Bert L. Hartnell

Bounds on the bondage number of a graph

Abstract The bondage number b ( G ) of a graph G is the minimum cardinality of a set of edges of G whose removal from G results in a graph with domination number larger than that of G . Several new sharp upper bounds for b ( G ) are established. In addition, we present an infinite class of graphs each of whose bondage number is greater than its maximum degree plus one, thus showing a previously conjectured upper bound to be incorrect.

On dominating the Cartesian product of a graph and K

In this paper we consider the Cartesian product of an arbitrary graph and a complete graph of order two. Although an upper and lower bound for the domination number of this product follow easily from known results, we are interested in the graphs that actually attain these bounds. In each case, we provide an infinite class of graphs to show that the bound is sharp. The graphs that achieve the lower bound are of particular interest given the special nature of their dominating sets and are investigated further.

A characterization of well-covered graphs that contain neither 4- nor 5-cycles

A graph is well covered if every maximal independent set has the same cardinality. A vertex x, in a well-covered graph G, is called extendable if G – {x} is well covered and β(G) = β(G – {x}). If G is a connected, well-covered graph containing no 4- nor 5-cycles as subgraphs and G contains an extendable vertex, then G is the disjoint union of edges and triangles together with a restricted set of edges joining extendable vertices. There are only 3 other connected, well-covered graphs of this type that do not contain an extendable vertex. Moreover, all these graphs can be recognized in polynomial time.

Graphs whose vertex independence number is unaffected by single edge addition or deletion

Abstract A graph is β + -stable (β − -stable) if its vertex independence number remains the same upon the addition (deletion) of any edge. We give a constructive characterization of β + -stable and β − -stable trees.

Limited packings in graphs

We define a k-limited packing in a graph, which generalizes a 2-packing in a graph, and give several bounds on the size of a k-limited packing. One such bound involves the domination number of the graph, and here we show all trees attaining the bound can be built via a simple sequence of operations. We consider graphs where every maximal 2-limited packing is a maximum 2-limited packing, and characterize their structure in a number of cases.

Connected domatic number in planar graphs

A dominating set in a graph G is a connected dominating set of G if it induces a connected subgraph of G. The connected domatic number of G is the maximum number of pairwise disjoint, connected dominating sets in V(G). We establish a sharp lower bound on the number of edges in a connected graph with a given order and given connected domatic number. We also show that a planar graph has connected domatic number at most 4 and give a characterization of planar graphs having connected domatic number 3.

A bound on the size of a graph with given order and bondage number

Abstract The domination number of a graph is the minimum number of vertices in a set S such that every vertex of the graph is either in S or adjacent to a member of S. The bondage number of a graph G is the cardinality of a smallest set of edges whose removal results in a graph with domination number greater than that of G. We prove that a graph with p vertices and bondage number b has at least p(b + 1)/4 edges, and for each b there is at least one p for which this bound is sharp.

On 4-connected claw-free well-covered graphs

Abstract Recognition of well-covered graphs is co-NP-complete, even when the graphs in question are k 1,4 -free. On the other hand, very recently, an algorithm has been found which recognizes k 1,3 -free well-covered graphs in polynomial time. Claw-free well-covered graphs having no 4-cycles have recently been characterized. In the present paper, we determine two different classes of claw-free well-covered graphs — those which are 4-connected and 4-regular as well as those which are 4-connected and planar.

Vizing's conjecture and the one-half argument

The domination number of a graph G is the smallest order, γ(G), of a dominating set for G. A conjecture of V. G. Vizing [5] states that for every pair of graphs G and H, γ(G H) ≥ γ(G)γ(H), where G H denotes the Cartesian product of G and H. We show that if the vertex set of G can be partitioned in a certain way then the above inequality holds for every graph H. The class of graphs G which have this type of partitioning includes those whose 2-packing number is no smaller than γ(G) − 1 as well as the collection of graphs considered by Barcalkin and German in [1]. A crucial part of the proof depends on the well-known fact that the domination number of any connected graph of order at least two is no more than half its order.

On designing a network to defend against random attacks of radius two

This paper considers the following variation on the construction of a reliable communication network. Whenever a vertex is attacked, all vertices within distance 2 are also destroyed (or fail) indirectly. We are interested in designing a connected graph (undirected, all edges of length one) on p vertices such that when a random subset of the vertices are attacked the expected number of vertices that are destroyed (directly and indirectly) is minimized. It is assumed that any of the 2p subsets of vertices is equally likely to be attacked. The optimal structure is determined for all p and is shown to be one of five patterns depending on r where p = 5t + r.