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  "body": "\n## Definition\n\nLet $G = (V, E)$ be a finite [[graph|0JD5C82QMB]] and let $\\Sigma$ be a [[surface|VGJJVBZ1MQ]].\nAn embedding of $G$ in $\\Sigma$ consists of a point $p_v \\in \\Sigma$ for each vertex $v$ and a subset $\\gamma_e \\subseteq \\Sigma$ for each edge $e$ such that:\n\n1. The points $p_v$ are distinct.\n2. For each edge $e = uv$, the set $\\gamma_e$ is the image of a [[topological embedding|08SVRVHJNY]] of $[0, 1]$ into $\\Sigma$ with ends $p_u$ and $p_v$.\n3. An arc $\\gamma_e$ contains no point $p_w$ other than its two ends.\n4. Two distinct arcs meet at most in common ends.\n\n## Remarks\n\nExample: drawing $K_4$ in the plane as a triangle with a central vertex joined to its three corners satisfies all four clauses.\nNon-example: drawing $K_4$ as a square with both diagonals is not an embedding, because the diagonals cross at an inner point, which violates clause 4.\nA graph is planar exactly when it has an embedding in the plane, or equivalently in the sphere.\nThe genus of $G$ is the least $g$ such that $G$ embeds in the [[orientable surface|0EKZW08FQ5]] of genus $g$; for example $K_5$ and $K_{3,3}$ have genus $1$.\n",
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