Elise Vandomme

Syntactic Complexity of Ultimately Periodic Sets of Integers and Application to a Decision Procedure

We compute the cardinality of the syntactic monoid of the language 0a repb(m

Leaf realization problem, caterpillar graphs and prefix normal words

\mathbb{N}

Fully Leafed Induced Subtrees

) made of base b expansions of the multiples of the integer m. We also give lower bounds for the syntactic complexity of any (ultimately) periodic set of integers written in base b. We apply our results to a well studied problem: decide whether or not a b-recognizable set of integers is ultimately periodic.

Nearest constrained circular words

Abstract Given a simple graph G with n vertices and a natural number i ≤ n , let L G ( i ) be the maximum number of leaves that can be realized by an induced subtree T of G with i vertices. We introduce a problem that we call the leaf realization problem, which consists in deciding whether, for a given sequence of n + 1 natural numbers ( l 0 , l 1 , … , l n ) , there exists a simple graph G with n vertices such that l i = L G ( i ) for i = 0 , 1 , … , n . We present basic observations on the structure of these sequences for general graphs and trees. In the particular case where G is a caterpillar graph, we exhibit a bijection between the set of the discrete derivatives of the form ( Δ L G ( i ) ) 1 ≤ i ≤ n − 3 and the set of prefix normal words.

Critical exponents of infinite balanced words

We consider the problem \(\mathrm {LIS}\) of deciding whether there exists an induced subtree with exactly \(i \le n\) vertices and \(\ell \) leaves in a given graph G with n vertices. We study the associated optimization problem, that consists in computing the maximal number of leaves, denoted by \(L_G(i)\), realized by an induced subtree with i vertices, for \(0 \le i \le n\). We begin by proving that the \(\mathrm {LIS}\) problem is NP-complete in general. Then, we describe a nontrivial branch and bound algorithm that computes the function \(L_G\) for any simple graph G. In the special case where G is a tree of maximum degree \(\varDelta \), we provide a \(\mathcal {O}(n^3\varDelta )\) time and \(\mathcal {O}(n^2)\) space algorithm to compute the function \(L_G\).

Constant 2-labellings and an application to (r,a,b)-covering codes

In this paper, we study circular words arising in the development of equipment using shields in brachytherapy. This equipment has physical constraints that have to be taken into consideration. From an algorithmic point of view, the problem can be formulated as follows: Given a circular word, find a constrained circular word of the same length such that the Manhattan distance between these two words is minimal. We show that we can solve this problem in pseudo polynomial time (polynomial time in practice) using dynamic programming.

Some Properties of Abelian Return Words

Over an alphabet of size 3 we construct an infinite balanced word with critical exponent 2+sqrt(2)/2. Over an alphabet of size 4 we construct an infinite balanced word with critical exponent (5+sqrt(5))/4. Over larger alphabets, we give some candidates for balanced words (found computationally) having small critical exponents. We also explore a method for proving these results using the automated theorem prover Walnut.

A New Approach to the 2-Regularity of the ℓ-Abelian Complexity of 2-Automatic Sequences

Abstract We introduce the concept of constant 2-labelling of a vertex-weighted graph and show how it can be used to obtain perfect weighted coverings. Roughly speaking, a constant 2-labelling of a vertex-weighted graph is a black and white colouring of its vertex set which preserves the sum of the weights of black vertices under some automorphisms. We study constant 2-labellings on four types of vertex-weighted cycles. Our results on cycles allow us to determine (r, a, b)-codes in Z2 whenever |a − b| > 4, r ≥ 2 and we give the precise values of a and b. This is a refinement of Axenovich’s theorem proved in 2003.