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In real-life scenarios, both the vertex weight and edge weight in a network are hard to define exactly. We can incorporate the fuzziness into a network to handle this type of uncertain situation. Here, we use triangular fuzzy number to describe the vertex weight and edge weight of a fuzzy network G. In this paper, we consider weighted k-domination problem in fuzzy network. The weighted k-domination (WKD) problem is to find a k dominating set D which minimizes the cost f (D) : = ∑u∈Dw (u) + ∑v∈V\D min {∑u∈Sw (uv) |S ⊆ N (v) ∩ D, |S| = k}. First, we put forward an integer linear programming model with a polynomial number of constrains for the WKD problem. If G is a cycle, we design a dynamic algorithm to determine its exact weighted 2-domination number. If G is a tree, we give a label algorithm to determine its exact weighted 2-domination number. Combining a primal-dual method and a greedy method, we put forward an approximation algorithm for general fuzzy network on the WKD problem. Finally, we describe an application of the WKD problem to police camp problem. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the fourth-order four-point boundary value problem where and . The upper and lower solution method and a new maximum principle are employed to establish existence results and we release the increasing condition imposed on . [Copyright &y& Elsevier]

Abstract: In this work, we apply a monotone iterative procedure to prove the existence of positive pseudo-symmetric solutions for a three-point second-order -Laplacian boundary value problem. [Copyright &y& Elsevier]

Abstract: In the paper, we obtain the existence of monotone positive solutions and establish acorresponding iterative scheme for the following multipoint boundary value problem with p-Laplacian operator, The main tool is the monotone iterative technique. [Copyright &y& Elsevier]

The independent domination number of a graph G, denoted by i(G), is the minimum cardinality of a maximal independent set of G. A maximal independent set of cardinality i(G) in G we call an i(G)-set. In this paper, we solve the following problem due to Rautenbach and Volkmann: characterization of the connected bipartite graphs G with i(G) = n/2, where n = |V(G)|. Furthermore, we provide a constructive characterization of tree T with i(T) = n/2, where n = |V(T)|. [Copyright &y& Elsevier]

According to acoustic measurement experiment,the refuge haven loose circle depth of 1.8-2.0 m and the combination support patterns to anchor rod, anchor cable and anchor net spray were determined. The total effective area of 136 m², effective length of 39 m, the effective width of 3.5 m, semicircle section width of 4.7 m, straight wall height of 1.3 m, invert depth of 0.85 m were determined, according to the maximum workers, personnel and equipment size, rock consolidating coefficient. The numerical simulation determined the haven floor force was larger than the roof and the side wall, verified that the combination support patterns could effectively reduce vertical and horizontal stress and displacement, affirmed the necessity of invert. Through the pressure calculation and structure safety analysis, the protective wall thickness of 1 m, protective structure consists of airtight buffer layer and concrete lining layer were determined. Through pressure and airtight experiments, the maximum positive pressure of 2000 Pa was measured and the air tightness was good. [ABSTRACT FROM AUTHOR]

A dominating set in a graph G is a connected dominating set of G if it induces a connected subgraph of G. The minimum number of vertices in a connected dominating set of G is called the connected domination number of G, and is denoted by γc(G). The purpose of this paper is to initiate an investigation of those graphs which are critical in the following sense: for each υ, μ ε V (G) with υ not adjacent to u, γc(G + υu) < γc(G). Thus, G is k-γc-critical if γc(G) = k and for each edge e ∉ E(G), γc(G + e) < k. First we discuss whether some particular classes of graphs are γc-critical. Then 2-γc-critical graphs are characterized. Finally, the properties of the 3-γc-critical graphs are studied. [Copyright &y& Elsevier]