Hence Duplicator Has a Distance Respecting K-round Winning Strategy on P I

and p j on one hand and on p i+1 and p j+1 on the other hand. Because all paths are pairwise disjoint, these winning strategies can be be combined to winning strategies on the two graphs consisting of p i ; (v i ; v i+1); p i+1 and p i ; (v i ; v j+1); p j+1 and on the two graphs consisting of p j ; (v j ; v j+1); p j+1 and p j ; (v j ; v i+1); p i+1. Let G 0 be deened as in (b) above, by deleting the edges (v i ; v i+1) and (v j ; v j+1) and inserting new edges (v i ; v j+1) and (v j ; v i+1). Let H 1 be the set of vertices fv i ; v j ; v i+1 ; v j+1 g and let H 0 1 be the respective set of vertices of G 0. We have to show that conditions (i) and (ii) of Theorem 5 are fulllled. But (i) was already shown above, and (ii) follows immediately from the construction. Hence, the statement now follows from Theorem 5. u t The graph G in the proof of Theorem 9 is hamiltonian and consists of a single cycle in contrast to G 0. Therefore we can conclude the following corollary. Corollary 10. Let f : IN ! IN with f(n) = n o(1). Neither the class of Hamilto-nian graphs nor the class of graphs that consist of a single cycle are in MonNP even in the presence of built-in relations of degree at most f(n). References AF90] M. Ajtai and R. Fagin. Reachability is harder for directed than for undirected nite graphs. l d+1 vertices and all vertices which have fB 1 ; : : : ; B m g-distance at most d from v i are included in p i. We note that, in contrast to the case of a single successor relation, here the paths p i are, in general, of diierent size. Again we construct E by concatenating p 1 ; p 2 ; p 3 ; p 4 ; : : :, adding all remaining vertices (i.e. those that are not contained in any p i) at the end of the path and closing the cycle by connecting the rst and the last vertex. For every odd number i …