Discrete Mathematics First Exam (Alternative evaluation). April 6, 2015 There are 5 problems in this exam. Problems 1 and 2 are shorter. Problem 5 is about implementing a function in SAGE. The Sage Worksheet Exam1.sws shows examples of use of the main commands of Sage for graphs, some other useful commands in Sage, an implementation of the Depth–First Search algorithm, and the data of the Problems 4 and 5. For problems 1 to 4, you are allowed to use Sage as a advanced calculator and for checking your answers. But your answers should be justified by mathematical reasonings and your knowledge of graph theory (i.e. do not just use built-in functions of Sage). Problem 1. For each of the following sequences, indicate whether or not it is the degree sequence of a (simple) graph. If not, explain clearly why. Else, describe a graph with this degree sequence. 1. (3, 3, 2, 1, 1, 1). 2. (5, 3, 3, 2, 1, 0). 3. (3, 3, 2, 2, 2). 4. (5, 5, 4, 2, 2, 2). Problem 2. For any graph G, let δ( G ) be the minimum degree for the vertices of G, κ ( G ) be the vertex connectivity of G and λ( G ) be the edge connectivity of G. 1. Prove that λ( G ) ≤ δ( G ). 2. Prove that κ ( G ) ≤ λ( G ). Problem 3. 1. List all connected graphs with 4 vertices, up to isomorphism. How many graphs do you obtain? 2. For each of them, indicate the vertex–connectivity and the edge– connectivity. Justify your answer, recall the definition or characterization that you use of connectivity. 3. List all digraphs with 3 vertices, up to isomorphism (no multiple arcs are allowed, nor loops x → x, but beware that there might be two arcs between vertices x and y: x → y and y → x). Use convenient invariants to show clearly that your list does not contain any pair of isomorphic digraphs. How many digraphs do you obtain? Problem 4. Figure 1 shows the flights offered by some airline company. Table 1 shows the distances of the flights, in kilometers. 1. Give, for each city of the map, the minimum number of flights necessary to reach it from Madrid. 2. List all the articulation points of the graph depicted in Figure 1.
Matemática Discreta. Grados en Ingeniería Infiormática. ETSII. Universidad de Sevilla 2014–2015. Emmanuel Briand
discrete mathematics first exam (alternative evaluation).
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3. List the cities of the map, in increasing order of distance (in km) from Madrid. Give also the distances. You should indicate clearly the method used and make explicit all steps. You can use a calculator or the computer for numerical computations, or to check your results, but do not use advanced built-in functions of Sage. Problem 5. 1. Implement your own function distance(G, v0) that takes as an input a graph with vertex set some set of the type {0, . . . , N }, and returns the list [d0, d1, ..., dN] where di is the distance of vertex i to v0 , in number of edges (no weights here). 2. Test your function on the graph of Problem 4. Remarks: • For this problem, do not hesitate to ask for help if you face a syntax error you cannot solve, or if you need to know a particular command. • If you find that you cannot give an implementation of the algorithm, give a description of your algorithm in pseudo–code instead.
Documents
Bangkok–Beijing Bangkok–New Dehli Bangkok–Sidney Beijing –Moscow Beijing –New Dehli Beijing –Tokyo Berlin –London Berlin –Madrid Berlin –Moscow Berlin –New Dehli Brasilia –Buenos Aires Brasilia –Lima Brasilia –Pretoria Brasilia –Washington DC London –Madrid London –Ottawa London –Washington DC Madrid –New Dehli Madrid –Pretoria Ottawa –Washington DC
3282 2908 7540 5807 3784 2103 929 1866 1619 5796 2314 3154 7906 6764 1261 5379 5915 7288 8033 734
Table 1: Distances between some of the world’s major cities (km). From Algorithmic Graph Theory, D. Joyner, M. Van Nguyen, N. Cohen, Version 0.7-r1901, 2011, http://code.google.com/p/
graph-theory-algorithms-book/
discrete mathematics first exam (alternative evaluation).
2.8. Problems
93 Figure 1: Some of the world’s major cities. From Algorithmic Graph Theory, D. Joyner, M. Van Nguyen, N. Cohen, Version 0.7-r1901, 2011, http:
Lima
Washington DC
Ottawa
Figure 2.15: Major capital cities of the world.
Buenos Aires
Brasilia
Madrid
London
Berlin
Pretoria
Moscow
New Delhi
Bangkok
Beijing
Tokyo
Sydney
//code.google.com/p/graph-theory-algorithms-book/
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