We show that any two tree-based rooted binary phylogenetic networks (with the same number of reticulation nodes and identical degrees at the root) are connected by at most (4k+8)⋅|X|+34k+|X|⋅⌈log2|X|⌉ rooted nearest-neighbour interchange (rNNI) moves. As a corollary of the proof we obtain that the diameter is at most a 2k additive term larger for distance-1 tail moves.

A joint degree matrix (JDM) specifies the number of connections between nodes of given degrees in a graph, for all degree pairs, and uniquely determines the degree sequence of the graph. We consider the space of all balanced realizations of an arbitrary JDM, realizations in which the links between any two fixed-degree groups of nodes are placed as uniformly as possible. We prove that a swap Markov chain Monte Carlo algorithm in the space of all balanced realizations of an arbitrary graphical JDM mixes rapidly, i.e., the relaxation time of the chain is bounded from above by a polynomial in the number of nodes $n$. To prove fast mixing, we first prove a general factorization theorem similar to the Martin--Randall method for disjoint decompositions (partitions). This theorem can be used to bound from below the spectral gap with the help of fast mixing subchains within every partition and a bound on an auxiliary Markov chain between the partitions. Our proof of the general factorization theorem is direct and uses conductance based methods (Cheeger inequality).

We characterize obstruction sets in caterpillar dualities in terms of regular languages and give a construction of the dual of a regular family of caterpillars. In particular, we prove that every monadic linear Datalog program with at most one extensional database per rule defines the complement of a contraint satisfaction problem.

This is a paper about the beauty of the permutation method. New and shorter proofs are given for the theorem [P. L. Erdos and G. O. H. Katona, J. Combin. Theory. Ser. A, 43 (1986), pp. 58--69; S. Shahriari, Discrete Math., 162 (1996), pp. 229--238] determining all extremal two-part Sperner families and for the uniqueness of k-Sperner families of maximum size [P. Erdos, Bull. Amer. Math. Soc., 51 (1945), pp. 898--902].