{"id":182799,"date":"2007-10-11T00:00:00","date_gmt":"2009-10-31T10:02:48","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/feedback-arc-sets-and-girth-in-digraphs\/"},"modified":"2016-09-09T09:44:30","modified_gmt":"2016-09-09T16:44:30","slug":"feedback-arc-sets-and-girth-in-digraphs","status":"publish","type":"msr-video","link":"https:\/\/www.microsoft.com\/en-us\/research\/video\/feedback-arc-sets-and-girth-in-digraphs\/","title":{"rendered":"Feedback Arc Sets and Girth in Digraphs"},"content":{"rendered":"<div class=\"asset-content\">\n<p>Given a directed graph G with girth at least m+1 (and no parallel edges), let b(G) denote the size of the smallest subset X of the edges of G so that G  X has no directed cycles, and let c(G) be the number of non-edges (unordered pairs of vertices with no edge between them).<br \/>\nPrior joint work with Maria Chudnovsky and Paul Seymour showed that when m = 3, b(G) is at most c(G), and we conjectured b(G) is actually at most c(G)\/2.<br \/>\nCan one say anything stronger if m is greater than three? In this talk, I will discuss a new conjecture giving a ratio between b(G) and c(G), namely<br \/>\nb(G) is at most 2c(G) \/ (m<sup>2<\/sup>-m-1), for all m which are at least three. The talk will also cover two new results in this direction: that b(G) is at most c(G)\/3 when m is at least four, and for circular interval graphs, a generalization of previous methods which gives a new bound for all m.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Given a directed graph G with girth at least m+1 (and no parallel edges), let b(G) denote the size of the smallest subset X of the edges of G so that G X has no directed cycles, and let c(G) be the number of non-edges (unordered pairs of vertices with no edge between them). Prior [&hellip;]<\/p>\n","protected":false},"featured_media":194752,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr_hide_image_in_river":0,"footnotes":""},"research-area":[],"msr-video-type":[],"msr-locale":[268875],"msr-post-option":[],"msr-session-type":[],"msr-impact-theme":[],"msr-pillar":[],"msr-episode":[],"msr-research-theme":[],"class_list":["post-182799","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/-6g7_euy6yg","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/182799","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video"}],"about":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-video"}],"version-history":[{"count":0,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/182799\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media\/194752"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=182799"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=182799"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=182799"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=182799"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=182799"},{"taxonomy":"msr-session-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-session-type?post=182799"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=182799"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=182799"},{"taxonomy":"msr-episode","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-episode?post=182799"},{"taxonomy":"msr-research-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-theme?post=182799"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}