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Unformatted text preview: 6.13 Exercises 343 6.53. Let D r be the digraph which is defined in the end of Subsection 6.10.2. Show that every strong spanning subdigraph of D r has cyclomatic number at least 2 r 1. Next show that every cyclic spanning subdigraph of D r with cyclomatic number r is an rcycle factor in which all cycles are 4cycles. 6.54. Prove that if a path P in an extended semicomplete digraph D contains two vertices from an independent set I of D , then there exists a path P and a cycle C in D with V ( P ) = V ( P ) ∪ V ( C ). 6.55. First derive a direct O ( n 3 ) algorithm from the proof of Theorem 6.11.2. Then show ( (+) exercise) how to improve this to O ( n 2 . 5 ) starting from a pcc ( D ) pathcycle factor. 6.56. Show that the proof of Theorem 6.11.6 can be turned into an O ( n 4 ) algo rithm for finding a minimum strong spanning subdigraph of a quasitransitive digraph. 6.57. (+) Prove Lemma 6.11.5. Hint: consider the way we argued in the proof of Proposition 6.11.1. 6.58. (+) Prove Theorem 6.11.7. Hint: use the same approach as in the proof of Theorem 5.9.1. 6.59. (+) Prove Theorem 6.11.8. Hint: use the same approach as in the proof of Theorem 5.9.4. 7. Global Connectivity The concept of connectivity is one of the most fundamental concepts in (di rected) graph theory. There are numerous practical problems which can be formulated as connectivity problems for digraphs and hence a significant part of this theory is also important from a practical point of view. Results on con nectivity are often quite difficult and a deep insight may be required before one can obtain results in the area. The purpose of this chapter is to con vey some of that insight by illustrating several important topics as well as techniques that have been successful in solving global connectivity problems. Several of these problems, such as the connectivity augmentation problems in Sections 7.6 and 7.7, are of significant practical interest. Because of the very large number of important results on connectivity, we will devote this chapter as well as Chapters 8 and 9 to this area. This chapter will mainly deal with global connectivity aspects. That is, the directed multigraph in question is k(arc)strong for some k ≥ 0, or we want to make it k(arc)strong by adding new arcs. We will often consider directed multigraphs rather than directed graphs, since several results on arcstrong connectivity hold for this larger class and also it becomes easier to prove many results. However, when we consider vertexstrong connectivity, multiple arcs play no role and then we may as sume that we are considering digraphs. Note that, unless we explicitly say otherwise, we will assume that we are working with a directed graph (i.e there are no multiple arcs)....
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 Spring '11
 Algorithms

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