"En teoria de grafs, un graf pla o planar \u00E9s un graf que pot ser dibuixat en un pla sense que les arestes s'intersequin (o utilitzant una definici\u00F3 m\u00E9s formal, que aquest graf pugui ser \"embegut\" en un pla). Un graf no \u00E9s pla si no pot ser dibuixat sobre un pla sense que les seves arestes s'intersequin. Hi ha dos grafs no plans minimals; K\u2085 i K3,3. Tots els grafs no plans contenen almenys un d'aquests subgrafs minimals. En canvi, el graf complet K\u2084 \u00E9s pla, perqu\u00E8 pot ser redibuixat sense que les arestes es creuin, passant una de les diagonals per l'exterior."@ca . . . . . "Planar graph"@en . . . . . "\uD3C9\uBA74 \uADF8\uB798\uD504(planar graph)\uB294 \uD3C9\uBA74 \uC0C1\uC5D0 \uADF8\uB798\uD504\uB97C \uADF8\uB838\uC744 \uB54C, \uB450 \uBCC0\uC774 \uAF2D\uC9D3\uC810 \uC774\uC678\uC5D0 \uB9CC\uB098\uC9C0 \uC54A\uB3C4\uB85D \uADF8\uB9B4 \uC218 \uC788\uB294 \uADF8\uB798\uD504\uB97C \uC758\uBBF8\uD55C\uB2E4. \uC608\uB97C \uB4E4\uC5B4 \uB2E4\uC74C\uC758 \uADF8\uB798\uD504\uB294 \uBAA8\uB450 \uD3C9\uBA74 \uADF8\uB798\uD504\uC774\uB2E4. \n* \uD3C9\uBA74 \uADF8\uB798\uD504\uC758 \uC608 \n* \n* \uC774 \uADF8\uB9BC \uC0C1\uC5D0\uC11C\uB294 \uB450 \uBCC0\uC774 \uB9CC\uB098\uC9C0\uB9CC, \uB9CC\uB098\uC9C0 \uC54A\uB3C4\uB85D \uADF8\uB9B4 \uC218 \uC788\uB2E4. \n* \uC55E\uC758 \uADF8\uB798\uD504\uC640 \uAC19\uC740 \uADF8\uB798\uD504\uC774\uB2E4. \uD55C\uD3B8 \uC544\uB798 \uADF8\uB798\uD504\uB294 \uD3C9\uBA74 \uADF8\uB798\uD504\uAC00 \uC544\uB2C8\uB2E4. \n* \uD3C9\uBA74 \uADF8\uB798\uD504\uAC00 \uC544\uB2CC \uADF8\uB798\uD504\uC758 \uC608 \n* \uAF2D\uC9D3\uC810\uC774 5\uAC1C\uC778 \uC644\uC804 \uADF8\uB798\uD504 \n* \uAF2D\uC9D3\uC810\uC774 3\uAC1C\uC529\uC778 \uC644\uC804 \uC774\uBD84 \uADF8\uB798\uD504 \n* \uD398\uD14C\uB974\uC13C \uADF8\uB798\uD504 \uC5C4\uBC00\uD558\uAC8C \uC815\uC758\uD558\uBA74, \uD3C9\uBA74 \uADF8\uB798\uD504\uB294 \uD3C9\uBA74\uC5D0 \uADF8\uB798\uD504 \uC784\uBCA0\uB529\uC774 \uAC00\uB2A5\uD55C \uADF8\uB798\uD504\uB97C \uC758\uBBF8\uD55C\uB2E4."@ko . . "Nella teoria dei grafi si definisce grafo planare un grafo che pu\u00F2 essere raffigurato in un piano in modo che non si abbiano archi che si intersecano. Ad esempio sono planari i seguenti grafi: Il secondo pu\u00F2 essere raffigurato senza archi che si intersecano spostando uno degli archi dati da una diagonale al di fuori del perimetro del quadrato. Vi sono invece grafi che posseggono solo raffigurazioni piane nelle quali si hanno coppie di archi che si intersecano. Le due seguenti figure forniscono raffigurazioni di due grafi non planari: K5 K3,3 Si tratta del grafo completo con 5 nodi e del grafo bipartito completo con 3+3 nodi ; questi due grafi sono chiamati anche grafi di Kuratowski, in onore del matematico polacco Kazimierz Kuratowski. Si constata infatti che non \u00E8 possibile ridisegnare queste raffigurazioni evitando che gli archi si intersechino. In effetti Kuratowski nel 1929 ha dimostrato che questi sono i due grafi non planari pi\u00F9 ridotti, con il seguente enunciato."@it . . . "\u041F\u043B\u0430\u043D\u0430\u0301\u0440\u043D\u044B\u0439 \u0433\u0440\u0430\u0444 \u2014 \u0433\u0440\u0430\u0444, \u043A\u043E\u0442\u043E\u0440\u044B\u0439 \u043C\u043E\u0436\u043D\u043E \u0438\u0437\u043E\u0431\u0440\u0430\u0437\u0438\u0442\u044C \u043D\u0430 \u043F\u043B\u043E\u0441\u043A\u043E\u0441\u0442\u0438 \u0431\u0435\u0437 \u043F\u0435\u0440\u0435\u0441\u0435\u0447\u0435\u043D\u0438\u0439 \u0440\u0451\u0431\u0435\u0440 \u043D\u0435 \u043F\u043E \u0432\u0435\u0440\u0448\u0438\u043D\u0430\u043C. \u041A\u0430\u043A\u043E\u0435-\u043B\u0438\u0431\u043E \u043A\u043E\u043D\u043A\u0440\u0435\u0442\u043D\u043E\u0435 \u0438\u0437\u043E\u0431\u0440\u0430\u0436\u0435\u043D\u0438\u0435 \u043F\u043B\u0430\u043D\u0430\u0440\u043D\u043E\u0433\u043E \u0433\u0440\u0430\u0444\u0430 \u043D\u0430 \u043F\u043B\u043E\u0441\u043A\u043E\u0441\u0442\u0438 \u0431\u0435\u0437 \u043F\u0435\u0440\u0435\u0441\u0435\u0447\u0435\u043D\u0438\u044F \u0440\u0451\u0431\u0435\u0440 \u043D\u0435 \u043F\u043E \u0432\u0435\u0440\u0448\u0438\u043D\u0430\u043C \u043D\u0430\u0437\u044B\u0432\u0430\u0435\u0442\u0441\u044F \u043F\u043B\u043E\u0441\u043A\u0438\u043C \u0433\u0440\u0430\u0444\u043E\u043C. \u0418\u043D\u0430\u0447\u0435 \u0433\u043E\u0432\u043E\u0440\u044F, \u043F\u043B\u0430\u043D\u0430\u0440\u043D\u044B\u0439 \u0433\u0440\u0430\u0444 \u0438\u0437\u043E\u043C\u043E\u0440\u0444\u0435\u043D \u043D\u0435\u043A\u043E\u0442\u043E\u0440\u043E\u043C\u0443 \u043F\u043B\u043E\u0441\u043A\u043E\u043C\u0443 \u0433\u0440\u0430\u0444\u0443, \u0438\u0437\u043E\u0431\u0440\u0430\u0436\u0451\u043D\u043D\u043E\u043C\u0443 \u043D\u0430 \u043F\u043B\u043E\u0441\u043A\u043E\u0441\u0442\u0438 \u0442\u0430\u043A, \u0447\u0442\u043E \u0435\u0433\u043E \u0432\u0435\u0440\u0448\u0438\u043D\u044B \u2014 \u044D\u0442\u043E \u0442\u043E\u0447\u043A\u0438 \u043F\u043B\u043E\u0441\u043A\u043E\u0441\u0442\u0438, \u0430 \u0440\u0451\u0431\u0440\u0430 \u2014 \u043A\u0440\u0438\u0432\u044B\u0435 \u043D\u0430 \u043F\u043B\u043E\u0441\u043A\u043E\u0441\u0442\u0438, \u043A\u043E\u0442\u043E\u0440\u044B\u0435 \u0435\u0441\u043B\u0438 \u0438 \u043F\u0435\u0440\u0435\u0441\u0435\u043A\u0430\u044E\u0442\u0441\u044F \u043C\u0435\u0436\u0434\u0443 \u0441\u043E\u0431\u043E\u0439, \u0442\u043E \u0442\u043E\u043B\u044C\u043A\u043E \u043F\u043E \u0432\u0435\u0440\u0448\u0438\u043D\u0430\u043C. \u041E\u0431\u043B\u0430\u0441\u0442\u0438, \u043D\u0430 \u043A\u043E\u0442\u043E\u0440\u044B\u0435 \u0433\u0440\u0430\u0444 \u0440\u0430\u0437\u0431\u0438\u0432\u0430\u0435\u0442 \u043F\u043B\u043E\u0441\u043A\u043E\u0441\u0442\u044C, \u043D\u0430\u0437\u044B\u0432\u0430\u044E\u0442\u0441\u044F \u0435\u0433\u043E \u0433\u0440\u0430\u043D\u044F\u043C\u0438. \u041D\u0435\u043E\u0433\u0440\u0430\u043D\u0438\u0447\u0435\u043D\u043D\u0430\u044F \u0447\u0430\u0441\u0442\u044C \u043F\u043B\u043E\u0441\u043A\u043E\u0441\u0442\u0438 \u2014 \u0442\u043E\u0436\u0435 \u0433\u0440\u0430\u043D\u044C, \u043D\u0430\u0437\u044B\u0432\u0430\u0435\u043C\u0430\u044F \u0432\u043D\u0435\u0448\u043D\u0435\u0439 \u0433\u0440\u0430\u043D\u044C\u044E. \u041B\u044E\u0431\u043E\u0439 \u043F\u043B\u043E\u0441\u043A\u0438\u0439 \u0433\u0440\u0430\u0444 \u043C\u043E\u0436\u0435\u0442 \u0431\u044B\u0442\u044C \u0441\u043F\u0440\u044F\u043C\u043B\u0451\u043D, \u0442\u043E \u0435\u0441\u0442\u044C \u043F\u0435\u0440\u0435\u0440\u0438\u0441\u043E\u0432\u0430\u043D \u043D\u0430 \u043F\u043B\u043E\u0441\u043A\u043E\u0441\u0442\u0438 \u0442\u0430\u043A, \u0447\u0442\u043E \u0432\u0441\u0435 \u0435\u0433\u043E \u0440\u0451\u0431\u0440\u0430 \u0431\u0443\u0434\u0443\u0442 \u043E\u0442\u0440\u0435\u0437\u043A\u0430\u043C\u0438 \u043F\u0440\u044F\u043C\u044B\u0445."@ru . . "Graphe planaire"@fr . . . . . . . . . . . . . . . . . . . . "In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points. Every graph that can be drawn on a plane can be drawn on the sphere as well, and vice versa, by means of stereographic projection. Plane graphs can be encoded by combinatorial maps or rotation systems. An equivalence class of topologically equivalent drawings on the sphere, usually with additional assumptions such as the absence of isthmuses, is called a planar map. Although a plane graph has an external or unbounded face, none of the faces of a planar map has a particular status. Planar graphs generalize to graphs drawable on a surface of a given genus. In this terminology, planar graphs have genus 0, since the plane (and the sphere) are surfaces of genus 0. See \"graph embedding\" for other related topics."@en . . . . . . "\u5E73\u9762\u30B0\u30E9\u30D5\uFF08\u3078\u3044\u3081\u3093\u30B0\u30E9\u30D5\u3001\u82F1: plane graph\uFF09\u306F\u3001\u5E73\u9762\u4E0A\u306E\u9802\u70B9\u96C6\u5408\u3068\u305D\u308C\u3092\u4EA4\u5DEE\u306A\u304F\u7D50\u3076\u8FBA\u96C6\u5408\u304B\u3089\u306A\u308B\u30B0\u30E9\u30D5\u3067\u3042\u308B\u3002\u5E73\u9762\u30B0\u30E9\u30D5\u3068\u540C\u578B\u306A\u30B0\u30E9\u30D5\u3092\u5E73\u9762\u7684\u30B0\u30E9\u30D5 (planar graph) \u3068\u3044\u3046\u3002\u5E73\u9762\u7684\u30B0\u30E9\u30D5\u3067\u3042\u3063\u3066\u3082\u3001\u63CF\u304D\u65B9\u306B\u3088\u3063\u3066\u306F\u5E73\u9762\u30B0\u30E9\u30D5\u306B\u306A\u3089\u306A\u3044\u3002 \u5E73\u9762\u7684\u30B0\u30E9\u30D5\u306F\u3001\u7403\u9762\u306A\u3069\u306E\u7A2E\u65700\u306E\u66F2\u9762\u306B\u63CF\u3051\u308B\u30B0\u30E9\u30D5\u3068\u540C\u5024\u3067\u3042\u308B\u3002\u6975\u5C0F\u306A\u975E\u5E73\u9762\u7684\u30B0\u30E9\u30D5\u306F\u3001K3,3\u3068K5\u3067\u3042\u308B\u3002"@ja . "\u041F\u043B\u0430\u043D\u0430\u0440\u043D\u044B\u0439 \u0433\u0440\u0430\u0444"@ru . "Grafo planare"@it . . . . . . . . . . . . . . . "En teoria de grafs, un graf pla o planar \u00E9s un graf que pot ser dibuixat en un pla sense que les arestes s'intersequin (o utilitzant una definici\u00F3 m\u00E9s formal, que aquest graf pugui ser \"embegut\" en un pla). Un graf no \u00E9s pla si no pot ser dibuixat sobre un pla sense que les seves arestes s'intersequin. Hi ha dos grafs no plans minimals; K\u2085 i K3,3. Tots els grafs no plans contenen almenys un d'aquests subgrafs minimals. En canvi, el graf complet K\u2084 \u00E9s pla, perqu\u00E8 pot ser redibuixat sense que les arestes es creuin, passant una de les diagonals per l'exterior. Per poder comprovar la planaritat d'un graf, primer s'ha d'adequar a una s\u00E8rie de criteris b\u00E0sics: \n* El graf \u00E9s connex. Si tenim un graf inconnex, podem considerar cada part com a graf connex independent. \n* No t\u00E9 cap v\u00E8rtex de grau 1. Si cont\u00E9 v\u00E8rtexs de grau 1, per tant amb una sola aresta, aquests es poden eliminar de l'estructura sense risc a modificar-ne la planaritat."@ca . . . . . . . "Planarer Graph"@de . . . . "\u5728\u5716\u8AD6\u4E2D\uFF0C\u5E73\u9762\u5716\u662F\u53EF\u4EE5\u753B\u5728\u5E73\u9762\u4E0A\u5E76\u4E14\u4F7F\u5F97\u4E0D\u540C\u7684\u908A\u53EF\u4EE5\u4E92\u4E0D\u4EA4\u758A\u7684\u5716\u3002\u800C\u5982\u679C\u4E00\u4E2A\u56FE\u65E0\u8BBA\u600E\u6837\u90FD\u65E0\u6CD5\u753B\u5728\u5E73\u9762\u4E0A\uFF0C\u5E76\u4F7F\u5F97\u4E0D\u540C\u7684\u8FB9\u4E92\u4E0D\u4EA4\u53E0\uFF0C\u90A3\u4E48\u8FD9\u6837\u7684\u56FE\u4E0D\u662F\u5E73\u9762\u56FE\uFF0C\u6216\u8005\u79F0\u4E3A\u975E\u5E73\u9762\u56FE\u3002\u5B8C\u5168\u56FE K5\u548C\u5B8C\u5168\u4E8C\u5206\u56FE K3,3\uFF08\u6E6F\u746A\u68EE\u5716\uFF09\u662F\u6700\u201C\u5C0F\u201D\u7684\u975E\u5E73\u9762\u56FE\u3002 \u4E00\u500B\u5C07\u5E73\u9762\u5716\u756B\u5728\u5E73\u9762\u4E0A\u7684\u65B9\u6CD5\u7A31\u70BA\u5E73\u7248\u5716\uFF0C\u53C8\u7A31\u70BA\u5716\u7684\u5E73\u9762\u5D4C\u5165\uFF0C\u66F4\u7CBE\u78BA\u5730\u8AAA\uFF0C\u5E73\u7248\u5716\u5305\u542B\u4E00\u500B\u5E73\u9762\u5716\u8207\u4E00\u500B\u6620\u5C04\uFF0C\u6B64\u6620\u5C04\u5C07\u5E73\u9762\u5716\u7684\u9802\u9EDE\u5C0D\u61C9\u5230\u5E73\u9762\u4E0A\u7684\u4E00\u9EDE\uFF0C\u908A\u5C0D\u61C9\u5230\u4E00\u689D\u5E73\u9762\u66F2\u7EBF\u6BB5\uFF0C\u6EFF\u8DB3\u908A\u5169\u7AEF\u9EDE\u5C0D\u61C9\u5230\u7DDA\u6BB5\u7684\u5169\u7AEF\u9EDE\uFF0C\u4E26\u4E14\u7DDA\u6BB5\u4E4B\u9593\u9664\u4E86\u5728\u7AEF\u9EDE\u4E4B\u5916\u90FD\u4E0D\u76F8\u4EA4\u3002 \u85C9\u7531\u7403\u6781\u6295\u5F71\u53EF\u77E5\u4E00\u500B\u5716\u53EF\u4EE5\u88AB\u5D4C\u5165\u5E73\u9762\u82E5\u4E14\u552F\u82E5\u53EF\u4EE5\u88AB\u5D4C\u5165\u7403\u9762\u3002\u5716\u7684\u7403\u9762\u5D4C\u5165\u5728\u95DC\u4FC2\u4E2D\u7684\u7B49\u50F9\u985E\u7A31\u70BA\u5E73\u9762\u6620\u5C04\u3002\u6CE8\u610F\u5230\u4E00\u500B\u5E73\u7248\u5716\u6703\u6709\u5916\u570D\u9762\uFF0C\u53C8\u7A31\u7121\u754C\u9762\uFF0C\u4F46\u56E0\u70BA\u5E73\u9762\u6620\u5C04\u5B9A\u7FA9\u662F\u5728\u7403\u9762\u4E0A\u7684\u7B49\u50F9\u985E\uFF0C\u4E0D\u6703\u6709\u4EFB\u4F55\u4E00\u500B\u9762\u6709\u9019\u500B\u7279\u6B8A\u7684\u5730\u4F4D\u3002 \u5E73\u9762\u5716\u53EF\u4EE5\u88AB\u8996\u70BA\u4E00\u500B\u3002"@zh . . "Graf planarny"@pl . "32775"^^ . "\uD3C9\uBA74 \uADF8\uB798\uD504(planar graph)\uB294 \uD3C9\uBA74 \uC0C1\uC5D0 \uADF8\uB798\uD504\uB97C \uADF8\uB838\uC744 \uB54C, \uB450 \uBCC0\uC774 \uAF2D\uC9D3\uC810 \uC774\uC678\uC5D0 \uB9CC\uB098\uC9C0 \uC54A\uB3C4\uB85D \uADF8\uB9B4 \uC218 \uC788\uB294 \uADF8\uB798\uD504\uB97C \uC758\uBBF8\uD55C\uB2E4. \uC608\uB97C \uB4E4\uC5B4 \uB2E4\uC74C\uC758 \uADF8\uB798\uD504\uB294 \uBAA8\uB450 \uD3C9\uBA74 \uADF8\uB798\uD504\uC774\uB2E4. \n* \uD3C9\uBA74 \uADF8\uB798\uD504\uC758 \uC608 \n* \n* \uC774 \uADF8\uB9BC \uC0C1\uC5D0\uC11C\uB294 \uB450 \uBCC0\uC774 \uB9CC\uB098\uC9C0\uB9CC, \uB9CC\uB098\uC9C0 \uC54A\uB3C4\uB85D \uADF8\uB9B4 \uC218 \uC788\uB2E4. \n* \uC55E\uC758 \uADF8\uB798\uD504\uC640 \uAC19\uC740 \uADF8\uB798\uD504\uC774\uB2E4. \uD55C\uD3B8 \uC544\uB798 \uADF8\uB798\uD504\uB294 \uD3C9\uBA74 \uADF8\uB798\uD504\uAC00 \uC544\uB2C8\uB2E4. \n* \uD3C9\uBA74 \uADF8\uB798\uD504\uAC00 \uC544\uB2CC \uADF8\uB798\uD504\uC758 \uC608 \n* \uAF2D\uC9D3\uC810\uC774 5\uAC1C\uC778 \uC644\uC804 \uADF8\uB798\uD504 \n* \uAF2D\uC9D3\uC810\uC774 3\uAC1C\uC529\uC778 \uC644\uC804 \uC774\uBD84 \uADF8\uB798\uD504 \n* \uD398\uD14C\uB974\uC13C \uADF8\uB798\uD504 \uC5C4\uBC00\uD558\uAC8C \uC815\uC758\uD558\uBA74, \uD3C9\uBA74 \uADF8\uB798\uD504\uB294 \uD3C9\uBA74\uC5D0 \uADF8\uB798\uD504 \uC784\uBCA0\uB529\uC774 \uAC00\uB2A5\uD55C \uADF8\uB798\uD504\uB97C \uC758\uBBF8\uD55C\uB2E4."@ko . "\u5728\u5716\u8AD6\u4E2D\uFF0C\u5E73\u9762\u5716\u662F\u53EF\u4EE5\u753B\u5728\u5E73\u9762\u4E0A\u5E76\u4E14\u4F7F\u5F97\u4E0D\u540C\u7684\u908A\u53EF\u4EE5\u4E92\u4E0D\u4EA4\u758A\u7684\u5716\u3002\u800C\u5982\u679C\u4E00\u4E2A\u56FE\u65E0\u8BBA\u600E\u6837\u90FD\u65E0\u6CD5\u753B\u5728\u5E73\u9762\u4E0A\uFF0C\u5E76\u4F7F\u5F97\u4E0D\u540C\u7684\u8FB9\u4E92\u4E0D\u4EA4\u53E0\uFF0C\u90A3\u4E48\u8FD9\u6837\u7684\u56FE\u4E0D\u662F\u5E73\u9762\u56FE\uFF0C\u6216\u8005\u79F0\u4E3A\u975E\u5E73\u9762\u56FE\u3002\u5B8C\u5168\u56FE K5\u548C\u5B8C\u5168\u4E8C\u5206\u56FE K3,3\uFF08\u6E6F\u746A\u68EE\u5716\uFF09\u662F\u6700\u201C\u5C0F\u201D\u7684\u975E\u5E73\u9762\u56FE\u3002 \u4E00\u500B\u5C07\u5E73\u9762\u5716\u756B\u5728\u5E73\u9762\u4E0A\u7684\u65B9\u6CD5\u7A31\u70BA\u5E73\u7248\u5716\uFF0C\u53C8\u7A31\u70BA\u5716\u7684\u5E73\u9762\u5D4C\u5165\uFF0C\u66F4\u7CBE\u78BA\u5730\u8AAA\uFF0C\u5E73\u7248\u5716\u5305\u542B\u4E00\u500B\u5E73\u9762\u5716\u8207\u4E00\u500B\u6620\u5C04\uFF0C\u6B64\u6620\u5C04\u5C07\u5E73\u9762\u5716\u7684\u9802\u9EDE\u5C0D\u61C9\u5230\u5E73\u9762\u4E0A\u7684\u4E00\u9EDE\uFF0C\u908A\u5C0D\u61C9\u5230\u4E00\u689D\u5E73\u9762\u66F2\u7EBF\u6BB5\uFF0C\u6EFF\u8DB3\u908A\u5169\u7AEF\u9EDE\u5C0D\u61C9\u5230\u7DDA\u6BB5\u7684\u5169\u7AEF\u9EDE\uFF0C\u4E26\u4E14\u7DDA\u6BB5\u4E4B\u9593\u9664\u4E86\u5728\u7AEF\u9EDE\u4E4B\u5916\u90FD\u4E0D\u76F8\u4EA4\u3002 \u85C9\u7531\u7403\u6781\u6295\u5F71\u53EF\u77E5\u4E00\u500B\u5716\u53EF\u4EE5\u88AB\u5D4C\u5165\u5E73\u9762\u82E5\u4E14\u552F\u82E5\u53EF\u4EE5\u88AB\u5D4C\u5165\u7403\u9762\u3002\u5716\u7684\u7403\u9762\u5D4C\u5165\u5728\u95DC\u4FC2\u4E2D\u7684\u7B49\u50F9\u985E\u7A31\u70BA\u5E73\u9762\u6620\u5C04\u3002\u6CE8\u610F\u5230\u4E00\u500B\u5E73\u7248\u5716\u6703\u6709\u5916\u570D\u9762\uFF0C\u53C8\u7A31\u7121\u754C\u9762\uFF0C\u4F46\u56E0\u70BA\u5E73\u9762\u6620\u5C04\u5B9A\u7FA9\u662F\u5728\u7403\u9762\u4E0A\u7684\u7B49\u50F9\u985E\uFF0C\u4E0D\u6703\u6709\u4EFB\u4F55\u4E00\u500B\u9762\u6709\u9019\u500B\u7279\u6B8A\u7684\u5730\u4F4D\u3002 \u5E73\u9762\u5716\u53EF\u4EE5\u88AB\u8996\u70BA\u4E00\u500B\u3002"@zh . . . . . . . . . . . . . . . . . . "Plan\u00E4r graf"@sv . . . . . . "\u5E73\u9762\u30B0\u30E9\u30D5\uFF08\u3078\u3044\u3081\u3093\u30B0\u30E9\u30D5\u3001\u82F1: plane graph\uFF09\u306F\u3001\u5E73\u9762\u4E0A\u306E\u9802\u70B9\u96C6\u5408\u3068\u305D\u308C\u3092\u4EA4\u5DEE\u306A\u304F\u7D50\u3076\u8FBA\u96C6\u5408\u304B\u3089\u306A\u308B\u30B0\u30E9\u30D5\u3067\u3042\u308B\u3002\u5E73\u9762\u30B0\u30E9\u30D5\u3068\u540C\u578B\u306A\u30B0\u30E9\u30D5\u3092\u5E73\u9762\u7684\u30B0\u30E9\u30D5 (planar graph) \u3068\u3044\u3046\u3002\u5E73\u9762\u7684\u30B0\u30E9\u30D5\u3067\u3042\u3063\u3066\u3082\u3001\u63CF\u304D\u65B9\u306B\u3088\u3063\u3066\u306F\u5E73\u9762\u30B0\u30E9\u30D5\u306B\u306A\u3089\u306A\u3044\u3002 \u5E73\u9762\u7684\u30B0\u30E9\u30D5\u306F\u3001\u7403\u9762\u306A\u3069\u306E\u7A2E\u65700\u306E\u66F2\u9762\u306B\u63CF\u3051\u308B\u30B0\u30E9\u30D5\u3068\u540C\u5024\u3067\u3042\u308B\u3002\u6975\u5C0F\u306A\u975E\u5E73\u9762\u7684\u30B0\u30E9\u30D5\u306F\u3001K3,3\u3068K5\u3067\u3042\u308B\u3002"@ja . . . "\u041F\u043B\u0430\u043D\u0430\u0440\u043D\u0438\u0439 \u0433\u0440\u0430\u0444 \u2014 \u0433\u0440\u0430\u0444, \u044F\u043A\u0438\u0439 \u043C\u043E\u0436\u0435 \u0431\u0443\u0442\u0438 \u0437\u043E\u0431\u0440\u0430\u0436\u0435\u043D\u0438\u0439 \u043D\u0430 \u043F\u043B\u043E\u0449\u0438\u043D\u0456 \u0431\u0435\u0437 \u043F\u0435\u0440\u0435\u0442\u0438\u043D\u0443 \u0440\u0435\u0431\u0435\u0440. \u0413\u0440\u0430\u0444 \u0437\u043E\u0431\u0440\u0430\u0436\u0435\u043D\u0438\u0439 \u043D\u0430 \u043F\u043B\u043E\u0449\u0438\u043D\u0456 \u043D\u0430\u0437\u0438\u0432\u0430\u0454\u0442\u044C\u0441\u044F \u043F\u043B\u043E\u0441\u043A\u0438\u043C, \u044F\u043A\u0449\u043E \u0439\u043E\u0433\u043E \u0440\u0435\u0431\u0440\u0430 \u043D\u0435 \u043F\u0435\u0440\u0435\u0442\u0438\u043D\u0430\u044E\u0442\u044C\u0441\u044F. \u0413\u0440\u0430\u0444 \u043D\u0430\u0437\u0438\u0432\u0430\u0454\u0442\u044C\u0441\u044F \u043F\u043B\u0430\u043D\u0430\u0440\u043D\u0438\u043C, \u044F\u043A\u0449\u043E \u0432\u0456\u043D \u0456\u0437\u043E\u043C\u043E\u0440\u0444\u043D\u0438\u0439 \u0434\u0435\u044F\u043A\u043E\u043C\u0443 \u043F\u043B\u043E\u0441\u043A\u043E\u043C\u0443 \u0433\u0440\u0430\u0444\u0443. \u0422\u043E\u0431\u0442\u043E \u0456\u0441\u043D\u0443\u0454 \u0432\u0456\u0434\u043E\u0431\u0440\u0430\u0436\u0435\u043D\u043D\u044F \u0432\u0435\u0440\u0448\u0438\u043D \u0433\u0440\u0430\u0444\u0430 \u043D\u0430 \u0434\u0435\u044F\u043A\u0456 \u0442\u043E\u0447\u043A\u0438 \u043F\u043B\u043E\u0449\u0438\u043D\u0438 \u0456 \u0440\u0435\u0431\u0435\u0440 \u0433\u0440\u0430\u0444\u0430 \u043D\u0430 \u043F\u0440\u043E\u0441\u0442\u0456 \u043A\u0440\u0438\u0432\u0456 \u0443 \u043F\u043B\u043E\u0449\u0438\u043D\u0456, \u0442\u0430\u043A \u0449\u043E \u043A\u0456\u043D\u0446\u044F\u043C\u0438 \u043A\u0440\u0438\u0432\u0438\u0445 \u0454 \u0442\u043E\u0447\u043A\u0438, \u0449\u043E \u0432\u0456\u0434\u043F\u043E\u0432\u0456\u0434\u0430\u044E\u0442\u044C \u0432\u0435\u0440\u0448\u0438\u043D\u0430\u043C \u0440\u0435\u0431\u0440\u0430 \u0456 \u0434\u0432\u0456 \u0440\u0456\u0437\u043D\u0456 \u043A\u0440\u0438\u0432\u0456 \u043D\u0435 \u043C\u0430\u044E\u0442\u044C \u0441\u043F\u0456\u043B\u044C\u043D\u0438\u0445 \u0442\u043E\u0447\u043E\u043A, \u043E\u043A\u0440\u0456\u043C \u043C\u043E\u0436\u043B\u0438\u0432\u043E \u043A\u0456\u043D\u0446\u0435\u0432\u0438\u0445."@uk . . "\u041F\u043B\u0430\u043D\u0430\u0440\u043D\u0438\u0439 \u0433\u0440\u0430\u0444 \u2014 \u0433\u0440\u0430\u0444, \u044F\u043A\u0438\u0439 \u043C\u043E\u0436\u0435 \u0431\u0443\u0442\u0438 \u0437\u043E\u0431\u0440\u0430\u0436\u0435\u043D\u0438\u0439 \u043D\u0430 \u043F\u043B\u043E\u0449\u0438\u043D\u0456 \u0431\u0435\u0437 \u043F\u0435\u0440\u0435\u0442\u0438\u043D\u0443 \u0440\u0435\u0431\u0435\u0440. \u0413\u0440\u0430\u0444 \u0437\u043E\u0431\u0440\u0430\u0436\u0435\u043D\u0438\u0439 \u043D\u0430 \u043F\u043B\u043E\u0449\u0438\u043D\u0456 \u043D\u0430\u0437\u0438\u0432\u0430\u0454\u0442\u044C\u0441\u044F \u043F\u043B\u043E\u0441\u043A\u0438\u043C, \u044F\u043A\u0449\u043E \u0439\u043E\u0433\u043E \u0440\u0435\u0431\u0440\u0430 \u043D\u0435 \u043F\u0435\u0440\u0435\u0442\u0438\u043D\u0430\u044E\u0442\u044C\u0441\u044F. \u0413\u0440\u0430\u0444 \u043D\u0430\u0437\u0438\u0432\u0430\u0454\u0442\u044C\u0441\u044F \u043F\u043B\u0430\u043D\u0430\u0440\u043D\u0438\u043C, \u044F\u043A\u0449\u043E \u0432\u0456\u043D \u0456\u0437\u043E\u043C\u043E\u0440\u0444\u043D\u0438\u0439 \u0434\u0435\u044F\u043A\u043E\u043C\u0443 \u043F\u043B\u043E\u0441\u043A\u043E\u043C\u0443 \u0433\u0440\u0430\u0444\u0443. \u0422\u043E\u0431\u0442\u043E \u0456\u0441\u043D\u0443\u0454 \u0432\u0456\u0434\u043E\u0431\u0440\u0430\u0436\u0435\u043D\u043D\u044F \u0432\u0435\u0440\u0448\u0438\u043D \u0433\u0440\u0430\u0444\u0430 \u043D\u0430 \u0434\u0435\u044F\u043A\u0456 \u0442\u043E\u0447\u043A\u0438 \u043F\u043B\u043E\u0449\u0438\u043D\u0438 \u0456 \u0440\u0435\u0431\u0435\u0440 \u0433\u0440\u0430\u0444\u0430 \u043D\u0430 \u043F\u0440\u043E\u0441\u0442\u0456 \u043A\u0440\u0438\u0432\u0456 \u0443 \u043F\u043B\u043E\u0449\u0438\u043D\u0456, \u0442\u0430\u043A \u0449\u043E \u043A\u0456\u043D\u0446\u044F\u043C\u0438 \u043A\u0440\u0438\u0432\u0438\u0445 \u0454 \u0442\u043E\u0447\u043A\u0438, \u0449\u043E \u0432\u0456\u0434\u043F\u043E\u0432\u0456\u0434\u0430\u044E\u0442\u044C \u0432\u0435\u0440\u0448\u0438\u043D\u0430\u043C \u0440\u0435\u0431\u0440\u0430 \u0456 \u0434\u0432\u0456 \u0440\u0456\u0437\u043D\u0456 \u043A\u0440\u0438\u0432\u0456 \u043D\u0435 \u043C\u0430\u044E\u0442\u044C \u0441\u043F\u0456\u043B\u044C\u043D\u0438\u0445 \u0442\u043E\u0447\u043E\u043A, \u043E\u043A\u0440\u0456\u043C \u043C\u043E\u0436\u043B\u0438\u0432\u043E \u043A\u0456\u043D\u0446\u0435\u0432\u0438\u0445."@uk . . . . "Dans la th\u00E9orie des graphes, un graphe planaire est un graphe qui a la particularit\u00E9 de pouvoir se repr\u00E9senter sur un plan sans qu'aucune ar\u00EAte (ou arc pour un graphe orient\u00E9) n'en croise une autre. Autrement dit, ces graphes sont pr\u00E9cis\u00E9ment ceux que l'on peut plonger dans le plan, ou encore les graphes dont le nombre de croisements est nul. Les m\u00E9thodes associ\u00E9es \u00E0 ces graphes permettent de r\u00E9soudre des probl\u00E8mes comme l'\u00E9nigme des trois maisons et d'autres plus difficiles comme le th\u00E9or\u00E8me des quatre couleurs."@fr . "Grafo planar"@pt . . . . . . . . . . . "Graf planarny \u2013 graf, kt\u00F3ry mo\u017Cna narysowa\u0107 na p\u0142aszczy\u017Anie (i ka\u017Cdej powierzchni genusu 0) tak, by krzywe obrazuj\u0105ce kraw\u0119dzie grafu nie przecina\u0142y si\u0119 ze sob\u0105. Odwzorowanie grafu planarnego na p\u0142aszczyzn\u0119 o tej w\u0142asno\u015Bci nazywane jest jego rysunkiem p\u0142askim. Graf planarny o zbiorze wierzcho\u0142k\u00F3w i kraw\u0119dzi zdefiniowanym poprzez rysunek p\u0142aski nazywany jest grafem p\u0142askim."@pl . . . "En teor\u00EDa de grafos, un grafo plano (o planar seg\u00FAn referencias) es un grafo que puede ser dibujado en el plano sin que ninguna arista se cruce (una definici\u00F3n m\u00E1s formal puede ser que este grafo pueda ser \"incrustado\" en un plano). Los grafos K5 y el K3,3 son los grafos no planos minimales, lo cual nos permitir\u00E1n caracterizar el resto de los grafos no planos."@es . "Rovinn\u00FD graf"@cs . . . . . . . . . "In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points."@en . . "Graf pla"@ca . "Inom grafteori \u00E4r en plan\u00E4r graf en graf som kan i planet, det vill s\u00E4ga ritas p\u00E5 planet p\u00E5 ett s\u00E5dant s\u00E4tt att kanterna inte sk\u00E4r varandra, utan bara m\u00F6ts i noderna. En s\u00E5dan avbildning kallas en plan graf eller plan\u00E4r inb\u00E4ddning av grafen. En plan graf kan definieras som en plan\u00E4r graf med en avbildning av varje nod till en punkt i planet, och fr\u00E5n varje kant till en i planet, s\u00E5 att varje kurvas \u00E4ndpunkter \u00E4r de punkter som avbildas av kantens \u00E4ndnoder och s\u00E5 att alla kurvor \u00E4r disjunkta utom i \u00E4ndpunkterna. Varje graf som kan avbildas p\u00E5 ett plan kan avbildas p\u00E5 en sf\u00E4r och vice versa. En generalisering av plan\u00E4ra grafer \u00E4r grafer som kan ritas p\u00E5 en yta av ett givet genus. Med denna terminologi har plan\u00E4ra grafer grafgenus 0, eftersom planet (och sf\u00E4ren) har genus 0."@sv . . "24314"^^ . . "\u041F\u043B\u0430\u043D\u0430\u0301\u0440\u043D\u044B\u0439 \u0433\u0440\u0430\u0444 \u2014 \u0433\u0440\u0430\u0444, \u043A\u043E\u0442\u043E\u0440\u044B\u0439 \u043C\u043E\u0436\u043D\u043E \u0438\u0437\u043E\u0431\u0440\u0430\u0437\u0438\u0442\u044C \u043D\u0430 \u043F\u043B\u043E\u0441\u043A\u043E\u0441\u0442\u0438 \u0431\u0435\u0437 \u043F\u0435\u0440\u0435\u0441\u0435\u0447\u0435\u043D\u0438\u0439 \u0440\u0451\u0431\u0435\u0440 \u043D\u0435 \u043F\u043E \u0432\u0435\u0440\u0448\u0438\u043D\u0430\u043C. \u041A\u0430\u043A\u043E\u0435-\u043B\u0438\u0431\u043E \u043A\u043E\u043D\u043A\u0440\u0435\u0442\u043D\u043E\u0435 \u0438\u0437\u043E\u0431\u0440\u0430\u0436\u0435\u043D\u0438\u0435 \u043F\u043B\u0430\u043D\u0430\u0440\u043D\u043E\u0433\u043E \u0433\u0440\u0430\u0444\u0430 \u043D\u0430 \u043F\u043B\u043E\u0441\u043A\u043E\u0441\u0442\u0438 \u0431\u0435\u0437 \u043F\u0435\u0440\u0435\u0441\u0435\u0447\u0435\u043D\u0438\u044F \u0440\u0451\u0431\u0435\u0440 \u043D\u0435 \u043F\u043E \u0432\u0435\u0440\u0448\u0438\u043D\u0430\u043C \u043D\u0430\u0437\u044B\u0432\u0430\u0435\u0442\u0441\u044F \u043F\u043B\u043E\u0441\u043A\u0438\u043C \u0433\u0440\u0430\u0444\u043E\u043C. \u0418\u043D\u0430\u0447\u0435 \u0433\u043E\u0432\u043E\u0440\u044F, \u043F\u043B\u0430\u043D\u0430\u0440\u043D\u044B\u0439 \u0433\u0440\u0430\u0444 \u0438\u0437\u043E\u043C\u043E\u0440\u0444\u0435\u043D \u043D\u0435\u043A\u043E\u0442\u043E\u0440\u043E\u043C\u0443 \u043F\u043B\u043E\u0441\u043A\u043E\u043C\u0443 \u0433\u0440\u0430\u0444\u0443, \u0438\u0437\u043E\u0431\u0440\u0430\u0436\u0451\u043D\u043D\u043E\u043C\u0443 \u043D\u0430 \u043F\u043B\u043E\u0441\u043A\u043E\u0441\u0442\u0438 \u0442\u0430\u043A, \u0447\u0442\u043E \u0435\u0433\u043E \u0432\u0435\u0440\u0448\u0438\u043D\u044B \u2014 \u044D\u0442\u043E \u0442\u043E\u0447\u043A\u0438 \u043F\u043B\u043E\u0441\u043A\u043E\u0441\u0442\u0438, \u0430 \u0440\u0451\u0431\u0440\u0430 \u2014 \u043A\u0440\u0438\u0432\u044B\u0435 \u043D\u0430 \u043F\u043B\u043E\u0441\u043A\u043E\u0441\u0442\u0438, \u043A\u043E\u0442\u043E\u0440\u044B\u0435 \u0435\u0441\u043B\u0438 \u0438 \u043F\u0435\u0440\u0435\u0441\u0435\u043A\u0430\u044E\u0442\u0441\u044F \u043C\u0435\u0436\u0434\u0443 \u0441\u043E\u0431\u043E\u0439, \u0442\u043E \u0442\u043E\u043B\u044C\u043A\u043E \u043F\u043E \u0432\u0435\u0440\u0448\u0438\u043D\u0430\u043C. \u041E\u0431\u043B\u0430\u0441\u0442\u0438, \u043D\u0430 \u043A\u043E\u0442\u043E\u0440\u044B\u0435 \u0433\u0440\u0430\u0444 \u0440\u0430\u0437\u0431\u0438\u0432\u0430\u0435\u0442 \u043F\u043B\u043E\u0441\u043A\u043E\u0441\u0442\u044C, \u043D\u0430\u0437\u044B\u0432\u0430\u044E\u0442\u0441\u044F \u0435\u0433\u043E \u0433\u0440\u0430\u043D\u044F\u043C\u0438. \u041D\u0435\u043E\u0433\u0440\u0430\u043D\u0438\u0447\u0435\u043D\u043D\u0430\u044F \u0447\u0430\u0441\u0442\u044C \u043F\u043B\u043E\u0441\u043A\u043E\u0441\u0442\u0438 \u2014 \u0442\u043E\u0436\u0435 \u0433\u0440\u0430\u043D\u044C, \u043D\u0430\u0437\u044B\u0432\u0430\u0435\u043C\u0430\u044F \u0432\u043D\u0435\u0448\u043D\u0435\u0439 \u0433\u0440\u0430\u043D\u044C\u044E. \u041B\u044E\u0431\u043E\u0439 \u043F\u043B\u043E\u0441\u043A\u0438\u0439 \u0433\u0440\u0430\u0444 \u043C\u043E\u0436\u0435\u0442 \u0431\u044B\u0442\u044C \u0441\u043F\u0440\u044F\u043C\u043B\u0451\u043D, \u0442\u043E \u0435\u0441\u0442\u044C \u043F\u0435\u0440\u0435\u0440\u0438\u0441\u043E\u0432\u0430\u043D \u043D\u0430 \u043F\u043B\u043E\u0441\u043A\u043E\u0441\u0442\u0438 \u0442\u0430\u043A, \u0447\u0442\u043E \u0432\u0441\u0435 \u0435\u0433\u043E \u0440\u0451\u0431\u0440\u0430 \u0431\u0443\u0434\u0443\u0442 \u043E\u0442\u0440\u0435\u0437\u043A\u0430\u043C\u0438 \u043F\u0440\u044F\u043C\u044B\u0445."@ru . "\u5E73\u9762\u30B0\u30E9\u30D5"@ja . . . . . . . . "\u0645\u062E\u0637\u0637 \u0645\u0633\u062A\u0648"@ar . "Dans la th\u00E9orie des graphes, un graphe planaire est un graphe qui a la particularit\u00E9 de pouvoir se repr\u00E9senter sur un plan sans qu'aucune ar\u00EAte (ou arc pour un graphe orient\u00E9) n'en croise une autre. Autrement dit, ces graphes sont pr\u00E9cis\u00E9ment ceux que l'on peut plonger dans le plan, ou encore les graphes dont le nombre de croisements est nul. Les m\u00E9thodes associ\u00E9es \u00E0 ces graphes permettent de r\u00E9soudre des probl\u00E8mes comme l'\u00E9nigme des trois maisons et d'autres plus difficiles comme le th\u00E9or\u00E8me des quatre couleurs."@fr . "\uD3C9\uBA74 \uADF8\uB798\uD504"@ko . "\u5E73\u9762\u56FE (\u56FE\u8BBA)"@zh . "Rovinn\u00FD graf (t\u00E9\u017E plan\u00E1rn\u00ED graf) je graf, pro kter\u00FD existuje takov\u00E9 rovinn\u00E9 nakreslen\u00ED, \u017Ee se \u017E\u00E1dn\u00E9 dv\u011B hrany nek\u0159\u00ED\u017E\u00ED."@cs . . . . . . . . . "Em Teoria dos Grafos, um grafo planar \u00E9 um grafo que pode ser imerso no plano de tal forma que suas arestas n\u00E3o se cruzem, esta \u00E9 uma idealiza\u00E7\u00E3o abstrata de um grafo plano, um grafo plano \u00E9 um grafo planar que foi desenhado no plano sem o cruzamento de arestas. Observe os dois exemplos, ambos isomorfos entre si, ambos planares, por\u00E9m apenas o que \u00E9 desenhado sem cruzamento de arestas \u00E9 um grafo plano."@pt . . . "En teor\u00EDa de grafos, un grafo plano (o planar seg\u00FAn referencias) es un grafo que puede ser dibujado en el plano sin que ninguna arista se cruce (una definici\u00F3n m\u00E1s formal puede ser que este grafo pueda ser \"incrustado\" en un plano). Los grafos K5 y el K3,3 son los grafos no planos minimales, lo cual nos permitir\u00E1n caracterizar el resto de los grafos no planos. Todo grafo plano puede ser dibujado sobre la esfera, y viceversa. Una generalizaci\u00F3n de los grafos planos son grafos dibujados e incrustados sobre superficies de g\u00E9nero arbitrario. En esta terminolog\u00EDa, los grafos planos tienen g\u00E9nero 0, por ser el plano y la esfera de g\u00E9nero 0."@es . . "\u0641\u064A \u0627\u0644\u0645\u062E\u0637\u0637\u0627\u062A\u060C \u0627\u0644\u0645\u062E\u0637\u0651\u0637 \u0627\u0644\u0645\u0633\u062A\u0648\u064A \u0647\u0648 \u0627\u0644\u0645\u062E\u0637\u0651\u0637 \u0627\u0644\u0630\u064A \u064A\u0642\u0628\u0644 \u062A\u0645\u062B\u064A\u0644\u0627 \u0641\u064A \u0627\u0644\u0645\u0633\u062A\u0648\u0649\u060C \u0628\u062D\u064A\u062B \u0644\u0627 \u064A\u062A\u0642\u0627\u0637\u0639 \u0623\u064A \u062D\u0631\u0641\u064A\u0646 \u0645\u0646 \u0627\u0644\u0645\u062E\u0637\u0651\u0637."@ar . . . . . . . "Rovinn\u00FD graf (t\u00E9\u017E plan\u00E1rn\u00ED graf) je graf, pro kter\u00FD existuje takov\u00E9 rovinn\u00E9 nakreslen\u00ED, \u017Ee se \u017E\u00E1dn\u00E9 dv\u011B hrany nek\u0159\u00ED\u017E\u00ED."@cs . . . . . . . . . . . . . . . . . . . "Ein planarer oder pl\u00E4ttbarer Graph ist in der Graphentheorie ein Graph, der auf einer Ebene, mit Punkten f\u00FCr die Knoten und Linien f\u00FCr die Kanten, dargestellt werden kann, sodass sich keine Kanten schneiden."@de . . . . . . . . . . . . "Grafo plano"@es . . . . . "1123061420"^^ . . . . . . . "\u0641\u064A \u0627\u0644\u0645\u062E\u0637\u0637\u0627\u062A\u060C \u0627\u0644\u0645\u062E\u0637\u0651\u0637 \u0627\u0644\u0645\u0633\u062A\u0648\u064A \u0647\u0648 \u0627\u0644\u0645\u062E\u0637\u0651\u0637 \u0627\u0644\u0630\u064A \u064A\u0642\u0628\u0644 \u062A\u0645\u062B\u064A\u0644\u0627 \u0641\u064A \u0627\u0644\u0645\u0633\u062A\u0648\u0649\u060C \u0628\u062D\u064A\u062B \u0644\u0627 \u064A\u062A\u0642\u0627\u0637\u0639 \u0623\u064A \u062D\u0631\u0641\u064A\u0646 \u0645\u0646 \u0627\u0644\u0645\u062E\u0637\u0651\u0637."@ar . . . . . . . . . . . . "\u041F\u043B\u0430\u043D\u0430\u0440\u043D\u0438\u0439 \u0433\u0440\u0430\u0444"@uk . . . . "Ein planarer oder pl\u00E4ttbarer Graph ist in der Graphentheorie ein Graph, der auf einer Ebene, mit Punkten f\u00FCr die Knoten und Linien f\u00FCr die Kanten, dargestellt werden kann, sodass sich keine Kanten schneiden."@de . . "Graf planarny \u2013 graf, kt\u00F3ry mo\u017Cna narysowa\u0107 na p\u0142aszczy\u017Anie (i ka\u017Cdej powierzchni genusu 0) tak, by krzywe obrazuj\u0105ce kraw\u0119dzie grafu nie przecina\u0142y si\u0119 ze sob\u0105. Odwzorowanie grafu planarnego na p\u0142aszczyzn\u0119 o tej w\u0142asno\u015Bci nazywane jest jego rysunkiem p\u0142askim. Graf planarny o zbiorze wierzcho\u0142k\u00F3w i kraw\u0119dzi zdefiniowanym poprzez rysunek p\u0142aski nazywany jest grafem p\u0142askim."@pl . . . . . . . . . . . . . . . "Nella teoria dei grafi si definisce grafo planare un grafo che pu\u00F2 essere raffigurato in un piano in modo che non si abbiano archi che si intersecano. Ad esempio sono planari i seguenti grafi: Il secondo pu\u00F2 essere raffigurato senza archi che si intersecano spostando uno degli archi dati da una diagonale al di fuori del perimetro del quadrato. Vi sono invece grafi che posseggono solo raffigurazioni piane nelle quali si hanno coppie di archi che si intersecano. Le due seguenti figure forniscono raffigurazioni di due grafi non planari: K5 K3,3"@it . . . . . . "Inom grafteori \u00E4r en plan\u00E4r graf en graf som kan i planet, det vill s\u00E4ga ritas p\u00E5 planet p\u00E5 ett s\u00E5dant s\u00E4tt att kanterna inte sk\u00E4r varandra, utan bara m\u00F6ts i noderna. En s\u00E5dan avbildning kallas en plan graf eller plan\u00E4r inb\u00E4ddning av grafen. En plan graf kan definieras som en plan\u00E4r graf med en avbildning av varje nod till en punkt i planet, och fr\u00E5n varje kant till en i planet, s\u00E5 att varje kurvas \u00E4ndpunkter \u00E4r de punkter som avbildas av kantens \u00E4ndnoder och s\u00E5 att alla kurvor \u00E4r disjunkta utom i \u00E4ndpunkterna. Varje graf som kan avbildas p\u00E5 ett plan kan avbildas p\u00E5 en sf\u00E4r och vice versa."@sv . . . "Em Teoria dos Grafos, um grafo planar \u00E9 um grafo que pode ser imerso no plano de tal forma que suas arestas n\u00E3o se cruzem, esta \u00E9 uma idealiza\u00E7\u00E3o abstrata de um grafo plano, um grafo plano \u00E9 um grafo planar que foi desenhado no plano sem o cruzamento de arestas. Aparentemente o estudo da planaridade de um grafo \u00E9 uma quest\u00E3o topol\u00F3gica que se baseia em resultados como o Teorema da Curva de Jordan que de forma simplificada diz que uma curva fechada simples no plano divide-o em duas partes, apesar deste ser um crit\u00E9rio muito importante, \u00E9 natural o questionamento se h\u00E1 algum resultado combinat\u00F3rio que caracterize um grafo plano. Observe os dois exemplos, ambos isomorfos entre si, ambos planares, por\u00E9m apenas o que \u00E9 desenhado sem cruzamento de arestas \u00E9 um grafo plano."@pt . .