Quiz - MO412 - Luis Vásquez

A mining site consists of \(n\) Extraction Mines and \(m\) Refining Plants (where \(n, m \geq 1\)). For logistical efficiency, every mine must be connected to every plant by a direct road. In mathematics, we represent this as a complete bipartite graph , denoted by \(K_{n, m}\). In this bipartite structure, the nodes are divided into two distinct sets: a set of \(n\) nodes representing the mines and a set of \(m\) nodes representing the plants. Connections only exist between nodes of different sets—never between two mines or two plants. To prevent traffic collisions, these roads are built on a single flat plain and must be planar (meaning no two roads are allowed to cross). Problem Statement Consider a "Feasibility Matrix" \(M\) for all \(n, m \geq 1\), where an entry \(M_{n,m} = 1\) if a planar layout for the graph \(K_{n, m}\) is possible, and \(0\) otherwise. Which of the following statements is true? a) \(M_{n, m} = 0\)...

As a resistance leader, you have obtained a schematic map of the dictatorship's communication cables. Because the government wants to control every piece of news, the signal is one-way only , flowing from the capital out to the military. To launch your attack, you must completely cut all connection between the Capital (s) and the Military Base (t) . Each one-way cable is guarded. The cost to destroy a cable (in C4 charges and personnel) is equal to its capacity. You have a total budget of B = 10 . s a b t 10 8 2 3 ...