Volume 185, Issue 1

1. On morphisms preserving palindromic richness

Francesco Dolce ; Edita Pelantová.
It is known that each word of length $n$ contains at most $n+1$ distinctpalindromes. A finite rich word is a word with maximal number of palindromicfactors. The definition of palindromic richness can be naturally extended toinfinite words. Sturmian words and Rote complementary symmetric sequences formtwo classes of binary rich words, while episturmian words and words codingsymmetric $d$-interval exchange transformations give us other examples onlarger alphabets. In this paper we look for morphisms of the free monoid, whichallow us to construct new rich words from already known rich words. We focus onmorphisms in Class $P_{ret}$. This class contains morphisms injective on thealphabet and satisfying a particular palindromicity property: for everymorphism $\varphi$ in the class there exists a palindrome $w$ such that$\varphi(a)w$ is a first complete return word to $w$ for each letter $a$. Wecharacterize $P_{ret}$ morphisms which preserve richness over a binaryalphabet. We also study marked $P_{ret}$ morphisms acting on alphabets withmore letters. In particular we show that every Arnoux-Rauzy morphism isconjugated to a morphism in Class $P_{ret}$ and that it preserves richness.

2. Self-stabilisation of cellular automata on tilings

Nazim Fatès ; Irène Marcovici ; Siamak Taati.
Given a finite set of local constraints, we seek a cellular automaton (i.e.,a local and uniform algorithm) that self-stabilises on the configurations thatsatisfy these constraints. More precisely, starting from a finite perturbationof a valid configuration, the cellular automaton must eventually fall back intothe space of valid configurations where it remains still. We allow the cellularautomaton to use extra symbols, but in that case, the extra symbols can alsoappear in the initial finite perturbation. For several classes of localconstraints (e.g., $k$-colourings with $k\neq 3$, and North-East deterministicconstraints), we provide efficient self-stabilising cellular automata with orwithout additional symbols that wash out finite perturbations in linear orquadratic time, but also show that there are examples of local constraints forwhich the self-stabilisation problem is inherently hard. We note that theoptimal self-stabilisation speed is the same for all local constraints that areisomorphic to one another. We also consider probabilistic cellular automatarules and show that in some cases, the use of randomness simplifies theproblem. In the deterministic case, we show that if finite perturbations arecorrected in linear time, then the cellular automaton self-stabilises evenstarting from a random perturbation of a valid configuration, that is, whenerrors in the initial configuration occur independently with a sufficiently lowdensity.

3. Spanning Structures in Walker--Breaker Games

Jovana Forcan ; Mirjana Mikalački.
We study the biased $(2:b)$ Walker--Breaker games, played on the edge set ofthe complete graph on $n$ vertices, $K_n$. These games are a variant of theMaker--Breaker games with the restriction that Walker (playing the role ofMaker) has to choose her edges according to a walk. We look at the two standardgraph games -- the Connectivity game and the Hamilton Cycle game and show thatWalker can win both games even when playing against Breaker whose bias is ofthe order of magnitude $n/ \ln n$.