PlanarGraphPlot[{{x1, y1}, {x2, y2}, ...}] plots the Delaunay triangulation of the points {{x1, y1}, ...}.
PlanarGraphPlot[{{x1, y1}, {x2, y2}, ...}, g]] plots the points {{x1, y1}, ...} as well as the planar graph described by the vertex adjacency list or circuit g.
The planar graph g may be a vertex adjacency list of the form {{i, {i1, i2, ...}}, ...} or a single circuit of the form {i, j, ...}, where i refers to the point {xi, yi}.
PlanarGraphPlot has the same options as Graphics, with the following addition:
Valid settings for LabelPoints are True and False. With the setting LabelPoints->True, each point is labeled according to its position in the input list.