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...

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 γt-critical. In this paper, we disprove a conjecture posed in a recent paper(On an open problem concerning total domination critical graphs, Expo. Mat...

A set S of vertices in a graph G(V,E) is called a total dominating set if every vertex v ∈ V is adjacent to an element of S. A set S of vertices in a graph G(V,E) is called a total restrained dominating set if every vertex v ∈ V is adjacent to an element of S and every vertex of V − S is adjacent to a vertex in V − S. The total domination number of a graph G denoted by γt(G) is the minimum card...

A subset $S$ of vertices a digraph $D$ is double dominating set (total $2$-dominating set) if every vertex not in adjacent from at least two $S$, and one (the subdigraph induced by has no isolated vertices). The domination number $2$-domination number) the minimum cardinality $D$. In this work, we investigate these concepts which can be considered as extensions graphs to digraphs, along with $2...

Let $G=(V,E)$ be a finite and simple graph of order $n$ maximumdegree $\Delta$. A signed strong total Roman dominating function ona $G$ is $f:V(G)\rightarrow\{-1, 1,2,\ldots, \lceil\frac{\Delta}{2}\rceil+1\}$ satisfying the condition that (i) forevery vertex $v$ $G$, $f(N(v))=\sum_{u\in N(v)}f(u)\geq 1$, where$N(v)$ open neighborhood (ii) every forwhich $f(v)=-1$ adjacent to at least one vertex...