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Jordan's classic theorem states that the center of every tree (the set of minimum eccentricity vertices) forms a complete subgraph. This property, which we refer to as the “Jordan property,” has been established for various definitions of eccentricity, the most popular being the maximum and average distances of a vertex to the others. In this note, we consider unimodal eccentricity functions, such...
We show that the feasibility of a booking in the European entry‐exit gas market can be decided in polynomial time on single‐cycle networks that are passive, i.e., do not contain controllable elements. The feasibility of a booking can be characterized by solving polynomially many nonlinear potential‐based flow models for computing so‐called potential‐difference maximizing load flow scenarios. We thus...
A distinctive feature of software‐defined networking (SDN) is a logically centralized control plane realized using multiple physical controllers. The placement of the controllers, the so‐called controller placement problem (CPP), is a crucial design issue. It influences network performance parameters such as latency, flow setup time, network availability, load balance of the controllers, and energy...
This article presents a novel shortest path algorithm for connected networks with nonnegative edge weights. The worst case running time of the single source shortest path version of the algorithm is O(max(m, )) where m is the number of edges of the input network and is the normalized eccentricity of the source vertex vs. The pseudo‐polynomial nature of the time complexity is overcome with...
Consider a graph with perfect nodes but independent edge failures with identical probability ρ. The reliability is the connectedness probability of the random graph. A graph with n nodes and e edges is uniformly optimally reliable (UOR) if it has the greatest reliability among all graphs with the same number of nodes and edges, for all values of ρ. In 1997, Gross and Saccoman proved that the simple...
Random graph models are a recurring tool‐of‐the‐trade for studying network structural properties and benchmarking community detection and other network algorithms. Moreover, they serve as test‐bed generators for studying diffusion and routing processes on networks. In this paper, we illustrate how to generate large random graphs having a power‐law degree distribution using the Chung–Lu model. In particular,...
Assume that the vertices of a graph G are always operational, but the edges of G are operational independently with probability p ∈ [0, 1]. For fixed vertices s and t, the two‐terminal reliability of G is the probability that the operational subgraph contains an (s, t)‐path, while the all‐terminal reliability of G is the probability that the operational subgraph contains a spanning tree. Both reliabilities...
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