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A structure-induced bifurcation of nonsmooth nonlinear systems is studied and illustrated on electric power system models. The consequence of structure-induced bifurcation is an immediate instability induced by generator reactive power limits. It is numerically shown that structureinduced bifurcation can occur at both small power systems and large-scale power systems. Without taking the structure-induced bifurcation into account in defining power system operating limits, the resulting operating limits can be overly optimistic.

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We study the Independent Feedback Vertex Set problem -a variant of the classic Feedback Vertex Set problem where, given a graph G and an integer k, the problem is to decide whether there exists a vertex set S ⊆ V (G) such that G \ S is a forest and S is an independent set of size at most k. We present an O * ((1 + ϕ 2 ) k )-time FPT algorithm for this problem, where ϕ < 1.619 is the golden ratio, improving the previous fastest O * (4.1481 k )-time algorithm given by Agrawal et al. [2]. The exponential factor in our time complexity bound matches the fastest deterministic FPT algorithm for the classic Feedback Vertex Set problem.On the technical side, the main novelty is a refined measure of an input instance in a branching process, that allows for a simpler and more concise description and analysis of the algorithm. .pl. 1 The O * -notation suppresses factors that are polynomial in the input size. 2 Actually in the randomized FPT algorithm for FVS, the parameter is the treewidth of the graph. Since the treewidth of a yes-instance (G, k) to FVS is at most k + 1, the randomized algorithm for FVS runs in time O * (3 k ).

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Minimum vertex cover problem(Min-VC) on a graph is a NP-hard problem. The neighborhoods of a vertex are analyzed in this paper, and so are the information they hold for judging whether the vertex is belong to Min-VC or not. Then, the concept of Max-I share degree is put forword in order to qualify the possibility of a vertex to be collected into Min-VC. Based on Max-I share degree, a heuristic algorithm is proposed. Our algorithm has two remarkable characteristics: one is that it is able to get almost the same Min-VC as that of an exact algorithm, and the approximation degree is obviously better than that of other approximation algorithms; the other is that it has polynomial time complexity. Emulational results show that the running time is significantly less than those of&nbsp; other algorithms

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