What is Graphs: Definition and 654 Discussions

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.

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  1. M

    Explanation of graphs involving capacitors (charging/discharging)

    Homework Statement I've tested the circuit above, when the switch is in the 2nd position (not the one on the picture) and got the below graph from the plotted data I received. The capacitor C1 has been charged to 4V, and will start to discharge through R3. I'll have to explain...
  2. S

    MHB Sketching graphs in polar coordinates

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  3. datafiend

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  4. G

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  5. sbstratos79

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  6. T

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  7. G

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  8. G

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  9. A

    MHB Class 1 vs class 2 of graphs

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  10. C

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  11. E

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  12. S

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  13. C

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  14. C

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  15. M

    MHB Exercise about connected graphs

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  16. S

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  17. T

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  18. LarryS

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  19. jk22

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  20. 1

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  21. 1

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  22. P

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  23. F

    MHB Strong induction hypothesis for statement about connected graphs

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  24. N

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  25. O

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  26. N

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  27. J

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  28. A

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  29. X

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  30. D

    Engineering How to plot output graphs of circuits involving diodes and capacitors?

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  31. mathmaniac1

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  32. Fernando Revilla

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  33. P

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  34. S

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  35. P

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  36. O

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  37. P

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  38. I

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  39. M

    MHB Polar Graphs and a question about their poles

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  40. K

    Graphs of Frequency for Car Moving in a Circle

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  41. M

    Sketch the graphs of the functions - Calculus question

    Homework Statement Sketch the graphs of the functions. Indicate intervals on which the function is increasing, decreasing, concave up, or concave down; indicate relative maximum points, relative minimum points, points of inflection, horizontal asympototes, vertical asymptotes, symmetry, and...
  42. M

    Sketch the graphs of the functions - Calculus question

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  43. P

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  44. L

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  45. R

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  46. B

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  47. B

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  48. R

    The slopes of these graphs represent what? (Magnetic Fields)

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  49. B

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  50. P

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