The @math -colouring reconfiguration problem asks whether, for a given graph @math , two proper @math -colourings @math and @math of @math , and a positive integer @math , there exists a sequence of at most @math proper @math -colourings of @math which starts with @math and ends with @math and where successive colourings in the sequence differ on exactly one vertex of @math . We give a complete picture of the parameterized complexity of the @math -colouring reconfiguration problem for each fixed @math when parameterized by @math . First we show that the @math -colouring reconfiguration problem is polynomial-time solvable for @math , settling an open problem of Cereceda, van den Heuvel and Johnson. Then, for all @math , we show that the @math -colouring reconfiguration problem, when parameterized by @math , is fixed-parameter tractable (addressing a question of Mouawad, Nishimura, Raman, Simjour and Suzuki) but that it has no polynomial kernel unless the polynomial hierarchy collapses.
Finally, the study of the diameter of @math raises interesting questions. @cite_12 it is shown that every component of @math has diameter polynomial (in fact quadratic) in the size of @math . On the other hand, for @math , explicit constructions @cite_5 are given of graphs @math for which @math has at least one component with diameter exponential in the size of @math . It is known that if @math is a @math -degenerate graph then @math is connected and it is conjectured that in this case @math has diameter polynomial in the size of @math @cite_3 ; for graphs of treewidth @math the conjecture has been proved in the affirmative @cite_14 .
@cite_5 @cite_14 @cite_3 @cite_12
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We show that, over Q, if an n-variate polynomial of degree d = nO(1) is computable by an arithmetic circuit of size s (respectively by an arithmetic branching program of size s) then it can also be computed by a depth three circuit (i.e. a ΣΠΣ-circuit) of size exp(O(√(d log n log d log s))) (respectively of size exp(O(√(d log n log s))). In particular this yields a ΣΠΣ circuit of size exp(O(√(d log d))) computing the d × d determinant Detd. It also means that if we can prove a lower bound of exp(omega(√(d log d))) on the size of any ΣΠΣ-circuit computing the d × d permanent Permd then we get super polynomial lower bounds for the size of any arithmetic branching program computing Permd. We then give some further results pertaining to derandomizing polynomial identity testing and circuit lower bounds. The ΣΠΣ circuits that we construct have the property that (some of) the intermediate polynomials have degree much higher than d. Indeed such a counterintuitive construction is unavoidable - it is known that in any ΣΠΣ circuit C computing either Detd or Perm_d, if every multiplication gate has fanin at most d (or any constant multiple thereof) then C must have size at least exp(Ω(d)). We derive a new upper bound on the diameter of a polyhedron @math P = x ? R n : A x ≤ b , where @math A ? Z m × n . The bound is polynomial in @math n and the largest absolute value of a sub-determinant of @math A , denoted by @math Δ . More precisely, we show that the diameter of @math P is bounded by @math O ( Δ 2 n 4 log n Δ ) . If @math P is bounded, then we show that the diameter of @math P is at most @math O ( Δ 2 n 3.5 log n Δ ) . For the special case in which @math A is a totally unimodular matrix, the bounds are @math O ( n 4 log n ) and @math O ( n 3.5 log n ) respectively. This improves over the previous best bound of @math O ( m 16 n 3 ( log m n ) 3 ) due to Dyer and Frieze (Math Program 64:1---16, 1994). We consider the following problem: Given positive integers k and D, what is the maximum diameter of the graph obtained by deleting k edges from a graph G with diameter D, assuming that the resulting graph is still connected? For undirected graphs G we prove an upper bound of (k + 1)D and a lower bound of (k + 1)D − k for even D and of (k + 1)D − 2k + 2 for odd D ⩾ 3. For the special cases of k = 2 and k = 3, we derive the exact bounds of 3D − 1 and 4D − 2, respectively. For D = 2 we prove exact bounds of k + 2 and k + 3, for k ⩽ 4 and k = 6, and k = 5 and k ⩾ 7, respectively. For the special case of D = 1 we derive an exact bound on the resulting maximum diameter of order θ(√k). For directed graphs G, the bounds depend strongly on D: for D = 1 and D = 2 we derive exact bounds of θ(√k) and of 2k + 2, respectively, while for D ⩾ 3 the resulting diameter is in general unbounded in terms of k and D. Finally, we prove several related problems NP-complete. Chapter 36 Decomposing Graphs into Regions of Small Diameter* Nathan Linialt Michael .lss A decomposition of a graph G = (V, E) is a partition of the vertex set into subsets (called lhks). The diameter of a decomposition is the least. d such that any two vertices belonging to the same connected component of a block are at distance < d. In this paper we prove (nearly best possible) statements of the form: .4ny n–vertex graph has a decomposition into a small number of blocks each having small diameter. Such decompositions provide a tool for efficiently decentralizing distributed computations. In [AGLP1 it was shown that every graph has a decomposition into at most s(n) blocks of diameter at most s(n) for s(n) = o( loglog d h n). usinga,technique of Awerbuch [A] and Awerbuch and Peleg [AP], we improve this result by showing that every graph has a decomposition of diameter ()(log n) into O(log n) blocks. In addition, we give a randomized distributed algorithm that produces such a decomposition and runs in time 0(log2 n). The construction can be parametrized to provide decompositions that trade-off between the number of blocks and the diameter. We show that this trade-off is nearly best possible for two families of graphs the first consists of skeletons of certain triangulations of a simplex and the second consists of grid graphs with added diagonals. The proofs in both cases rely on basic results in combinatorial topology, Sperner’s lemma for the first class and Tucker’s lemma for the second. *This work was supported in part by NSF contracts DMS87-03541 and CCR-8911388 tDepartment of Computer Science, Hebrew University, Jerusalem, Israel. + ‘Department of Computer Science and Engineering, Mail Code C-014, University of California, San Diego, La Jolla, CA 92093-0114.