Graphs with one hole and competition number one
전체 글
Then it is obvious that D is acyclic, V (C(D)) = V (G) ∪ I k0
Lemma 7. If a path u = u 0 u 1 · · · u q = v, denoted by P ∗ , is a shortest hole-avoiding path from u to v for some u, v ∈ N , then for some q ∗ ∈ {0, . . . , q}, N ∗ (u 0 )∪ · · · ∪N ∗ (u q ) = N ∗ (u q∗
∪N ∗ (u r ) = N ∗ (u r∗
Lemma 8. For any r ∈ {1, . . . , s}, there exists a vertex z r in N ∩ V (Q r ) such that ∪ w∈N ∩V (Qr
Proof. Let N ∩ V (Q r ) = {w 1 , . . . , w nr
path in Q r from w 1 to w 2 . Let P 1 = u 0 u 1 · · · u q−1 u q (u 0 = w 1 , u q = w 2 ) be the shortest among such paths. By Lemma 7, there is a vertex w ψ(2) ∈ {u 0 , u 1 , . . . , u q } such that N ∗ (w ψ(2) ) = N ∗ (u 0 ) ∪ · · · ∪ N ∗ (u q ) where ψ is a mapping on {1, 2, . . . , n r }. By Lemma 6, {u 0 , u 1 , . . . , u q } ⊆ N ∩V (Q r ) and so w ψ(2) ∈ N ∩ V (Q r ). Since {w 1 , w 2 } ⊂ {u 0 , u 1 , . . . , u q }, it is true that N ∗ (w 1 ) ∪ N ∗ (w 2 ) ⊆ N ∗ (w ψ(2) ). By applying a similar argument for vertices w ψ(2) and w 3 , we can show that there exists vertex w ψ(3) in N ∩ V (Q r ) such that N ∗ (w ψ(2) ) ∪ N ∗ (w 3 ) ⊆ N ∗ (w ψ(3) ). Then clearly N ∗ (w 1 ) ∪ N ∗ (w 2 ) ∪ N ∗ (w 3 ) ⊆ N ∗ (w ψ(3) ). We repeat the argument as above until we find vertex w ψ(nr
N ∗ (z r ) = ∪ u∈V (Qr
difficult to check that closed walk formed by paths P 1 −1 , xy, P 2 , and v j v i
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