{"componentChunkName":"component---src-lekoarts-gatsby-theme-minimal-blog-core-templates-post-query-tsx","path":"/minimum-spanning-tree","result":{"data":{"post":{"slug":"/minimum-spanning-tree","title":"Minimum spanning tree","date":"April 07, 2019","tags":[{"name":"Algorithm","slug":"algorithm"}],"description":null,"body":"function _extends() { _extends = Object.assign || function (target) { for (var i = 1; i < arguments.length; i++) { var source = arguments[i]; for (var key in source) { if (Object.prototype.hasOwnProperty.call(source, key)) { target[key] = source[key]; } } } return target; }; return _extends.apply(this, arguments); }\n\nfunction _objectWithoutProperties(source, excluded) { if (source == null) return {}; var target = _objectWithoutPropertiesLoose(source, excluded); var key, i; if (Object.getOwnPropertySymbols) { var sourceSymbolKeys = Object.getOwnPropertySymbols(source); for (i = 0; i < sourceSymbolKeys.length; i++) { key = sourceSymbolKeys[i]; if 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\"span\"\n  }, {\n    \"className\": \"mclose\"\n  }), \")\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mrel\"\n  }), \"=\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"base\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\"\n  }), \"m\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\"\n  }), \"i\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\"\n  }), \"n\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"msupsub\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist-t vlist-t2\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist-r\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.33610799999999996em\"\n    }\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"0em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mtight\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault mtight\"\n  }), \"a\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault mtight\",\n    \"style\": {\n      \"marginRight\": \"0.01968em\"\n    }\n  }), \"l\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault mtight\",\n    \"style\": {\n      \"marginRight\": \"0.01968em\"\n    }\n  }), \"l\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mspace mtight\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mtight\"\n  }), \"\\xA0\")), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault mtight\",\n    \"style\": {\n      \"marginRight\": \"0.13889em\"\n    }\n  }), \"T\"))))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist-s\"\n  }), \"\\u200B\")), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist-r\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\"\n  }), \"L\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mopen\"\n  }), \"(\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\",\n    \"style\": {\n      \"marginRight\": \"0.13889em\"\n    }\n  }), \"T\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mclose\"\n  }), \")\"))))), mdx(\"p\", null, mdx(\"span\", _extends({\n    parentName: \"p\"\n  }, {\n    \"className\": \"katex\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-mathml\"\n  }), mdx(\"math\", _extends({\n    parentName: \"span\"\n  }, {\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }), mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"Proof:\")), mdx(\"annotation\", _extends({\n    parentName: \"semantics\"\n  }, {\n    \"encoding\": \"application/x-tex\"\n  }), \"\\\\text{Proof:}\")))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"base\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.69444em\",\n      \"verticalAlign\": \"0em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord text\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord\"\n  }), \"Proof:\")))))), mdx(\"p\", null, mdx(\"img\", _extends({\n    parentName: \"p\"\n  }, {\n    \"src\": \"https://cdn.jsdelivr.net/gh/Deerhound579/image-hosting/img/H7pKqwO.png\",\n    \"alt\": \"mst.png\"\n  }))), mdx(\"p\", null, \"\\u201D\", mdx(\"span\", _extends({\n    parentName: \"p\"\n  }, {\n    \"className\": \"katex\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-mathml\"\n  }), mdx(\"math\", _extends({\n    parentName: \"span\"\n  }, {\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }), mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2265\")), mdx(\"annotation\", _extends({\n    parentName: \"semantics\"\n  }, {\n    \"encoding\": \"application/x-tex\"\n  }), \"\\\\geq\")))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"base\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.7719400000000001em\",\n      \"verticalAlign\": \"-0.13597em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mrel\"\n  }), \"\\u2265\")))), \"\\u201D\"), mdx(\"p\", null, \"MST\", mdx(\"span\", _extends({\n    parentName: \"p\"\n  }, {\n    \"className\": \"katex\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-mathml\"\n  }), mdx(\"math\", _extends({\n    parentName: \"span\"\n  }, {\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }), mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2208\")), mdx(\"annotation\", _extends({\n    parentName: \"semantics\"\n  }, {\n    \"encoding\": \"application/x-tex\"\n  }), \"\\\\in\")))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"base\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.5782em\",\n      \"verticalAlign\": \"-0.0391em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mrel\"\n  }), \"\\u2208\")))), \"T, so L(MST) cannot be less than the \", mdx(\"strong\", {\n    parentName: \"p\"\n  }, \"minimum\"), \" L(T) of all T\\u2019s.\"), mdx(\"p\", null, \"\\u201D\", mdx(\"span\", _extends({\n    parentName: \"p\"\n  }, {\n    \"className\": \"katex\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-mathml\"\n  }), mdx(\"math\", _extends({\n    parentName: \"span\"\n  }, {\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }), mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2264\")), mdx(\"annotation\", _extends({\n    parentName: \"semantics\"\n  }, {\n    \"encoding\": \"application/x-tex\"\n  }), \"\\\\leq\")))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"base\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.7719400000000001em\",\n      \"verticalAlign\": \"-0.13597em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mrel\"\n  }), \"\\u2264\")))), \"\\u201D\"), mdx(\"p\", null, \"In the figure above, we assume the red arrow is the longest edge in MST. We construct the MST by adding the shortest \\u201Cbridge\\u201D each time we cross two sets, \", mdx(\"span\", _extends({\n    parentName: \"p\"\n  }, {\n    \"className\": \"katex\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-mathml\"\n  }), mdx(\"math\", _extends({\n    parentName: \"span\"\n  }, {\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }), mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"A\")), mdx(\"annotation\", _extends({\n    parentName: \"semantics\"\n  }, {\n    \"encoding\": \"application/x-tex\"\n  }), \"A\")))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"base\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.68333em\",\n      \"verticalAlign\": \"0em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\"\n  }), \"A\")))), \" and \", mdx(\"span\", _extends({\n    parentName: \"p\"\n  }, {\n    \"className\": \"katex\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-mathml\"\n  }), mdx(\"math\", _extends({\n    parentName: \"span\"\n  }, {\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }), mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"V\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2212\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"A\")), mdx(\"annotation\", _extends({\n    parentName: \"semantics\"\n  }, {\n    \"encoding\": \"application/x-tex\"\n  }), \"V-A\")))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"base\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.76666em\",\n      \"verticalAlign\": \"-0.08333em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\",\n    \"style\": {\n      \"marginRight\": \"0.22222em\"\n    }\n  }), \"V\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mbin\"\n  }), \"\\u2212\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"base\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.68333em\",\n      \"verticalAlign\": \"0em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\"\n  }), \"A\")))), \". Thus, all other grey edges are \", mdx(\"strong\", {\n    parentName: \"p\"\n  }, \"longer\"), \" than or equal to the red edge. These grey edges are in some other spanning trees.\"), mdx(\"p\", null, \"For convenience, take one grey edge and call it \", mdx(\"span\", _extends({\n    parentName: \"p\"\n  }, {\n    \"className\": \"katex\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-mathml\"\n  }), mdx(\"math\", _extends({\n    parentName: \"span\"\n  }, {\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }), mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mo\", _extends({\n    parentName: \"mrow\"\n  }, {\n    \"stretchy\": \"false\"\n  }), \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"x\"), mdx(\"mo\", _extends({\n    parentName: \"mrow\"\n  }, {\n    \"separator\": \"true\"\n  }), \",\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"y\"), mdx(\"mo\", _extends({\n    parentName: \"mrow\"\n  }, {\n    \"stretchy\": \"false\"\n  }), \")\")), mdx(\"annotation\", _extends({\n    parentName: \"semantics\"\n  }, {\n    \"encoding\": \"application/x-tex\"\n  }), \"(x,y)\")))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"base\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mopen\"\n  }), \"(\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\"\n  }), \"x\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mpunct\"\n  }), \",\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }), \"y\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mclose\"\n  }), \")\")))), \" because it\\u2019s from some vertex \", mdx(\"span\", _extends({\n    parentName: \"p\"\n  }, {\n    \"className\": \"katex\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-mathml\"\n  }), mdx(\"math\", _extends({\n    parentName: \"span\"\n  }, {\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }), mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"x\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2208\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"A\")), mdx(\"annotation\", _extends({\n    parentName: \"semantics\"\n  }, {\n    \"encoding\": \"application/x-tex\"\n  }), \"x\\\\in A\")))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"base\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.5782em\",\n      \"verticalAlign\": \"-0.0391em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\"\n  }), \"x\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mrel\"\n  }), \"\\u2208\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"base\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.68333em\",\n      \"verticalAlign\": \"0em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\"\n  }), \"A\")))), \" to \", mdx(\"span\", _extends({\n    parentName: \"p\"\n  }, {\n    \"className\": \"katex\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-mathml\"\n  }), mdx(\"math\", _extends({\n    parentName: \"span\"\n  }, {\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }), mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"y\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2208\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"V\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2212\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"A\")), mdx(\"annotation\", _extends({\n    parentName: \"semantics\"\n  }, {\n    \"encoding\": \"application/x-tex\"\n  }), \"y \\\\in V-A\")))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"base\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.7335400000000001em\",\n      \"verticalAlign\": \"-0.19444em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }), \"y\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mrel\"\n  }), \"\\u2208\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"base\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.76666em\",\n      \"verticalAlign\": \"-0.08333em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\",\n    \"style\": {\n      \"marginRight\": \"0.22222em\"\n    }\n  }), \"V\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mbin\"\n  }), \"\\u2212\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"base\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.68333em\",\n      \"verticalAlign\": \"0em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\"\n  }), \"A\")))), \". If it\\u2019s the max edge of its spanning tree, \", mdx(\"span\", _extends({\n    parentName: \"p\"\n  }, {\n    \"className\": \"katex\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-mathml\"\n  }), mdx(\"math\", _extends({\n    parentName: \"span\"\n  }, {\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }), mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"T\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"x\"))), mdx(\"annotation\", _extends({\n    parentName: \"semantics\"\n  }, {\n    \"encoding\": \"application/x-tex\"\n  }), \"T_x\")))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"base\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.83333em\",\n      \"verticalAlign\": \"-0.15em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\",\n    \"style\": {\n      \"marginRight\": \"0.13889em\"\n    }\n  }), \"T\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"msupsub\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist-t vlist-t2\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist-r\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.151392em\"\n    }\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.13889em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault mtight\"\n  }), \"x\")))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist-s\"\n  }), \"\\u200B\")), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist-r\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\"\n  }))))))))), \". \", mdx(\"span\", _extends({\n    parentName: \"p\"\n  }, {\n    \"className\": \"katex\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-mathml\"\n  }), mdx(\"math\", _extends({\n    parentName: \"span\"\n  }, {\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }), mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"L\"), mdx(\"mo\", _extends({\n    parentName: \"mrow\"\n  }, {\n    \"stretchy\": \"false\"\n  }), \"(\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"T\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"x\")), mdx(\"mo\", _extends({\n    parentName: \"mrow\"\n  }, {\n    \"stretchy\": \"false\"\n  }), \")\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2265\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"L\"), mdx(\"mo\", _extends({\n    parentName: \"mrow\"\n  }, {\n    \"stretchy\": \"false\"\n  }), \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"M\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"S\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"T\"), mdx(\"mo\", _extends({\n    parentName: \"mrow\"\n  }, {\n    \"stretchy\": \"false\"\n  }), \")\")), mdx(\"annotation\", _extends({\n    parentName: \"semantics\"\n  }, {\n    \"encoding\": \"application/x-tex\"\n  }), \"L(T_x) \\\\geq L(MST)\")))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"base\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\"\n  }), \"L\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mopen\"\n  }), \"(\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\",\n    \"style\": {\n      \"marginRight\": \"0.13889em\"\n    }\n  }), \"T\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"msupsub\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist-t vlist-t2\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist-r\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.151392em\"\n    }\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.13889em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault mtight\"\n  }), \"x\")))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist-s\"\n  }), \"\\u200B\")), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist-r\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mclose\"\n  }), \")\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mrel\"\n  }), \"\\u2265\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"base\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\"\n  }), \"L\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mopen\"\n  }), \"(\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\",\n    \"style\": {\n      \"marginRight\": \"0.10903em\"\n    }\n  }), \"M\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\"\n    }\n  }), \"S\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\",\n    \"style\": {\n      \"marginRight\": \"0.13889em\"\n    }\n  }), \"T\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mclose\"\n  }), \")\")))), \".\\nIf it\\u2019s not, there exists a max edge \", mdx(\"span\", _extends({\n    parentName: \"p\"\n  }, {\n    \"className\": \"katex\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-mathml\"\n  }), mdx(\"math\", _extends({\n    parentName: \"span\"\n  }, {\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }), mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mo\", _extends({\n    parentName: \"mrow\"\n  }, {\n    \"stretchy\": \"false\"\n  }), \"(\"), mdx(\"msup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msup\"\n  }, \"x\"), mdx(\"mo\", _extends({\n    parentName: \"msup\"\n  }, {\n    \"mathvariant\": \"normal\",\n    \"lspace\": \"0em\",\n    \"rspace\": \"0em\"\n  }), \"\\u2032\")), mdx(\"mo\", _extends({\n    parentName: \"mrow\"\n  }, {\n    \"separator\": \"true\"\n  }), \",\"), mdx(\"msup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msup\"\n  }, \"y\"), mdx(\"mo\", _extends({\n    parentName: \"msup\"\n  }, {\n    \"mathvariant\": \"normal\",\n    \"lspace\": \"0em\",\n    \"rspace\": \"0em\"\n  }), \"\\u2032\")), mdx(\"mo\", _extends({\n    parentName: \"mrow\"\n  }, {\n    \"stretchy\": \"false\"\n  }), \")\")), mdx(\"annotation\", _extends({\n    parentName: \"semantics\"\n  }, {\n    \"encoding\": \"application/x-tex\"\n  }), \"(x',y')\")))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"base\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1.001892em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mopen\"\n  }), \"(\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\"\n  }), \"x\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"msupsub\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist-t\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist-r\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.751892em\"\n    }\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"style\": {\n      \"top\": \"-3.063em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mtight\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mtight\"\n  }), \"\\u2032\"))))))))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mpunct\"\n  }), \",\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }), \"y\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"msupsub\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist-t\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist-r\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.751892em\"\n    }\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"style\": {\n      \"top\": \"-3.063em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mtight\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mtight\"\n  }), \"\\u2032\"))))))))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mclose\"\n  }), \")\")))), \" in \", mdx(\"span\", _extends({\n    parentName: \"p\"\n  }, {\n    \"className\": \"katex\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-mathml\"\n  }), mdx(\"math\", _extends({\n    parentName: \"span\"\n  }, {\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }), mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"T\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"x\"))), mdx(\"annotation\", _extends({\n    parentName: \"semantics\"\n  }, {\n    \"encoding\": \"application/x-tex\"\n  }), \"T_x\")))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"base\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.83333em\",\n      \"verticalAlign\": \"-0.15em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\",\n    \"style\": {\n      \"marginRight\": \"0.13889em\"\n    }\n  }), \"T\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"msupsub\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist-t vlist-t2\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist-r\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.151392em\"\n    }\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.13889em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault mtight\"\n  }), \"x\")))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist-s\"\n  }), \"\\u200B\")), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist-r\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\"\n  }))))))))), \" such that \", mdx(\"span\", _extends({\n    parentName: \"p\"\n  }, {\n    \"className\": \"katex\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-mathml\"\n  }), mdx(\"math\", _extends({\n    parentName: \"span\"\n  }, {\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }), mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mo\", _extends({\n    parentName: \"mrow\"\n  }, {\n    \"stretchy\": \"false\"\n  }), \"(\"), mdx(\"msup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msup\"\n  }, \"x\"), mdx(\"mo\", _extends({\n    parentName: \"msup\"\n  }, {\n    \"mathvariant\": \"normal\",\n    \"lspace\": \"0em\",\n    \"rspace\": \"0em\"\n  }), \"\\u2032\")), mdx(\"mo\", _extends({\n    parentName: \"mrow\"\n  }, {\n    \"separator\": \"true\"\n  }), \",\"), mdx(\"msup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msup\"\n  }, \"y\"), mdx(\"mo\", _extends({\n    parentName: \"msup\"\n  }, {\n    \"mathvariant\": \"normal\",\n    \"lspace\": \"0em\",\n    \"rspace\": \"0em\"\n  }), \"\\u2032\")), mdx(\"mo\", _extends({\n    parentName: \"mrow\"\n  }, {\n    \"stretchy\": \"false\"\n  }), \")\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \">\"), mdx(\"mo\", _extends({\n    parentName: \"mrow\"\n  }, {\n    \"stretchy\": \"false\"\n  }), \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"x\"), mdx(\"mo\", _extends({\n    parentName: \"mrow\"\n  }, {\n    \"separator\": \"true\"\n  }), \",\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"y\"), mdx(\"mo\", _extends({\n    parentName: \"mrow\"\n  }, {\n    \"stretchy\": \"false\"\n  }), \")\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2265\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"L\"), mdx(\"mo\", _extends({\n    parentName: \"mrow\"\n  }, {\n    \"stretchy\": \"false\"\n  }), \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"M\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"S\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"T\"), mdx(\"mo\", _extends({\n    parentName: \"mrow\"\n  }, {\n    \"stretchy\": \"false\"\n  }), \")\")), mdx(\"annotation\", _extends({\n    parentName: \"semantics\"\n  }, {\n    \"encoding\": \"application/x-tex\"\n  }), \"(x',y') \\\\gt (x, y) \\\\geq L(MST)\")))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"base\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1.001892em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mopen\"\n  }), \"(\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\"\n  }), \"x\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"msupsub\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist-t\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist-r\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.751892em\"\n    }\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"style\": {\n      \"top\": \"-3.063em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mtight\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mtight\"\n  }), \"\\u2032\"))))))))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mpunct\"\n  }), \",\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }), \"y\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"msupsub\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist-t\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist-r\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.751892em\"\n    }\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"style\": {\n      \"top\": \"-3.063em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mtight\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mtight\"\n  }), \"\\u2032\"))))))))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mclose\"\n  }), \")\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mrel\"\n  }), \">\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"base\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mopen\"\n  }), \"(\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\"\n  }), \"x\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mpunct\"\n  }), \",\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }), \"y\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mclose\"\n  }), \")\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mrel\"\n  }), \"\\u2265\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"base\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\"\n  }), \"L\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mopen\"\n  }), \"(\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\",\n    \"style\": {\n      \"marginRight\": \"0.10903em\"\n    }\n  }), \"M\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\"\n    }\n  }), \"S\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\",\n    \"style\": {\n      \"marginRight\": \"0.13889em\"\n    }\n  }), \"T\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mclose\"\n  }), \")\")))), \". So \", mdx(\"span\", _extends({\n    parentName: \"p\"\n  }, {\n    \"className\": \"katex\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-mathml\"\n  }), mdx(\"math\", _extends({\n    parentName: \"span\"\n  }, {\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }), mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"L\"), mdx(\"mo\", _extends({\n    parentName: \"mrow\"\n  }, {\n    \"stretchy\": \"false\"\n  }), \"(\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"T\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"x\")), mdx(\"mo\", _extends({\n    parentName: \"mrow\"\n  }, {\n    \"stretchy\": \"false\"\n  }), \")\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2265\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"L\"), mdx(\"mo\", _extends({\n    parentName: \"mrow\"\n  }, {\n    \"stretchy\": \"false\"\n  }), \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"M\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"S\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"T\"), mdx(\"mo\", _extends({\n    parentName: \"mrow\"\n  }, {\n    \"stretchy\": \"false\"\n  }), \")\")), mdx(\"annotation\", _extends({\n    parentName: \"semantics\"\n  }, {\n    \"encoding\": \"application/x-tex\"\n  }), \"L(T_x) \\\\geq L(MST)\")))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"base\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\"\n  }), \"L\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mopen\"\n  }), \"(\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\",\n    \"style\": {\n      \"marginRight\": \"0.13889em\"\n    }\n  }), \"T\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"msupsub\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist-t vlist-t2\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist-r\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.151392em\"\n    }\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.13889em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault mtight\"\n  }), \"x\")))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist-s\"\n  }), \"\\u200B\")), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist-r\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mclose\"\n  }), \")\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mrel\"\n  }), \"\\u2265\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }))), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"base\"\n  }), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  })), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\"\n  }), \"L\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mopen\"\n  }), \"(\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\",\n    \"style\": {\n      \"marginRight\": \"0.10903em\"\n    }\n  }), \"M\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\",\n    \"style\": {\n      \"marginRight\": \"0.05764em\"\n    }\n  }), \"S\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mord mathdefault\",\n    \"style\": {\n      \"marginRight\": \"0.13889em\"\n    }\n  }), \"T\"), mdx(\"span\", _extends({\n    parentName: \"span\"\n  }, {\n    \"className\": \"mclose\"\n  }), \")\")))), \".\"), mdx(\"p\", null, \"Q.E.D\"));\n}\n;\nMDXContent.isMDXComponent = true;","excerpt":"MST has the min max edge","banner":null}},"pageContext":{"slug":"/minimum-spanning-tree"}}}