No abstract
Recently in several papers, graphs with maximum neighborhood orderings were characterized and turned out to be algorithmically useful. This paper gives a unified framework for characterizations of those graphs in terms of neighborhood and clique hypergraphs which have the Helly property and whose line graph is chordal. These graphs are dual (in the sense of hypergraphs) to chordal graphs. By using the hypergraph approach in a systematical way new results are obtained, some of the old results are generalized, and some of the proofs are simplified. Introduction.The class of chordal graphs is a by now classical and wellunderstood graph class which is algorithmically useful and has several interesting characterizations. In the theory of relational database schemes there are close relationships between desirable properties of database schemes, acyclicity of corresponding hypergraphs, and chordality of graphs which corresponds to tree and Helly properties of hypergraphs [2], [5], [25]. Chordal graphs arise also in solving large sparse systems of linear equations [28], [36] and in facility location theory [13]. Recently a new class of graphs was introduced and characterized in [20], [6], [21], [39] which is defined by the existence of a maximum neighborhood ordering. These graphs appeared first in [20] and [16] under the name HT -graphs but only a few results have been published in [21]. [34] also introduces maximum neighborhoods but only in connection with chordal graphs (chordal graphs with maximum neighborhood ordering were called there doubly chordal graphs).It is our intention here to attempt to provide a unified framework for characterizations of those graph classes in terms of neighborhood and clique hypergraphs. These graphs are dual (in the sense of hypergraphs) to chordal graphs (this is why we call them dually chordal) but have very different properties-thus they are in general not perfect and not closed under taking induced subgraphs. By using the hypergraph approach in a systematical way new results are obtained, a part of the previous results are generalized, and some of the proofs are simplified. The present paper improves the results of the unpublished manuscripts [20] and [6].Graphs with maximum neighborhood orderings (alias dually chordal graphs) are a generalization of strongly chordal graphs (a well-known subclass of chordal graphs * 2. Standard hypergraph notions and properties. We mainly use the hypergraph terminology of Berge [7]. A finite hypergraph E is a family of nonempty subsets (the edges of E) from some finite underlying set V (the vertices of E). The subhypergraph induced by a set A ⊆ V is the hypergraph E A defined on A by the edge set E A = {e ∩ A : e ∈ E}. The dual hypergraph E * has E as its vertex set and {e ∈ E : v ∈ e} (v ∈ V ) as its edges. The 2-section graph 2SEC(E) of the hypergraph E has vertex set V , and two distinct vertices are adjacent if and only if they are contained in a common edge of E.
Transmission of Plasmodium falciparum malaria parasites occurs when nocturnal Anopheles mosquito vectors feed on human blood. In Africa, where malaria burden is greatest, bednets treated with pyrethroid insecticide were highly effective in preventing mosquito bites and reducing transmission, and essential to achieving unprecedented reductions in malaria until 2015 1. Since then, progress has stalled 2 and with insecticidal bednets losing efficacy against pyrethroid-resistant Anopheles vectors 3,4 , methods that restore performance are urgently needed to eliminate any risk of malaria returning to the levels seen prior to their widespread use throughout sub-Saharan Africa 5. Here we show that the primary malaria vector Anopheles gambiae is targeted and killed by small insecticidal net barriers positioned above a standard bednet, in a spatial region of high mosquito activity but zero contact with sleepers, opening the way for deploying many more insecticides on bednets than currently possible. Tested against wild pyrethroid-resistant Anopheles gambiae in Burkina Faso, pyrethroid bednets with organophosphate barriers achieved significantly higher killing rates than bednets alone. Treated barriers on untreated bednets were equally effective, without significant loss of personal protection. Mathematical modelling of transmission dynamics predicted reductions in clinical malaria incidence with barrier bednets that matched those of 'next-generation' nets recommended by WHO against resistant vectors. Mathematical
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