Suppose we have a connected graph G = (V, E), and a specific vertex u ? V . Suppose we compute a depth-first search tree rooted at u and obtain a tree T that includes all nodes of G. Suppose we then compute a breadth-first search tree rooted at u and obtain the same tree T. Prove that G = T. (In other words, if T is both a depth-first search tree and a breadth-first search tree rooted at u, then G cannot contain any edges that do not belong to T.) 2. [20 points] Suppose you and your friend Alanis live together with n – 2 other people at a popular cooperative apartment. Over the next n

View complete question »Suppose we have a connected graph G = (V, E), and a specific vertex u ? V . Suppose we compute a depth-first search tree rooted at u and obtain a tree T that includes all nodes of G. Suppose we then compute a breadth-first search tree rooted at u and obtain the same tree T. Prove that G = T. (In other words, if T is both a depth-first search tree and a breadth-first search tree rooted at u, then G cannot contain any edges that do not belong to T.) 2. [20 points] Suppose you and your friend Alanis live together with n – 2 other people at a popular cooperative apartment. Over the next n nights, each of you is supposed to cook dinner for the co-op exactly once, so some one cooks on each of the nights. To make things interesting, everyone has scheduling conflicts with some of the nights (e.g., exams, deadlines at work, basketball games, etc.), so deciding who should cook on which night becomes a tricky task. For concreteness, lets label the people {p1, …, pn} and the nights {d1, …, dn}. Then for person pi , associate a set of nights Si ? {d1, …, dn} when there are not available to cook. A feasible dinner schedule is defined to be an assignment of each person in the co-op to a different night such that each person cooks on exactly one night, there is there is someone to cook on each night, and if pi cooks on night dj , then dj 6? S

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