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 st...

Let γt(G) and γ2(G) be the total domination number and the 2domination number of a graph G, respectively. It has been shown that: γt(T ) ≤ γ2(T ) for any tree T . In this paper, we provide a constructive characterization of those trees with equal total domination number and 2-domination number.

A total Roman dominating function on a graph $G$ is a function $f: V(G) rightarrow {0,1,2}$ such that for every vertex $vin V(G)$ with $f(v)=0$ there exists a vertex $uin V(G)$ adjacent to $v$ with $f(u)=2$, and the subgraph induced by the set ${xin V(G): f(x)geq 1}$ has no isolated vertices. The total Roman domination number of $G$, denoted $gamma_{tR}(G)$, is the minimum weight $omega(f)=sum_...

‎For any integer $kgeq 1$‎, ‎a set $S$ of vertices in a graph $G=(V,E)$ is a $k$-‎tuple total dominating set of $G$ if any vertex‎ ‎of $G$ is adjacent to at least $k$ vertices in $S$‎, ‎and any vertex‎ ‎of $V-S$ is adjacent to at least $k$ vertices in $V-S$‎. ‎The minimum number of vertices of such a set‎ ‎in $G$ we call the $k$-tuple total restrained domination number of $G$‎. ‎The maximum num...

A graph with no isolated vertices is edge critical with respect to total restrained domination if for any non-edge e of G, the total restrained domination number of G+ e is less than the total restrained domination number of G. We call these graphs γtr-edge critical. In this paper, we characterize all γtr-edge critical unicyclic graphs.

This paper consists of two loosely related notes on the domination number of graphs. In the first part, we provide a new upper bound for the domination number of d-regular graphs. Our bound is the best known for d ≥ 6. In the second part, we compute the exact domination number and total domination number of certain Kneser graphs, and we provide some bounds on the domination number of other Knes...

Let G = (V, E) be a graph. A set D ⊆ V is a total restrained dominating set of G if every vertex in V has a neighbor in D and every vertex in V −D has a neighbor in V −D. The cardinality of a minimum total restrained dominating set in G is the total restrained domination number of G. In this paper, we define the concept of total restrained domination edge critical graphs, find a lower bound for...

Abstract. In this paper we give tight upper bounds on the total domination number, the weakly connected domination number and the connected domination number of a graph in terms of order and Euler characteristic. We also present upper bounds for the restrained bondage number, the total restrained bondage number and the restricted edge connectivity of graphs in terms of the orientable/nonorienta...

A subset S of V is called a total dominating set if every vertex in V is adjacent to some vertex in S. The total domination number γt (G) of G is the minimum cardinality taken over all total dominating sets of G. A dominating set is called a connected dominating set if the induced subgraph 〈S〉 is connected. The connected domination number γc(G) of G is the minimum cardinality taken over all min...