Computer Science

Extending the idea in [Impagliazzo, R., Moore, C. and Russell, A., An entropic proof of Chang's inequality. SIAM Journal on Discrete Mathematics, 28(1), pp.173-176.] we give a short information theoretic proof for Chang's lemma that is based on Pinsker's inequality.

A contact B 0 -VPG graph is a graph for which there exists a collection of nontrivial pairwise interiorly disjoint horizontal and vertical segments in one-to-one correspondence with its vertex set such that two vertices are adjacent if and only if the corresponding segments touch. It was shown by Deniz et al. that Recognition is NP -complete for contact B 0 -VPG graphs. In this paper we present a minimal forbidden induced subgraph characterisation of contact B 0 -VPG graphs within the class of circular-arc graphs and provide a polynomial-time algorithm for recognising these graphs.

Following earlier work by Aldo de Luca and others, we study trapezoidal words and their prefixes, with respect to their characteristic parameters K and R (length of shortest unrepeated suffix, and shortest length without right special factors, respectively), as well as their symmetric versions H and L . We consider the distinction between closed (i.e., periodic-like) and open prefixes, and between Sturmian and non-Sturmian ones. Our main results characterize right special and strictly bispecial trapezoidal words, as done by de Luca and Mignosi for Sturmian words.

We propose an algorithm for classification of linear codes over different finite fields based on canonical augmentation. We apply this algorithm to obtain classification results over fields with 2, 3 and 4 elements.

We consider minimally unsatisfiable 2-CNFs (short 2-MUs). Characterisations of 2-MUs in the literature have been restricted to the nonsingular case (where every variable occurs positively and negatively at least twice), and those with a unit-clause. We provide the full classification of 2-MUs F. The main tool is the implication digraph, and we show that the implication digraph of F is a "weak double cycle" (WDC), a big cycle of small cycles (with possible overlaps). Combining logical and graph-theoretical methods, we prove that WDCs have at most one skew-symmetry, and thus we obtain that the isomorphisms between 2-MUs F, F' are exactly the isomorphisms between their implication digraphs. We obtain a variety of applications. For fixed deficiency k, the difference of the number of clauses of F and the number n of variables of F, the automorphism group of F is a subgroup of the Dihedral group with 4k elements. The isomorphism problem restricted to 2-MUs F is decidable in linear time for fixed k. The number of isomorphism types of 2-MUs for fixed k is Theta(n^(3k-1)). The smoothing (removal of linear vertices) of skew-symmetric WDCs corresponds exactly to the canonical normal form of F obtained by 1-singular DP-reduction, a restricted form of DP-reduction (or "variable elimination") only reducing variables of degree 2. The isomorphism types of these normal forms, i.e., the homeomorphism types of skew-symmetric WDCs, are in one-to-one correspondence with binary bracelets (or "turnover necklaces") of length k.

Many NP-complete graph problems are polynomial-time solvable on graph classes of bounded clique-width. Several of these problems are polynomial-time solvable on a hereditary graph class G if they are so on the atoms (graphs with no clique cut-set) of G . Hence, we initiate a systematic study into boundedness of clique-width of atoms of hereditary graph classes. A graph G is H -free if H is not an induced subgraph of G , and it is ( H 1 , H 2 ) -free if it is both H 1 -free and H 2 -free. A class of H -free graphs has bounded clique-width if and only if its atoms have this property. This is no longer true for ( H 1 , H 2 ) -free graphs, as evidenced by one known example. We prove the existence of another such pair ( H 1 , H 2 ) and classify the boundedness of clique-width on ( H 1 , H 2 ) -free atoms for all but 18 cases.

We prove that if G is a sparse graph --- it belongs to a fixed class of bounded expansion C --- and d∈N is fixed, then the d th power of G can be partitioned into cliques so that contracting each of these clique to a single vertex again yields a sparse graph. This result has several graph-theoretic and algorithmic consequences for powers of sparse graphs, including bounds on their subchromatic number and efficient approximation algorithms for the chromatic number and the clique number.

This paper studies the mathematical properties of collectively canalizing Boolean functions, a class of functions that has arisen from applications in systems biology. Boolean networks are an increasingly popular modeling framework for regulatory networks, and the class of functions studied here captures a key feature of biological network dynamics, namely that a subset of one or more variables, under certain conditions, can dominate the value of a Boolean function, to the exclusion of all others. These functions have rich mathematical properties to be explored. The paper shows how the number and type of such sets influence a function's behavior and define a new measure for the canalizing strength of any Boolean function. We further connect the concept of collective canalization with the well-studied concept of the average sensitivity of a Boolean function. The relationship between Boolean functions and the dynamics of the networks they form is important in a wide range of applications beyond biology, such as computer science, and has been studied with statistical and simulation-based methods. But the rich relationship between structure and dynamics remains largely unexplored, and this paper is intended as a contribution to its mathematical foundation.

Recent results show that the structural similarity of graphs can be characterized by counting homomorphisms to them: the Tree Theorem states that the well-known color-refinement algorithm does not distinguish two graphs G and H if and only if, for every tree T, the number of homomorphisms Hom(T,G) from T to G is equal to the corresponding number Hom(T,H) from T to H (Dell, Grohe, Rattan 2018). We show how this approach transfers to hypergraphs by introducing a generalization of color refinement. We prove that it does not distinguish two hypergraphs G and H if and only if, for every connected Berge-acyclic hypergraph B, we have Hom(B,G) = Hom(B,H). To this end, we show how homomorphisms of hypergraphs and of a colored variant of their incidence graphs are related to each other. This reduces the above statement to one about vertex-colored graphs.

In this work we present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations. This approach provides a unified view on many known results and allows us to prove many new ones. In particular, we obtain four classical Gray codes for permutations, bitstrings, binary trees and set partitions as special cases. We present two distinct applications for our new framework: The first main application is the generation of pattern-avoiding permutations, yielding new Gray codes for different families of permutations that are characterized by the avoidance of certain classical patterns, (bi)vincular patterns, barred patterns, boxed patterns, Bruhat-restricted patterns, mesh patterns, monotone and geometric grid classes, and many others. We also obtain new Gray codes for all the combinatorial objects that are in bijection to these permutations, in particular for five different types of geometric rectangulations, also known as floorplans, which are divisions of a square into n rectangles subject to certain restrictions. The second main application of our framework are lattice congruences of the weak order on the symmetric group S n . Recently, Pilaud and Santos realized all those lattice congruences as (n−1) -dimensional polytopes, called quotientopes, which generalize hypercubes, associahedra, permutahedra etc. Our algorithm generates the equivalence classes of each of those lattice congruences, by producing a Hamilton path on the skeleton of the corresponding quotientope, yielding a constructive proof that each of these highly symmetric graphs is Hamiltonian. We thus also obtain a provable notion of optimality for the Gray codes obtained from our framework: They translate into walks along the edges of a polytope.