertices of X – {x}. In the context of a communications network, this means that any vertex that may receive communications from X may also be informed from X – {x}. The irredundance number ir (G) is the minimum cardinality taken over all maximal sets of vertices having no redundancies. The domination number γ(G) is the minimum cardinality taken over all dominating sets of G, and the independent domination number i(G) is the minimum cardinality taken over all maximal independent sets of vertices of G. The paper contians results that relate these parameters. For example, we prove that for any graph G, ir (G) > γ(G)/2 and for any grpah Gwith p vertices and no isolated vertices, i(G) ≤ p-γ(G) + 1 - ⌈(p - γ(G))/γ(G)⌉.

Information Dissemination in Trees

In large organizations there is frequently a need to pass information from one place, e.g., the president’s office or company headquarters, to all other divisions, departments or employees. This is often done along organizational reporting lines. Insofar as most organizations are structured in a hierarchical or treelike fashion, this can be described as a process of information dissemination in trees. In this paper we present an algorithm which determines the amount of time required to pass, or to broadcast, a unit of information from an arbitrary vertex to every other vertex in a tree. As a byproduct of this algorithm we determine the broadcast center of a tree, i.e., the set of all vertices from which broadcasting can be accomplished in the least amount of time. It is shown that the subtree induced by the broadcast center of a tree is always a star with two or more vertices. We also show that the problem of determining the minimum amount of time required to broadcast from an arbitrary vertex in an arbit...

Roman domination in graphs

Abstract A Roman dominating function on a graph G=(V,E) is a function f : V→{0,1,2} satisfying the condition that every vertex u for which f(u)=0 is adjacent to at least one vertex v for which f(v)=2. The weight of a Roman dominating function is the value f(V)=∑u∈Vf(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. In this paper, we study the graph theoretic properties of this variant of the domination number of a graph.

Contributions to the theory of domination, independence and irredundance in graphs

A vertex x in a subset X of vertices of an undirected graph is redundant if its closed neighbourhood is contained in the union of closed neighbourhoods of vertices of X-{x}. In the context of a communications network, this means that any vertex which may receive communications from X may also be informed from X-{x}. The lower and upper irredundance numbers ir(G) and IR(G) are respectively the minimum and maximum cardinalities taken over all maximal sets of vertices having no redundancies. The domination number @c(G) and upper domination number @C(G) are respectively the minimum and maximum cardinalities taken over all minimal dominating sets of G. The independent domination number i(G) and the independence number @b(G) are respectively the minimum and maximum cardinalities taken over all maximal independent sets of vertices of G. A variety of inequalities involving these quantities are established and sufficient conditions for the equality of the three upper parameters are given. In particular a conjecture of Hoyler and Cockayne [9], namely i+@b=<2p + 2@d - 22p@d, is proved.

Chessboard domination problems

A graph may be formed from n × n chessboard by taking the squares as the vertices and two vertices are adjacent if a chess piece situated on one square covers the other. In this paper we survey some recent results concerning domination parameters for certain graphs constructed in this way.

An upper bound for the k-domination number of a graph

The k-domination number of a graph G, γk(G), is the least cardinality of a set U of verticies such that any other vertex is adjacent to at least k vertices of U. We prove that if each vertex has degree at least k, then γk(G) ≤ kp/(k + 1).

Extremal graphs for inequalities involving domination parameters

Abstract A characterization of n-vertex isolate-free connected graphs G whose domination number γ ( G ) satisfies γ(G)=⌊n/2⌋ is obtained. This result enables us to obtain extremal graphs of inequalities which bound the sum of two domination parameters in isolate-free graphs.

Linear algorithms on recursive representations of trees

A recursive labeling of a tree T with M vertices is any assignment of the labels 1, 2,…, M to the vertices of T which has the property that every vertex, except the vertex labeled ‘1’ is adjacent to exactly one vertex with a smaller label. A corresponding recursive representation of T is the array C(2), C(3),…, C(M), where C(i) is the unique vertex adjacent to i having a smaller label. In this paper we discuss the feasibility, advantages and relative efficiency of using this representation to design algorithms on trees.