In this paper, we continue the study of the domination game in graphs introduced by Brešar, Klavžar, and Rall [SIAM J. Discrete Math. 24 (2010) 979–991]. We study the total version of the domination game and show that these two versions differ significantly. We present a key lemma, known as the Total Continuation Principle, to compare the Dominator-start total domination game and the Staller-st...

Let $G = (V, E)$ be a simple graph of order $n$. The total dominating set is a subset $D$ of $V$ that every vertex of $V$ is adjacent to some vertices of $D$. The total domination number of $G$ is equal to minimum cardinality of total dominating set in $G$ and denoted by $gamma_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=sum d_t(G,i)$, where $d_t(G,i)$ is the numbe...

In this paper, we present new upper bounds for the global domination and Roman domination numbers and also prove that these results are asymptotically best possible. Moreover, we give upper bounds for the restrained domination and total restrained domination numbers for large classes of graphs, and show that, for almost all graphs, the restrained domination number is equal to the domination num...

In this article we give a new definition of direct product of two arbitrary fuzzy graphs. We define the concepts of domination and total domination in this new product graph. We obtain an upper bound for the total domination number of the product fuzzy graph. Further we define the concept of total α-domination number and derive a lower bound for the total domination number of the product fuzzy ...

Let / be an integer-valued function defined on the vertex set V(G) of a graph G. A subset D of V(G) is an /-dominating set if each vertex x outside D is adjacent to at least f(x) vertices in D. The minimum number of vertices in an /-dominating set is denned to be the /-domination number, denoted by 7/(G). In a similar way one can define the connected and total /-domination numbers 7 C| /(G) and...

A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G− v is less than the total domination number of G. We call these graphs total domination critical or just γt-critical. If such a graph G has total domination number k, we call it k-γt-critical. We study an open problem of ...