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Graph Theory

Q1) This first question is about equivalence relations. The questions below have to do with the properties (reflectivity, symmetry, and transitivity) of an equivalence relation. Let be an equivalence relation on X. The equivalence class of an element z is the set (a) Explain which property ensures that every element a E X belongs to some equivalence class. (b) Explain which property ensures that if z є [1] then [z]. (c) Explain which properties imply that f e then x ] (d) Suppose there is some element w that lies in two equivalence classes, ie. [ ] Q2) For different classes of graphs, we need to use a different definition of isomorphism we [y). Explain/prove that Consider the class of simple digraphs. Two directed simple graphs G, H are iso- morphic if and only if there is a bijection f V(G)-V(H) such that there is a directed edge (u, v) E E(G) if and only if there is the corresponding directed edge (f(u),f(v)) E(H) Two of the digraphs below are isomorphic. Which two are they? Informally explain your reasoning.

Q4) A digraph G is said to be strongly connected if for any two vertices u, v there is a directed walk starting at u and ending at v. Given an undirected graph G an orientation on G, is a digraph obtained by orienting each edge of G so that it becomes strongly directed. (a) The streets of a city can be represented by the following graph (where vertices are intersections City council has decided that all streets should be one-way. Draw an orientation on this graph which will give an assignment of traffic direction that will permit everyhody to get from any origin to any destination. b) Draw a graph that is not orientable (ie. which doesnt admit an orientation.)

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