By Xu B.G.
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Additional resources for A 3-color Theorem on Plane Graphs without 5-circuits
Conversely, assume G to have a T -embedding G with the set F of inner facial cycles and outer facial cycle F0 . We have to show that the algorithm always generates a cycle F0 as required. Let for any C, InC , OutC ∈ T FC denote the sum of those cycles in F that sum up to C. FC and (F ∪ F0 ) \ FC are exactly the components UC∗ , WC∗ of G∗ − C ∗ and, by Theorem 1, F ∈FC F ∗ contains all edges of InC but no edge of OutC . Thus, the algorithm defines ZC ∗ to be (F ∪ F0 ) \ FC . At least F0 belongs to each ZC such that the algorithm ∗ .
3. 4. Determine the set B of blocks of H and the blockgraph Hb of H. Determine the sets C-InB,B , C-OutB,B , C-InB,Li and C-OutB,Li , the constraints TB := TH |B and the relation ≺. If ≺ is cyclic, stop. Find a block B0 maximal in ≺ such that order blocks(B ∪ LH ) results in a labelled tree R using the extention ≺ of ≺ by B ≺ B iff B0 , . . , B, q, B is a path in Hb . If this fails, stop. e. for any edge from B to B labelled with F and for the cutvertex p with path B, p, B in Hb embed B into face F at p of B.
Thus step 2 of order blocks can be done in (O(n2 + nl) + O((n + l)kj ) + O(p (nj , kj , tj , lj , xj ))) = O (n2 + nl) + (n + l)kj + p nj , kj , tj , lj , xj = O n3 + n2 l + nk + kl + p (n, k, t, l, n + l) . Since for a fixed B0 the number of calls is bounded by n, the time needed by the call order blocks(B ∪ LH ) in step 3 of algorithm reduce to blocks can be estimated by n O(n) + O n3 + n2 l + nk + kl + p (n, k, t, l, n + l) , that is O n4 + n3 l + n2 k + nkl + n · p (n, k, t, l, n + l) . In the worst case, however, all maximal blocks B0 in ≺ must be checked.