The total domination number of a simple, undirected graph G is the minimum cardinality of a subset D of the vertices of G such that each vertex of G is adjacent to some vertex in D. In 2007 Graffiti.pc, a program that makes graph theoretical conjectures, was used to generate conjectures on the total domination number of connected graphs. More recently, the program was used to generate conjectur...

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.

This paper deals the graphs for which the removal of any edge changes the majority domination number of the graph. γM -critical edges, γM -redundant edges, γM -durable graphs and γM -critical graphs are studied. Further, majority domination critical edges and majority domination critical graphs are characterized.

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.

In this paper we determine the exact values of the signed domination number, signed total domination number, and minus domination number of complete multipartite graphs, which substantially generalizes some previous results obtained for special subclasses of complete multipartite graphs such as cliques and complete bipartite graphs.

In this paper, we introduce a fractional domination game arising from fractional domination problems on graphs and focus on its balancedness and concavity. We first characterize the core of the fractional domination game and show that its core is always non-empty taking use of dual theory of linear programming. Furthermore we study concavity of this game.

In this paper, sharp upper bounds for the domination number, total domination number and connected domination number for the Cayley graph G = Cay(D2n, Ω) constructed on the finite dihedral group D2n, and a specified generating set Ω of D2n. Further efficient dominating sets in G = Cay(D2n, Ω) are also obtained. More specifically, it is proved that some of the proper subgroups of D2n are efficie...

The fractional analogues of domination and packing in a graph form an interesting pair of dual linear programs in that the feasible vectors for both LPs have interpretations as functions from the vertices of the graph to the unit interval; efficient (fractional) domination is accomplished when a function simultaneously solves both LPs. We investigate some structural properties of the functions ...