Was not sure how Euler Convex Polyhedron Formula is derived and found this link with detailed explanation for it: http://www.ams.org/publicoutreach/feature-column/fcarc-eulers-formula.
I found it easier to understand with the visualization of diagrams provided in this link.
Basically, first imagine the polyhedron as a planar graph, remove edges so that the graph becomes a connected spanning tree. Let the number of removed edges be Er and the remaining edges be Et. For any tree, V = E + 1 and therefore V = Et + 1. On the other hand, Er = F - 1 as each of these edges helps complete each of the faces, except the infinite face (resulting from the spread of the polyhedron into a planar graph). Therefore, E = Et + Er = V - 1 + F - 1 = V + F - 2.
Was not sure how Euler Convex Polyhedron Formula is derived and found this link with detailed explanation for it: http://www.ams.org/publicoutreach/feature-column/fcarc-eulers-formula.
I found it easier to understand with the visualization of diagrams provided in this link.
Basically, first imagine the polyhedron as a planar graph, remove edges so that the graph becomes a connected spanning tree. Let the number of removed edges be Er and the remaining edges be Et. For any tree, V = E + 1 and therefore V = Et + 1. On the other hand, Er = F - 1 as each of these edges helps complete each of the faces, except the infinite face (resulting from the spread of the polyhedron into a planar graph). Therefore, E = Et + Er = V - 1 + F - 1 = V + F - 2.