Aistis Atminas

Linear time algorithm for computing a small biclique in graphs without long induced paths

The biclique problem asks, given a graph G and a parameter k, whether G has a complete bipartite subgraph of k vertices in each part (a biclique of order k). Fixed-parameter tractability of this problem is a longstanding open question in parameterized complexity that received a lot of attention from the community. In this paper we consider a restricted version of this problem by introducing an additional parameter s and assuming that G does not have induced (i.e. chordless) paths of length s. We prove that under this parameterization the problem becomes fixed-parameter linear. The main tool in our proof is a Ramsey-type theorem stating that a graph with a long (not necessarily induced) path contains either a long induced path or a large biclique.

Labelled Induced Subgraphs and Well-Quasi-Ordering

It is known that the set of all simple graphs is not well-quasi-ordered by the induced subgraph relation, i.e. it contains infinite antichains (sets of incomparable elements) with respect to this relation. However, some particular graph classes are well-quasi-ordered by induced subgraphs. Moreover, some of them are well-quasi-ordered by a stronger relation called labelled induced subgraphs. In this paper, we conjecture that a hereditary class X which is well-quasi-ordered by the induced subgraph relation is also well-quasi-ordered by the labelled induced subgraph relation if and only if X is defined by finitely many minimal forbidden induced subgraphs. We verify this conjecture for a variety of hereditary classes that are known to be well-quasi-ordered by induced subgraphs and prove a number of new results supporting the conjecture.

Implicit representations and factorial properties of graphs

The idea of implicit representation of graphs was introduced in Kannan et?al. (1992) and can be defined as follows. A representation of an n -vertex graph G is said to be implicit if it assigns to each vertex of G a binary code of length O ( log n ) so that the adjacency of two vertices is a function of their codes. Since an implicit representation of an n -vertex graph uses O ( n log n ) bits, any class of graphs admitting such a representation contains 2 O ( n log n ) labelled graphs with n vertices. In the terminology of Balogh et?al. (2000) such classes have at most factorial speed of growth. In this terminology, the implicit graph conjecture can be stated as follows: every class with at most factorial speed of growth which is hereditary admits an implicit representation. The question of deciding whether a given hereditary class has at most factorial speed of growth is far from being trivial. In the present paper, we introduce a number of tools simplifying this question. Some of them can be used to obtain a stronger conclusion on the existence of an implicit representation. We apply our tools to reveal new hereditary classes with the factorial speed of growth. For many of them we show the existence of an implicit representation.

Deciding WQO for Factorial Languages

A language is factorial if it is closed under taking factors (i.e. contiguous subwords). Every factorial language can be described by an antidictionary, i.e. a minimal set of forbidden factors. We show that the problem of deciding whether a factorial language given by a finite antidictionary is well-quasi-ordered under the factor containment relation can be solved in polynomial time.

Universal graphs and universal permutations

Let X be a family of graphs and Xn the set of n-vertex graphs in X. A graph U(n) containing all graphs from Xn as induced subgraphs is called n-universal for X. Moreover, we say that U(n) is a propern-universal graph for X if it belongs to X. In the present paper, we construct a proper n-universal graph for the class of split permutation graphs. Our solution includes two ingredients: a proper universal 321-avoiding permutation and a bijection between 321-avoiding permutations and symmetric split permutation graphs. The n-universal split permutation graph constructed in this paper has 4n3 vertices, which means that this construction is order-optimal.

Scattered packings of cycles

We consider the problem Scattered Cycles which, given a graph G and two positive integers r and ź, asks whether G contains a collection of r cycles that are pairwise at distance at least ź. This problem generalizes the problem Disjoint Cycles which corresponds to the case ź = 1 . We prove that when parameterized by r, ź, and the maximum degree Δ, the problem Scattered Cycles admits a kernel on 24 ź 2 Δ ź r log ź ( 8 ź 2 Δ ź r ) vertices. We also provide a ( 16 ź 2 Δ ź ) -kernel for the case r = 2 and a ( 148 Δ r log ź r ) -kernel for the case ź = 1 . Our proofs rely on two simple reduction rules and a careful analysis.

Deciding the Bell Number for Hereditary Graph Properties

The paper [J. Balogh, B. Bollobas, D. Weinreich, J. Combin. Theory Ser. B, 95 (2005), pp. 29--48] identifies a jump in the speed of hereditary graph properties to the Bell number

Deciding the Bell number for hereditary graph properties

B_n

Linear Ramsey Numbers

and provides a partial characterization of the family of minimal classes whose speed is at least

Characterising inflations of monotone grid classes of permutations

B_n