In this paper, we continue the study of the total domination game in graphs introduced in [Graphs Combin. 31(5) (2015), 1453–1462], where the players Dominator and Staller alternately select vertices of G. Each vertex chosen must strictly increase the number of vertices totally dominated, where a vertex totally dominates another vertex if they are neighbors. This process eventually produces a t...

a set $s$ of vertices in a graph $g=(v,e)$ is called a total$k$-distance dominating set if every vertex in $v$ is withindistance $k$ of a vertex in $s$. a graph $g$ is total $k$-distancedomination-critical if $gamma_{t}^{k} (g - x) < gamma_{t}^{k}(g)$ for any vertex $xin v(g)$. in this paper,we investigate some results on total $k$-distance domination-critical of graphs.

For a graph G = (V, E), a set S ⊆ V (G) is a total dominating set if it is dominating and both 〈S〉 has no isolated vertices. The cardinality of a minimum total dominating set in G is the total domination number. A set S ⊆ V (G) is a total restrained dominating set if it is total dominating and 〈V (G) − S〉 has no isolated vertices. The cardinality of a minimum total restrained dominating set in ...

‎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...

given a graph $g$, the total dominator coloring problem seeks aproper coloring of $g$ with the additional property that everyvertex in the graph is adjacent to all vertices of a color class. weseek to minimize the number of color classes. we initiate to studythis problem on several classes of graphs, as well as findinggeneral bounds and characterizations. we also compare the totaldominator chro...