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.

In recent years, Multimetric Indices (MMIs) have received a lot of attention thanks to their ability to develop integrative evaluations of water quality, particularly in lagoons. In this article, we propose a new MMI for determining the water quality in lagoons. The proposed index is composed of biotic and abiotic indicators, in particular macroinvertebrates, macrophytes and morphological indicators. The proposed index is based on a geometric representation of a phenomenon associated with an ecological system, the ecosystem elements are mapped as vertices of a network and the relationship between them is represented by the corresponding edges. We classify the status of water bodies, from very low to very high using the ecological quality ratio. We compare our index with different different indices that measure water quality, such as General Biotic Index (JP(G)), Macrophyte Index for River (MIR) and Shannon diversity index (H’) and validate our index with Pearson’s correlation coefficient. A strong correlation with the JP(G) and MIR indices (R2 = 0.8605 and R2=0.7661, respectively) is obtained. Although the proposed index is composed of other indices, the independence of the proposed index with respect to its component indices is proven and the structure of the geometric model associated to the proposed network is studied. A close relationship between the measure called medium articulation and the geometric model associated with the proposed index is highlighted, which allows to determine the missing relationships in the network using structural analysis. The proposed index presents a more comprehensive measure than most indices currently used and has the advantage in the scalability, since other existing indicators can be integrated into our model.