New Bounds on the Double Total Domination Number of Graphs

Let G be a graph of minimum degree at least two. A set $$D\subseteq V(G)$$ D ⊆ V ( G ) is said to be a double total dominating set of G if $$|N(v)\cap D|\ge 2$$ | N ( v ) ∩ D | ≥ 2 for every vertex $$v\in V(G)$$ v ∈ V ( G ) . The minimum cardinality among all double total dominating sets of G is the double total domination number of G. In this article, we continue with the study of this parameter. In particular, we provide new bounds on the double total domination number in terms of other domination parameters. Some of our results are tight bounds that improve some well-known results.

Computing locating-total domination number in some rotationally symmetric graphs

Let [Formula: see text] be a connected graph. A locating-total dominating set in a graph G is a total dominating set S of a G, for every pair of vertices [Formula: see text], such that [Formula: see text]. The minimum cardinality of a locating-total dominating set is called locating-total domination number and represented as [Formula: see text]. In this paper, locating-total domination number is determined for some cycle-related graphs. Furthermore, some well-known graphs of convex polytopes from the literature are also considered for the locating-total domination number.

Correction: Cabrera Martínez et al. On the Secure Total Domination Number of Graphs. Symmetry 2019, 11, 1165

The authors wish to make the following corrections on paper [...]

On double edge-domination and total domination of trees

In a graph G, a vertex v is dominated by an edge e, if e is incident with v or e is incident with a vertex which is a neighbor of v. An edge-vertex dominating set D is a subset of the edge set of G such that every vertex of G is edge-vertex dominated by an edge of D. The ev-domination number equals to the number of an edge-vertex dominating set of G which has minimum cardinality and it is denoted by γev (G). We here analyze double edge-vertex domination such that a double edge-vertex dominating set D is a subset of the edge set of G, provided that all vertices in G are ev-dominated by at least two edges of D. The double ev-domination number equals to the number of an double edge-vertex dominating set of G which has minimum cardinality and it is denoted by γdev (G). We demonstrate that the enumeration of the double ev-domination number of chordal graphs is NP-complete. Moreover several results about total domination number and double ev-domination number are obtained for trees.