vertices with degree 4 or more. Vertex G can be removed so that edges AG, GF become a single edge AF .
, GF is formed from AF by subdivision.
AG
This leaves a graph with six vertices:A, B, C , D, E and F . If K is a subgraph, these vertices can be split into two sets of three, where there is an edge joining each vertex in the first set to each vertex in the second set. 3, 3
PL E
There is no edge C E or C F so {C , E, F } would need to be taken together as a set, with {A, B, D} as the other set. But there is no edge AE so K is not a subgraph. 3, 3
Hence the graph is planar.
The graph does not contain a subgraph that is a subdivision of K or K . 5
3,3
b Move edge AC so that it passes outside of B (and does not cross other edges) and the edge DF so that it passes outside of E (and does not cross other edges).
M
c Edge C E is added and vertex G is removed so that edges AG , GF become a single edge AF . Vertices A, C , E and F each have degree 4. is now a subgraph, with the two sets as {A, B, E} and K {C , D, F } .
SA
3, 3
There are edges from each of A, B and E to each of C , D and F .
So the graph is non-planar.
Two graphs might look very different from one another but have the same structure, in the sense that they can be transformed into one another by relabelling and moving the vertices and edges around without cutting any of the edges.
Key point 1.20
Two graphs are isomorphic if they have the same structure.
Tip From the ancient Greek: isos = equal, morphe = form or shape.
Original material © Cambridge University Press 2018