Vertices of Attachment Question

[cept]

I am currently reading "Graph Theory" by Tutte.

Just to make sure I understand the definition of vertex of attachment:

By Tutte's definition, "A vertex of attachment of H in G is a vertex of H that is incident with some edge of G that is not an edge of H. We write W(G, H) for the set of vertices of attachment of H in G..."

So if have a graph G(V) = {A, B, C, D} and G(E) = {a, b, c} drawn below

A ..a... B ...b ...C
0 ----- 0 -------0
|
|c
|
0
D

and subgraph H of G, where H(V) = {A, B} and H(E) = {a}

A.. a ...B
0-------0

then the set of vertices of attachment is: W(G, H) = { B }

Tutte futher writes:"If H and K are subgraphs of G, let u write Q(G; H, K) for the set of all vertices x of G such that x belongs to W(G, H) and W(G, K) but not to W(G, H U K). Alternatively we may characterize x as incident with an edge of H not belonging to K and with an edge of K not belonging to H, but not incident with any edge of G outside both E(H) and E(K)."

I cannot picture this last statement.

QUESTION:
How can there be vertices of attachment that belong to W(G, H) and W(G, K), but not their union W(G, H U K)?? Thanks in advance,

Carlos

 

[Valkarie]

To answer your question, consider the following example. Let's say we have a graph G with vertices A, B, C, and D, and edges a, b, and c. Let's also say that we have two subgraphs H and K of G such that H is the subgraph containing vertices A and B, and edges a and b, and K is the subgraph containing vertices B and C, and edges b and c. In this case, W(G, H) = {B} and W(G, K) = {B}, so W(G, H U K) = {B}. However, there are no vertices in G that belong to both W(G, H) and W(G, K) but not W(G, H U K). This is because any vertex that belongs to W(G, H) must also belong to W(G, H U K), and similarly for W(G, K). So Q(G; H, K) = ∅.In other words, for a vertex x of G to belong to W(G, H) and W(G, K) but not W(G, H U K), x must be incident with an edge of H not belonging to K, and an edge of K not belonging to H, but not incident with any edge of G outside both E(H) and E(K).