(3,k)-Factor-Critical Graphs and Toughness by Shi M., Yuan X., Cai M.

By Shi M., Yuan X., Cai M.

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The result is a quadrilateralization of P, establishing that P is reducible. Case 2 (x is right of b (Figs. ). The two situations illustrated in Figs. 10c are handled with the same reduction; we will use the case where x is above e (Fig. 10b) as illustrated. The replacements made are the same as in Case 1, but the argument is a bit different. Perform the same alterations as in Case 1, resulting, when P has no holes, in Px and P2 as illustrated in Fig. 12a. As in Case 1, P' is smaller, and so can be quadrilateralized.

2. KAHN, KLAWE, KLEITMAN PROOF 43 n«(E) n(E) Fig. 14. The chain induced by n(E) leads to tab(E). symmetric it must be the case that ni+2(E) falls between or at the heights of n\E) and ni+1(E): otherwise ni+\E) would map to nl{E), which would be closer and visible by symmetry. Therefore, the sequence cannot terminate with n\E) = nJ\E) with j-i>2; it must terminate with nk(E) = nk+2(E) for some k, as illustrated in Fig. 14. As we observed above, this implies that nk{E) and nk+1(E) are neighbors. 4.

End Pop. Push vt. Push ph end else if Pi is adjacent to vt then {Fig. 21b} begin while t > 0 and vt is not reflex do begin Draw diagonalp t -*v t -xPop. end Push pt. end The stack contents as the algorithm processes the polygon shown in Fig. 2. , is adjacent to both v0 and vt. Rather than insert special code to handle this case, we permit the redundancy of drawing one diagonal [(17,16) in the above example] that is superfluous. We now establish that each diagonal output by the algorithm lies entirely within the polygon.

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