Similarly, we can show that there is a value x + y ? t 2 < x + 2y such that ?(S 2 (t 2 )) ? 2c(c + 1) + 1. Denote by T (t 1 , t 2 ) the vertices of M (t 1 , t 2 ) on level less than d 0 We claim that ?(T (t 1 , t 2 )) ? 3. First, T (t 1 , t 2 ) can contain at most 3 vertices on each level 3·2 d?d 0 ? 3y ? t 2 ?t 1 , contradicting the assumption on the ?-values. Every v i,j ? T (t 1 , t 2 ) has a descendant v i ? ,j ? ? T (t 1 , t 2 ) for every i < i ? < d 0 , namely v i ? ,j ? with j ? = j2 i ? ?i . Thus by covering the at most 3 vertices of T (t 1 , t 2 ) on level d 0 ? 1 by at most 3 large edges, we can cover T (t 1 , t 2 ), and ?(T (t 1 , t 2 )) ? 3 follows. Define S := S(t 1 )?S(t 2 )?T (t 1 , t 2 ) Clearly, ?(S) ? 2(2c(c+1)+1)+3 = 4c(c+1)+5. We show that S separates M (t 1 , t 2 ) from the rest of the vertices. Suppose that v i,j , v i ? ,j ? ? S are adjacent vertices such that v i,j ? M (t 1 , t 2 ) and v i ? ,j ? ? M (t 1 , t 2 ) We have i ? d 0 (otherwise v i,j ? T (t 1 , t 2 )), hence v i,j ? A(t 1 ) If ?(v i ? ,j ? ) < t 1 , then v i ? ,j ? ? S(t 1 ) ? S, a contradiction. Moreover, if ?(v i ? ,j ? ) > t 2 , then i ? ? d 0 as v i,j and v i ? ,j ? are not neighbors if ?(v i,j ) < ?(v i ? ,j ? ) and i > i ? . Thus v i ? ,j ? ? A(t 2 ) and v i,j ? S(t 2 ) ? S, a contradiction, By the definition of x and y, we have ?(M (t 1 , t 2 )?W ) ? ?(M (x, x+y)?W ) ? ?(W )/4. To complete the proof that ?(W ? C) ? 3?(W )/4 for every component C of H(d, c) \ S, we show that ?(M (t 1 , t 2 ) ? W ) ? 3?(W )/4: as we have seen that every such component C is fully contained in either M (t 1 , t 2 ) or V \ M (t 1 this means that no component C can have ?(W ? C) > 3?(W )/4. Since x? t 1 < y, the minimality of y implies ?(M (t 1 , x)? W ) ? ?(W )/4. Similarly, it follows from t 2 ?(x+y) < y that ?(M (x+y, t 2 )?W ) ? ?(W )/4. Now ?(M (t 1 , t 2 )? W ) = ?(M (t 1 , x)? W )+ ?(M (x, x+ y)? W )+ ?(M (x+ y, t 2 )? W ) ? 3 4 ?(W ) ,

Constraint satisfaction problems with succinctly specified relations, Downey and M. R. Fellows. Parameterized Complexity, 1999. ,

Parameterized Complexity Theory Complexity of k-tree structured constraint satisfaction problems, Proc. of AAAI-90, pp.4-9, 1990. ,

Fixed-parameter complexity in AI and nonmonotonic reasoning, Artificial Intelligence, vol.138, issue.1-2, pp.55-86, 2002. ,

DOI : 10.1016/S0004-3702(02)00182-0

The complexity of homomorphism and constraint satisfaction problems seen from the other side, J. ACM, vol.54, issue.11, 2007. ,

Constraint solving via fractional edge covers, SODA '06, pp.289-298, 2006. ,

Which problems have strongly exponential complexity?, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280), pp.512-530, 2001. ,

DOI : 10.1109/SFCS.1998.743516

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.29.8482

Approximating fractional hypertree width, SODA '09, pp.902-911, 2009. ,

DOI : 10.1145/1721837.1721845

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.215.3266

Can you beat treewidth?, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07), pp.169-179, 2007. ,

DOI : 10.1109/FOCS.2007.27