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  "body": "\n## Definition\n\nLet $G = (V, E)$ be a finite simple [[graph|0JD5C82QMB]] with $n = \\abs{V} \\ge 2$ vertices.\nThe graph $G$ is a *Laman graph* if both conditions hold.\n\n1. $\\abs{E} = 2n - 3$.\n2. For every set $S \\subseteq V$ with $k = \\abs{S} \\ge 2$, the [[induced subgraph|0V05FC4GKQ]] on $S$ has at most $2k - 3$ edges.\n\n## Remarks\n\nExample: the complete graph $K_4$ minus one edge has $4$ vertices and $5 = 2 \\cdot 4 - 3$ edges.\nEach $3$-vertex subset spans at most $3$ edges, and each pair spans at most $1$ edge, so condition 2 holds.\n\nNon-example: two triangles that share one vertex have $5$ vertices and $6$ edges, but $2 \\cdot 5 - 3 = 7$, so condition 1 fails.\nSuch a framework in the plane can rotate one triangle about the shared vertex.\n\nLaman's theorem concerns a generic placement of the vertices in $\\R^2$.\nIt states that $G$ is a Laman graph if and only if this framework is rigid and becomes flexible when any edge is removed.\nHilda Pollaczek-Geiringer found this result earlier, so it is also called the Geiringer–Laman theorem.\n",
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