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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
00
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
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
00
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