In this paper, we introduce and study a new coloring problem of graphs called the double total dominator coloring. A double total dominator coloring of a graph [Formula: see text] with minimum degree at least 2 is a proper vertex coloring of [Formula: see text] such that each vertex has to dominate at least two color classes. The minimum number of colors among all double total dominator coloring of [Formula: see text] is called the double total dominator chromatic number, denoted by [Formula: see text]. Therefore, we establish the close relationship between the double total dominator chromatic number [Formula: see text] and the double total domination number [Formula: see text]. We prove the NP-completeness of the problem. We also examine the effects on [Formula: see text] when [Formula: see text] is modified by some operations. Finally, we discuss the [Formula: see text] number of square of trees by giving some bounds.

In this paper, we introduce and study a new coloring problem of graphs called the double total dominator coloring. A double total dominator coloring of a graph G with minimum degree at least 2 is a proper vertex coloring of G such that each vertex has to dominate at least two color classes. The minimum number of colors among all double total dominator coloring of G is called the double total dominator chromatic number, denoted by χ t dd (G). Therefore, we establish the close relationship between the double total dominator chromatic number χ t dd (G) and the double total domination number γ ×2,t (G). We prove the NP-completeness of the problem. We also examine the effects on χ t dd (G) when G is modified by some operations. Finally, we discuss the χ t dd (G) number of square of trees by giving some bounds.

In a graph [Formula: see text], a module is a vertex subset [Formula: see text] of [Formula: see text] such that every vertex outside [Formula: see text] is adjacent to all or none of [Formula: see text]. For example, [Formula: see text], [Formula: see text][Formula: see text] and [Formula: see text] are modules of [Formula: see text], called trivial modules. A graph, all the modules of which are trivial, is prime; otherwise, it is decomposable. A vertex [Formula: see text] of a prime graph [Formula: see text] is critical if [Formula: see text] is decomposable. Moreover, a prime graph with [Formula: see text] noncritical vertices is called [Formula: see text]-critical graph. A prime graph [Formula: see text] is [Formula: see text]-minimal if there is some [Formula: see text]-vertex set [Formula: see text] of vertices such that there is no proper induced subgraph of [Formula: see text] containing [Formula: see text] is prime. From this perspective, Boudabbous proposes to find the [Formula: see text]-critical graphs and [Formula: see text]-minimal graphs for some integer [Formula: see text] even in a particular case of graphs. This research paper attempts to answer Boudabbous’s question. First, we describe the [Formula: see text]-critical tree. As a corollary, we determine the number of nonisomorphic [Formula: see text]-critical tree with [Formula: see text] vertices where [Formula: see text]. Second, we provide a complete characterization of the [Formula: see text]-minimal tree. As a corollary, we determine the number of nonisomorphic [Formula: see text]-minimal tree with [Formula: see text] vertices where [Formula: see text].