a connecred graph has 3 regionsand 8 edges find the number of vertices
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A connected graph has 3 regions and 8 edges find number of vertices Get the answers you need, now!
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When is it possible to draw a graph so that none of the edges cross? If this is possible, we say the graph is planar (since you can draw it on the plane).
Notice that the definition of planar includes the phrase “it is possible to.” This means that even if a graph does not look like it is planar, it still might be. Perhaps you can redraw it in a way in which no edges cross. For example, this is a planar graph:
That is because we can redraw it like this:
The graphs are the same, so if one is planar, the other must be too. However, the original drawing of the graph was not a planar representation of the graph.
When a planar graph is drawn without edges crossing, the edges and vertices of the graph divide the plane into regions. We will call each region a face. The graph above has 3 faces (yes, we do include the “outside” region as a face). The number of faces does not change no matter how you draw the graph (as long as you do so without the edges crossing), so it makes sense to ascribe the number of faces as a property of the planar graph.
WARNING: you can only count faces when the graph is drawn in a planar way. For example, consider these two representations of the same graph:
If you try to count faces using the graph on the left, you might say there are 5 faces (including the outside). But drawing the graph with a planar representation shows that in fact there are only 4 faces.
There is a connection between the number of vertices (
v
), the number of edges (
e
) and the number of faces (
f
) in any connected planar graph. This relationship is called Euler's formula.
Euler's Formula for Planar Graphs.
For any (connected) planar graph with
v
vertices,
e
edges and
f
faces, we have