By Khuller S.

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**Sample text**

Jf j( mm; 1 )k = 1 Using the approximation log m ; log(m ; 1) = ( m1 ) we can obtain a bound on k. k m log jf j: This gives a bound on the number of iterations of the algorithm. Taking into a consideration that a path with the highest residual capacity can be picked up in time O(m + n logn) (HW 4), the overall time complexity of the algorithm is O((m + n logn)m log jf j). Tarjan's book gives a slightly di erent proof and obtains the same bound on the number of augmenting paths that are found by the algorithm.

There is an older algorithm due to Tutte that nds a straight line embedding of a triconnected planar graph (if one exists) by reducing the problem to a system of linear equations. 2 Kuratowski's characterization De nition: A subdivision of a graph H is obtained by adding nodes of degree two on edges of H. G contains H as a homeomorph if G contains a subdivision of H as a subgraph. The following theorem very precisely characterizes the class of planar graphs. Note that K5 is the complete graph on ve vertices.

The paths may be of odd or even length. t M. t M. t M). jM N j = jP P 0j = jP j + jP 0j ; 2jP \ P 0 j jP1j + jP2j 2jP j: Simplifying we get jP 0j jP j + 2jP \ P 0j: 2 We still need to argue that after each phase, the shortest augmenting path is strictly longer than the shortest augmenting paths of the previous phase. ) 36 Suppose that in some phase we augmented the current matching by a maximal set of vertex disjoint paths of length k (and k was the length of the shortest possible augmenting path).