{"id":68,"date":"2022-02-20T14:58:04","date_gmt":"2022-02-20T14:58:04","guid":{"rendered":"http:\/\/localhost\/kpv\/?p=68"},"modified":"2025-02-14T17:41:45","modified_gmt":"2025-02-14T17:41:45","slug":"directed-line-graph","status":"publish","type":"post","link":"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/directed-line-graph\/","title":{"rendered":"Directed line graph"},"content":{"rendered":"\n<p>If <strong>G<\/strong> is a directed graph, any edges in <strong>L(G)<\/strong> that do not correspond to valid directional paths in <strong>G<\/strong> should be removed. Specifically, if no feasible path exists between two edges in <strong>G<\/strong>\u2014even if they share a common vertex\u2014they should not be connected in <strong>L(G)<\/strong>.<\/p>\n\n\n\n<p>For example, consider two edges, <strong>A<\/strong> and <strong>B<\/strong>, in <strong>G<\/strong> that are both directed outward from node <strong>1<\/strong>. Although they share a common vertex, there is no possible way to traverse from <strong>A<\/strong> to <strong>B<\/strong> through node <strong>1<\/strong>, nor in the reverse direction. Consequently, an edge between <strong>A<\/strong> and <strong>B<\/strong> should not exist in <strong>L(G)<\/strong>.<\/p>\n\n\n\n<p>Furthermore, the directionality of edges in <strong>L(G)<\/strong> must be consistent with the directional constraints of <strong>G<\/strong>. For instance, if a path exists from edge <strong>E<\/strong> to node <strong>4<\/strong>, which subsequently leads to edge <strong>C<\/strong>, then <strong>L(G)<\/strong> should include a directed edge <strong>E \u2192 C<\/strong>. However, if starting from <strong>C<\/strong> allows traversal to node <strong>4<\/strong> but does not permit reaching edge <strong>E<\/strong>, then the reverse connection <strong>C \u2192 E<\/strong> should not be included in <strong>L(G)<\/strong>. This ensures that <strong>L(G)<\/strong> faithfully preserves the directional flow of the original graph <strong>G<\/strong>.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"642\" height=\"553\" src=\"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-content\/uploads\/2022\/02\/lineGraph4-1.png\" alt=\"\" class=\"wp-image-231\" style=\"width:321px;height:277px\" srcset=\"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-content\/uploads\/2022\/02\/lineGraph4-1.png 642w, https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-content\/uploads\/2022\/02\/lineGraph4-1-300x258.png 300w\" sizes=\"auto, (max-width: 642px) 100vw, 642px\" \/><\/figure>\n<\/div>\n\n\n<p>Finally, we proceed to remove the original edges and nodes. As a result, the line graph will take the following form:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"530\" height=\"458\" src=\"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-content\/uploads\/2022\/02\/lineGraph5-1.png\" alt=\"\" class=\"wp-image-232\" style=\"width:265px;height:229px\" srcset=\"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-content\/uploads\/2022\/02\/lineGraph5-1.png 530w, https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-content\/uploads\/2022\/02\/lineGraph5-1-300x259.png 300w\" sizes=\"auto, (max-width: 530px) 100vw, 530px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"has-text-align-right\"><a href=\"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/pipeline\/\" data-type=\"URL\" data-id=\"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/index.php\/2022\/02\/20\/pipeline\/\">Go to pipeline information<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>If G is a directed graph, any edges in L(G) that do not correspond to valid directional paths in G should be removed. Specifically, if&#8230;<\/p>\n<div class=\"more-link-wrapper\"><a class=\"more-link\" href=\"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/directed-line-graph\/\">Continue reading<span class=\"screen-reader-text\">Directed line graph<\/span><\/a><\/div>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[],"class_list":["post-68","post","type-post","status-publish","format-standard","hentry","category-linegraph","entry"],"_links":{"self":[{"href":"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-json\/wp\/v2\/posts\/68","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-json\/wp\/v2\/comments?post=68"}],"version-history":[{"count":8,"href":"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-json\/wp\/v2\/posts\/68\/revisions"}],"predecessor-version":[{"id":352,"href":"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-json\/wp\/v2\/posts\/68\/revisions\/352"}],"wp:attachment":[{"href":"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-json\/wp\/v2\/media?parent=68"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-json\/wp\/v2\/categories?post=68"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-json\/wp\/v2\/tags?post=68"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}