No abstract

In this paper, we study the total outer-connected domination number of the middle graph of a simple graph and we obtain tight bounds for this number in terms of the order of the middle graph. We also compute the total outer-connected domination number of some families of graphs, explicitly. Moreover, some Nordhaus-Gaddum-like relations are presented for the total outer-connected domination number of middle graphs.

A k-tuple total dominating set (kTDS) of a graph G is a set S of vertices in which every vertex in G is adjacent to at least k vertices in S; the minimum size of a kTDS is denoted γ ×k,t (G). We give a Vizing-like inequality for Cartesian product graphs, namelywhere ρ is the packing number. We also give bounds on γ ×k,t (G H) in terms of (open) packing numbers, and consider the extremal case of γ ×k,t (Kn Km), i.e., the rook's graph, giving a constructive proof of a general formula for γ×2,t(Kn Km).

A total dominator coloring of a graph [Formula: see text] is a proper coloring of [Formula: see text] in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number of a graph is the minimum number of color classes in a total dominator coloring. In this paper, we study the total dominator coloring on middle graphs by giving several bounds for the case of general graphs and trees. Moreover, we calculate explicitly the total dominator chromatic number of the middle graph of several known families of graphs.

A k-tuple total dominating set (kTDS) of a graph G is a set S of vertices in which every vertex in G is adjacent to at least k vertices in S. The minimum size of a kTDS is called the ktuple total dominating number and it is denoted by γ ×k,t (G). We give a constructive proof of a general formula for γ ×3,t (K n K m ).Date: October 1, 2018.