Matheus - MO412

Evaluate the following affirmations regarding planarity and planarity testing algorithms: I. A multi-graph is considered planar when it admits a plane drawing . II. To postulate Euler's formula ( $N-L+F=2$ ), one can start with a single node and observe        the  effect on $N$ , $L$ , and $F$ when adding a new node connected to an existing node, or when    adding a link between two existing nodes . III.  Kuratowski's Theorem states that a graph is planar if and only if it contains a subdivision (the process of replacing a link by a path) of $K_{3,3}$ or of $K_{5}$ , effectively making them mandatory subgraphs for planarity . IV. The pseudo-code for the DMP algorithm begins by starting with a single node and iteratively selects a fragment $B$ with the maximum number of faces $F(B)$ that contain all its vertices of attachment . V.  When using the DMP algorithm, a fragment of a graph $G$ with respect to a subgraph $H$ can be an...

In the heart of a bustling kitchen, Chef Leo is famous for his "Variable Soufflé." The height of the soufflé, $H$ , depends on two main factors: the temperature of the oven, $x$ , and the amount of whisking time, $y$ . Leo has discovered a mathematical model for his perfect bake: $$H(x, y) = x^2y + 3xy^2 + 5x - 2y$$ One morning, while the oven is preheating, Leo needs to know exactly how the height of the soufflé changes if he increases the temperature ( $x$ ) while keeping the whisking time ( $y$ ) perfectly constant. He needs to calculate the partial derivative of $H$ with respect to $x$ at the point where $x = 2$ and $y = 3$ . Given the function $H(x, y) = x^2y + 3xy^2 + 5x - 2y$ , what is the value of the partial derivative $\frac{\partial H}{\partial x}$ at the point $(2, 3)$? a) 37 b) 44 c) 39 d) 41 e) None of the above. Original idea by: Matheus de Oliveira Saldanha