Douglas F. Rall

Associative graph products and their independence, domination and coloring numbers

Associative products are defined using a scheme of Imrich & Izbicki [18]. These include the Cartesian, categorical, strong and lexicographic products, as well as others. We examine which product ⊗ and parameter p pairs are multiplicative, that is, p(G ⊗H) ≥ p(G)p(H) for all graphs G and H or p(G⊗H) ≤ p(G)p(H) for all graphs G and H. The parameters are related to independence, domination and irredundance. This includes Vizing’s conjecture directly, and indirectly the Shannon capacity of a graph and Hedetniemi’s coloring conjecture.

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.

Total domination in categorical products of graphs

Several of the best known problems and conjectures in graph theory arise in studying the behavior of a graphical invariant on a graph product. Examples of this are Vizing’s conjecture, Hedetniemi’s conjecture and the calculation of the Shannon capacity of graphs, where the invariants are the domination number, the chromatic number and the independence number on the Cartesian, categorical and strong product, respectively. In this paper we begin an investigation of the total domination number on the categorical product of graphs. In particular, we show that the total domination number of the categorical product of a nontrivial tree and any graph without isolated vertices is equal to the product of their total domination numbers. In the process we establish a packing and covering equality for trees analogous to the well-known result of Meir and Moon. Specifically, we prove equality between the total domination number and the open packing number of any tree of order at least two.

On the Total Domination Number of Cartesian Products of Graphs

Abstract.The most famous open problem involving domination in graphs is Vizing’s conjecture which states the domination number of the Cartesian product of any two graphs is at least as large as the product of their domination numbers. In this paper, we investigate a similar problem for total domination. In particular, we prove that the product of the total domination numbers of any nontrivial tree and any graph without isolated vertices is at most twice the total domination number of their Cartesian product, and we characterize the extremal graphs.

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.

On the upper total domination number of Cartesian products of graphs

Abstract In this paper we continue the investigation of total domination in Cartesian products of graphs first studied in (Henning, M.A., Rall, D.F. in Graphs Comb. 21:63–69, 2005). A set S of vertices in a graph G is a total dominating set of G if every vertex in G is adjacent to some vertex in S. The maximum cardinality of a minimal total dominating set of G is the upper total domination number of G, denoted by Γt(G). We prove that the product of the upper total domination numbers of any graphs G and H without isolated vertices is at most twice the upper total domination number of their Cartesian product; that is, Γt(G)Γt(H)≤2Γt(G □ H).

The 4/5 upper bound on the game total domination number

The recently introduced total domination game is studied. This game is played on a graph G by two players, named Dominator and Staller. They alternately take turns choosing vertices of G such that each chosen vertex totally dominates at least one vertex not totally dominated by the vertices previously chosen. Dominator’s goal is to totally dominate the graph as fast as possible, and Staller wishes to delay the process as much as possible. The game total domination number, γtg(G), of G is the number of vertices chosen when Dominator starts the game and both players play optimally. The Staller-start game total domination number, γ′tg(G), of G is the number of vertices chosen when Staller starts the game and both players play optimally. In this paper it is proved that if G is a graph on n vertices in which every component contains at least three vertices, then γtg(G)≤4n/5 and γ′tg(G)≤(4n+2)/5. As a consequence of this result, we obtain upper bounds for both games played on any graph that has no isolated vertices.

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.

The Ultimate Categorical Independence Ratio of a Graph

Let