NP [LPS]

Given a graph G, a set S subset of V (G) is Delta-convex if there is no vertex u is an element of V (G) \ S forming a triangle with two vertices of S. The Delta-convex hull of S is the minimum Delta-convex set containing S. The Delta-hull number of a graph G is the cardinality of a minimum set such that its Delta-convex hull is V (G). We show that the problem of deciding whether the Delta-hull number of a general graph is at most k is an NP-complete problem and present polynomial-time algorithms for computing the Delta-hull number of some graph classes including chordal graphs, dually chordal graphs, and cographs. (C) 2020 Elsevier B.V. All rights reserved.

Significant grain refinement of AZ91 alloys was achieved by ceramic nanoparticle(NP)-induced growth control. Advanced characterization techniques including 3D tomography and in-situ observation were carried out to examine the microstructure evolution. The results show that the NPs are distributed in matrix in three ways, i.e. along the grain boundaries, inside the alpha-Mg grains and within the interior of 13 phases. The intergranular NPs that are distributed along the grain boundaries can form a multilayer of NPs covering the surface of alpha-Mg, greatly restricting the grain growth. A numerical model is proposed to predict the grain size, in which the solute concentration gradient at dendrite tip and the impingement of grains are recognized as two pivotal factors in modelling. It is demonstrated from the model prediction that a reduced growing rate of alpha-Mg can be obtained upon NP addition due to the lowered solute concentration gradient at dendrite tip, which could increase the nucleation events, accelerate the grain impingement and thus induce significant grain refinement. Given the chemically and thermally stability of ceramic NPs, which can avoid the intrinsic drawbacks of conventional inoculation, e.g. element poisoning, NP-induced growth control (NIGC) may provide a new avenue for grain refinement in magnesium alloys. (c) 2020 The Author(s). Published by Elsevier Ltd.