Randomly removing g handles at once

AbstractIndyk and Sidiropoulos (2007) proved that any orientable graph of genus g can be probabilistically embedded into a graph of genus g−1 with constant distortion. Viewing a graph of genus g as embedded on the surface of a sphere with g handles attached, Indyk and Sidiropoulos' method gives an embedding into a distribution over planar graphs with distortion 2O(g), by iteratively removing the handles. By removing all g handles at once, we present a probabilistic embedding with distortion O(g2) for both orientable and non-orientable graphs. Our result is obtained by showing that the minimum-cut graph of Erickson and Har-Peled (2004) has low dilation, and then randomly cutting this graph out of the surface using the Peeling Lemma of Lee and Sidiropoulos (2009)

Probabilistic embeddings of bounded genus graphs into planar graphs

Ì��ÓÒ×Ø�ÒØC�×�ÐÐ��Ø����×ØÓÖØ�ÓÒÓ�Ø���Ñ�����Ò � ÄÓÛ��×ØÓÖØ�ÓÒÔÖÓ����Ð�×Ø��Ñ�����Ò�×�Ò��Ð�Ö��Ù�Ò��Ð �ÓÖ�Ø�Ñ�ÔÖÓ�Ð�Ñ×ÓÚ�Ö���Ö���Ù�×ØÑ�ØÖ�×�ÒØÓ���×Ý� i�×ÒÓØ�Ö��Ø�ÖØ��Ò×ÓÑ�ÓÒ×Ø�ÒØCØ�Ñ� × �Ð�××��Ð�Ø�Ö�ÒØ��××�Ø�ÓÒ�ÓÖ��ÓÖÑ�Ð���Ò�Ø�ÓÒ�ÓÖ �Ò�ØÙÖ�Ð��Ò�Ö�Ð�Þ�Ø�ÓÒÓ�ÔÐ�Ò�Ö�ØÝ�Ò��Ó�Ø��Ò��Ù× �Ü�ÑÔÐ��ÔÐ�Ò�Ö�Ö�Ô���×��ÒÙ×0�Ò���Ö�Ô�Ø��Ø�Ò �Ò�Ø��ÒÓØ�ÓÒÓ�Ø����ÒÙ×Ó���Ö�Ô�ÁÒ�ÓÖÑ�ÐÐÝ��Ö�Ô � ��×��ÒÙ×g�ÓÖ×ÓÑ�g≥0���Ø�Ò���Ö�ÛÒÛ�Ø�ÓÙØ�ÒÝ ÖÓ××�Ò�×ÓÒØ��×ÙÖ���Ó��×Ô��Ö�Û�Ø�g����Ø�ÓÒ�Ð��Ò �Ó×ØÑ�ØÖ� × ����ÒØÐÝ��Ú�Ò��Ö�Û�Ò�Ó�Ø���Ö�Ô�ÓÒ���ÒÙ×g×ÙÖ ��ÒÙ×�Ò��ÔÖÓ����Ð�×Ø��ÐÐÝ�Ñ�������ÒØÓÔÐ�Ò�Ö�Ö�Ô� × Û�Ø�ÓÒ×Ø�ÒØ��×ØÓÖØ�ÓÒÌ���Ñ�����Ò��Ò��ÓÑÔÙØ� � Ï�×�ÓÛØ��Ø�Ú�ÖÝÑ�ØÖ��Ò�Ù���Ý��Ö�Ô�Ó��ÓÙÒ�� � �ÜÔÖ�××�Ò��ÓÛ��Ö��Ö�Ô��×�ÖÓÑ���Ò�ÔÐ�Ò�ÖÌÓØ��Ø �ÜØ�Ò��Ö�Ô�×Ó�×Ñ�ÐÐ��ÒÙ×Ù×Ù�ÐÐÝ�Ü����ØÒ���Ð�ÓÖ�Ø � Ì����ÒÙ×Ó���Ö�Ô��Ò���ÒØ�ÖÔÖ�Ø���×�ÕÙ�ÒØ�ØÝ Ø��ÓÖ���Ò�Ð��×Ø�Ò��ÓÖC ≥ 1 �� � ÀÓÛ�Ú�Ö×Ù��ÜØ�Ò×�ÓÒ×�Ò�Ò×ÓÑ��×�×��ÓÑÔÐ��Ø� � �×Ñ�ÐÐÐÓ××�Ò����ÒÝÓÖ�ÒØ��ÕÙ�Ð�ØÝÓ�Ø��×ÓÐÙØ�ÓÒ Ñ�ÔÖÓÔ�ÖØ��×Ñ��ÒÐÝ�Ù�ØÓØ���Ö�Ò��Ö�ÒØ×�Ñ�Ð�Ö�Ø��×Û�Ø � ���Ö�ÛÒÓÒ�ØÓÖÙ×��×��ÒÙ×1 �Ò���×��ÓÒ���ÓØ��Ò�ÕÙ� × ÔÐ�Ò�Ö�Ö�Ô�×ÅÓÖ�ÔÖ��×�ÐÝ�Ð�ÓÖ�Ø�Ñ×�ÓÖÔÐ�Ò�Ö�Ö�Ô� × �ÒÙ×Ù�ÐÐÝ���ÜØ�Ò���ØÓ�Ö�Ô�×Ó��ÓÙÒ�����ÒÙ×Û�Ø� �ØÝ℄���Ò�Ö�Ð ���Ò�ÐÝ×�×Ó��Ð�ÓÖ�Ø�Ñ×�Ò�ÈÖÓ�Ð�Ñ�ÓÑÔÐ�Ü Ð�Ñ×ÓÒ�Ö�Ô�×Ó��ÓÙÒ�����ÒÙ×�ÝÖ��Ù�Ò�Ø��ÑØÓÓÖ Ö�×ÔÓÒ��Ò�ÔÖÓ�Ð�Ñ×ÓÒÔÐ�Ò�Ö�Ö�Ô�×ÇÙÖ�ÔÔÖÓ��� × ÁÒØ��×Ô�Ô�ÖÛ���Ú����Ò�Ö�ÐÑ�Ø�Ó��ÓÖ×ÓÐÚ�Ò�ÔÖÓ� �Ð�ÓÖ�Ø�Ñ×Ì��ÓÖÝ Ô����Ö�Ô�H�ÖÓÑFØ��ÒØ���ÜÔ�Ø����×Ø�Ò���ØÛ��Ò �ÓÙÒ�����ÒÙ×Ø��Ö��Ü�×Ø×���×ØÖ��ÙØ�ÓÒFÓÚ�ÖÔÐ�Ò�Ö Ñ�ØÖ�×�ÝØÖ��×��℄Ï�×�ÓÛØ��Ø�ÓÖ�ÒÝ�Ö�Ô�GÓ � �Ò×Ô�Ö���Ý��ÖØ�Ð×ÔÖÓ����Ð�×Ø��ÔÔÖÓÜ�Ñ�Ø�ÓÒÓ���Ò�Ö�Ð (G)��Û� Categories and Subject Descriptor