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Question about Robustness

Consider a large scale-free network with average degree ⟨k⟩ = 5. According to robustness concepts, which of the following options best describes the critical fraction of nodes that would need to be removed to cause the collapse of the giant component (i) in the case of random failures and (ii) in the case of targeted attack, respectively? a) The network is equally robust to random failures and to attacks on hubs: in both cases, it is necessary to remove about 50% of the nodes to collapse the giant component. b) Under random failures, the network collapses after removing only about 15% of the nodes; in contrast, a targeted attack requires removing about 80% of the nodes (including the hubs) to fragment the giant component. c) In random failures, it is necessary to remove about 80% of the nodes to eliminate the giant component; in targeted attacks (removing hubs), removing about 15% of the nodes is enough to drastically fragment the network. d) There is no critical threshold for random f...
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Question about planarity

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The Unicamp Campus administration plans to create paths connecting all restaurants to optimize pedestrian mobility. Due to environmental restrictions, it's not allowed to build tunnels, and no path can cross another. Considering the Unicamp's map below, where the red dots represent the restaurants, analyze the statements: a) The project is feasible, since it is possible to create a path that connects all the 8 restaurants without crossing. b) The project would only be feasible if two of the restaurants were not directly connected to the others. c) The project would be feasible if the restaurants were divided into 2 groups of 4, and only one path would connect both groups. d) There is no way that the project would be feasible. e) None of the above. Original idea by: Ingrid Barbosa
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Question about Network Flow + Calculus

A research data center needs to transfer data to a remote backup facility during a 4‑hour maintenance window. The data transfer process is modeled as a directed network, where nodes represent logical components (functional stages) of the data transfer pipeline, and edges represent logical communication channels, with limited capacity. In this model, node A represents the point where data is generated in the primary data center. The intermediate nodes (B, C, D) represent functional stages of the transfer process, such as internal processing, aggregation, or interfaces to external networks. And the node E represents the logical destination where data is finally stored at the backup site. All edges have constant transmission capacity, except for one critical link, whose capacity varies over time due to shared usage with other services. The network is described as follows, with capacities measured in GB/hour: \( A→B: Capacity = 8 \) \( B→C: Capacity = 4 + 2sin(\frac{πt}{4}) \) \( C...
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Question about Kosaraju-Sharir Algorithm

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A delivery driver is working in a neighborhood where the streets are one-way. He serves one restaurant and has received 6 deliveries from different customers in that neighborhood. The graph below represents the routes, respecting the directions of the streets.  The driver must be able to return to the restaurant after serving customers to receive his payment, without violating the street directions. After picking up the orders at the restaurant, and using the Kosaraju–Sharir algorithm, which of the following alternatives is correct?  (a) All customers would receive their orders. (b) Only customers C1 and C3 would receive their orders. (c) Customers C4, C5 and C6 would not receive their orders. (d) Only customer C1 and C2 would receive their order. (e) None of the alternatives. Original idea by Ingrid Barbosa
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Question about DFS - Ingrid Barbosa - 13/03/2026 - MO412

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Consider the DFS time diagram below and analyze which of the graphs shown in the alternatives could correctly represent it.  a)                                                                      b)               c)                                                                       d)              e) None of the above Original idea by: Ingrid Barbosa   
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Question about Graph Theory - Ingrid Barbosa - 06/03/2026 - MO412

Consider an undirected network, where the node degree vector is \(K = [3, 2, 2, 3, 2, 2]\), and determine if the statements are True or False. I. The maximum possible number of links in this network is 30. II. The average degree of the network is 2.33. III. This network has 6 links. IV. It is possible that the diameter of the network is 3. V. The degree distribution is \( P(2) =  \frac{4}{7} \) , and  \( P(3) =  \frac{2}{7} \). Which of the statements are correct? A) I and II B) II, III and V C) II and IV D) II only E) None of the above. Original idea by: Ingrid Barbosa
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