Annika Huch

Duncan Adamson, Amanita Dietz, Pamela Fleischmann et al.

This paper investigates the new notion of $2$-word-$π$-repre\-sentable graphs: the nodes of the graph correspond to the letters of the two words and there exists an edge between two nodes if the projections of any two letters of both words are equal. The benefit of not only using one word for a representation as introduced by Kitaev and Pyatkin is that every graph is $2$-word-$π$-representable. We present an algorithm that returns two representing words for any graph. Aside, we show that every permutation graph is representable by two $1$-uniform words and give constructions how graph operations on $2$-word-$π$-representable graphs can be realised on their representing words which give further insights into the representation of cographs.

Pamela Fleischmann, Jonas Höfer, Annika Huch et al.

In 1991 Hébrard introduced a factorization of words that turned out to be a powerful tool for the investigation of a word's scattered factors (also known as (scattered) subwords or subsequences). Based on this, first Karandikar and Schnoebelen introduced the notion of $k$-richness and later on Barker et al. the notion of $k$-universality. In 2022 Fleischmann et al. presented a generalization of the arch factorization by intersecting the arch factorization of a word and its reverse. While the authors merely used this factorization for the investigation of shortest absent scattered factors, in this work we investigate this new $α$-$β$-factorization as such. We characterize the famous Simon congruence of $k$-universal words in terms of $1$-universal words. Moreover, we apply these results to binary words. In this special case, we obtain a full characterization of the classes and calculate the index of the congruence. Lastly, we start investigating the ternary case, present a full list of possibilities for $αβα$-factors, and characterize their congruence.

Duncan Adamson, Pamela Fleischmann, Annika Huch

Starting in the 1970s with the fundamental work of Imre Simon, \emph{scattered factors} (also known as subsequences or scattered subwords) have remained a consistently and heavily studied object. The majority of work on scattered factors can be split into two broad classes of problems: given a word, what information, in the form of scattered factors, are contained, and which are not. In this paper, we consider an intermediary problem, introducing the notion of \emph{complement scattered factors}. Given a word $w$ and a scattered factor $u$ of $w$, the complement scattered factors of $w$ with regards to $u$, $C(w, u)$, is the set of scattered factors in $w$ that can be formed by removing any embedding of $u$ from $w$. This is closely related to the \emph{shuffle} operation in which two words are intertwined, i.e., we extend previous work relating to the shuffle operator, using knowledge about scattered factors. Alongside introducing these sets, we provide combinatorial results on the size of the set $C(w, u)$, an algorithm to compute the set $C(w, u)$ from $w$ and $u$ in $O(\vert w \vert \cdot \vert u \vert \binom{w}{u})$ time, where $\binom{w}{u}$ denotes the number of embeddings of $u$ into $w$, an algorithm to construct $u$ from $w$ and $C(w, u)$ in $O(\vert w \vert^2 \binom{\vert w \vert}{\vert w \vert - \vert u \vert})$ time, and an algorithm to construct $w$ from $u$ and $C(w, u)$ in $O(\vert u \vert \cdot \vert w \vert^{\vert u \vert + 1})$ time.

Duncan Adamson, Moritz Dudey, Pamela Fleischmann et al.

In 2011, Fici and Lipták introduced prefix normal words. A binary word is prefix normal if it has no factor (substring) that contains more occurrences of the letter 1 than the prefix of the same length. Among the open problems regarding this topic are the enumeration of prefix normal words and efficient testing methods. We show a range of characteristics of prefix normal words. These include properties of factors that are responsible for a word not being prefix normal. With word chains and generators, we introduce new ways of relating words of the same length to each other.

Pamela Fleischmann, Lukas Haschke, Annika Huch et al.

Determining the index of the Simon congruence is a long outstanding open problem. Two words $u$ and $v$ are called Simon congruent if they have the same set of scattered factors, which are parts of the word in the correct order but not necessarily consecutive, e.g., $\mathtt{oath}$ is a scattered factor of $\mathtt{logarithm}$. Following the idea of scattered factor $k$-universality, we investigate $m$-nearly $k$-universality, i.e., words where $m$ scattered factors of length $k$ are absent, w.r.t. Simon congruence. We present a full characterisation as well as the index of the congruence for $m=1$. For $m\neq 1$, we show some results if in addition $w$ is $(k-1)$-universal as well as some further insights for different $m$.