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Pamela Fleischmann

Publications and source records attributed to Pamela Fleischmann.

At least 19 recordsLinked to original sources
arXiv2026

Language-Representability: Possibilities and Limitations

The study of word-representability was initiated by the seminal work of Kitaev and Pyatkin in 2008 that has later led to the monograph by Kitaev and Lozin in 2015. In this paper, we build on the very recent work by Fernau et al. who proposed a general framework that generalizes certain aspects of word-representability, so that any binary language describes a graph class. In this work, we systematically study particularly small languages and observe that they characterize well-known graph classes, e.g., interval, permutation, circle, and bipartite chain graphs. Thus, we strengthen the bond between formal languages and graph classes, even for small binary languages. We also show some limitations of our approach by proving that, e.g., families of sparse graphs like planar graphs cannot be characterized by any language following this approach.

cs.FL↗
arXiv2026

$2$-word-$π$-representable Graphs

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.

math.CO↗
arXiv2026

On Languages Describing Large Graph Classes

In this work, we introduce a new notion for representing graph classes with formal languages. In contrast to the seminal work by Kitaev and Pyatkin to represent graphs by words, we use formal binary languages in order to have a set of patterns (given by the languages' words) defining the edges in the graph. In particular, we investigate famous languages like the palindromes, copy-words, Lyndon words, and Dyck words to represent all graphs or specific graph classes by restricting these languages.

cs.FL↗
arXiv2026

(Sets of ) Complement Scattered Factors

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.

cs.DS↗
arXiv2026

Determining Factorial Speed Fast

The speed of a graph class $\cal G$ measures how many labeled graphs on $n$ vertices one can find in $\cal G$. This graph class complexity function is explicitly provided on graphclasses.org. However, for many graph classes, their speed status is classified as \emph{unknown}. In this paper, w}\shortversion{W}e show that any graph class representable by a finite binary language has at most factorial speed, meaning that its speed function behaves like $2^{Θ(n\log n)}$, and we use this criterion to classify many graph classes whose speed was previously unknown as factorial. As a consequence, inclusions between several graph classes can now be seen to be proper. We also prove that $k$-letter graphs have exponential speed, i.e., the speed function lies in $2^{Θ(n)}$.

cs.DM↗
arXiv2025

Tight Bounds for the Number of Absent Subsequences

A {\em subsequence} of a word $w$ is a word $u$ that can be obtained by deleting some letters from $w$ while maintaining the relative order of the remaining letters, e.g., $\mathtt{lala}$ is a subsequence of $\mathtt{alfalfa}$. A word, over some alphabet $Σ$, which has all possible words of length $ι$ over $Σ$ as subsequences is called $ι$-universal, and the largest $ι$ for which this holds is called the universality index of $w$, and denoted $ι(w)$. Moreover, words that are not subsequences of $w$ are called absent subsequences (AS) of $w$, and their investigation was started in (Kosche et al., 2022). In this paper, we present tight bounds on the number of AS of a given length $k$ among all words with the same universality index $ι$. For both the lower and upper bound, we construct words that have, respectively, a minimal and maximal number of absent subsequences of the respective length $k$, and, in the case of the lower bound, we provide the exact number of missing subsequences as a closed form. Finally, we present efficient enumeration algorithms for the set of subsequences of given length of a word: we give a novel, optimal enumeration algorithm with output linear delay of this set of subsequences, with preprocessing time $O(|w|)$, which is further improved to an incremental enumeration algorithm with $O(1)$ delay of this set of subsequences, with preprocessing time $O(|w|)$.

cs.FL↗
arXiv2025

Word Chain Generators for Prefix Normal Words

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.

math.CO↗
arXiv2025

Word-Representable Graphs and Locality of Words

In this work, we investigate the relationship between $k$-repre\-sentable graphs and graphs representable by $k$-local words. In particular, we show that every graph representable by a $k$-local word is $(k+1)$-representable. A previous result about graphs represented by $1$-local words is revisited with new insights. Moreover, we investigate both classes of graphs w.r.t. hereditary and in particular the speed as a measure. We prove that the latter ones belong to the factorial layer and that the graphs in this classes have bounded clique-width.

math.CO↗
arXiv2025

Jumbled Scattered Factors

In this work, we combine the research on (absent) scattered factors with the one of jumbled words. For instance, $\mathtt{wolf}$ is an absent scattered factor of $\mathtt{cauliflower}$ but since $\mathtt{lfow}$, a jumbled (or abelian) version of $\mathtt{wolf}$, is a scattered factor, $\mathtt{wolf}$ occurs as a jumbled scattered factor in $\mathtt{cauliflower}$. A \emph{jumbled scattered factor} $u$ of a word $w$ is constructed by letters of $w$ with the only rule that the number of occurrences per letter in $u$ is smaller than or equal to the one in $w$. We proceed to partition and characterise the set of jumbled scattered factors by the number of jumbled letters and use the latter as a measure. For this new class of words, we relate the folklore longest common subsequence (scattered factor) to the number of required jumbles. Further, we investigate the smallest possible number of jumbles alongside the jumbled scattered factor relation as well as Simon's congruence from the point of view of jumbled scattered factors and jumbled universality.

math.CO↗
arXiv2025

$k$-local Graphs

In 2017 Day et al. introduced the notion of locality as a structural complexity-measure for patterns in the field of pattern matching established by Angluin in 1980. In 2019 Casel et al. showed that determining the locality of an arbitrary pattern is NP-complete. Inspired by hierarchical clustering, we extend the notion to coloured graphs, i.e., given a coloured graph determine an enumeration of the colours such that colouring the graph stepwise according to the enumeration leads to as few clusters as possible. Next to first theoretical results on graph classes, we propose a priority search algorithm to compute the $k$-locality of a graph. The algorithm is optimal in the number of marking prefix expansions, and is faster by orders of magnitude than an exhaustive search. Finally, we perform a case study on a DBLP subgraph to demonstrate the potential of $k$-locality for knowledge discovery.

math.CO↗
arXiv2025

$k$-Universality of Regular Languages Revisited

A subsequence of a word $w$ is a word $u$ such that $u = w[i_1] w[i_2] \cdots w[i_k]$, for some set of indices $1 \leq i_1 < i_2 < \dots < i_k \leq \vert w \vert$. A word $w$ is \emph{$k$-subsequence universal} over an alphabet $Σ$ if every word over $Σ$ up to length $k$ appears in $w$ as a subsequence. In this paper, we revisit the problem $k$-ESU of deciding, for a given integer $k$, whether a regular language, given either as nondeterministic finite automaton or as a regular expression, contains a $k$-universal word. [Adamson et al., ISAAC 2023] showed that this problem is NP-hard, even in the case when $k=1$, and an FPT algorithm w.r.t. the size of the input alphabet was given. In this paper, we improve the aforementioned algorithmic result and complete the analysis of this problem w.r.t. other parameters. That is, we propose a more efficient FPT algorithm for $k$-ESU, with respect to the size of the input alphabet, and propose new FPT algorithms for this problem w.r.t.~the number of states of the input automaton and the length of the input regular expression. We also discuss corresponding lower bounds. Our results significantly improve the understanding of this problem.

cs.FL↗
arXiv2024

Generalized Word-Representable Graphs

The literature on word-representable graphs is quite rich, and a number of variations of the original definition have been proposed over the years. We are initiating a systematic study of such variations based on formal languages. In our framework, we can associate a graph class to each language over the binary alphabet \{0,1\}. All graph classes that are language-representable in this sense are hereditary and enjoy further common properties. Besides word-representable graphs and, more generally, 1^k- or k-11-representable graphs, we can identify many more graph classes in our framework, like (co)bipartite graphs, (co)comparability graphs, to name a few. It was already known that any graph is 111- or 2-11-representable. When such representations are considered for storing graphs, 111- or 2-11-representability bears the disadvantage of being significantly inferior to standard adjacency matrices or lists. We prove that quite famous languages like the palindromes, the copy language or the Lyndon words can match the efficiency of standard graph representations. The perspective of language theory allows us to prove general results that hold for all graph classes that can be defined in this way. This includes certain closure properties (e.g., all language-definable graph classes are hereditary) as well as certain limitations (e.g., all language-representable graph classes contain graphs of arbitrarily large treewidth and of arbitrarily large degeneracy, except a trivial case). As each language describes a graph class, we can also ask decidability questions concerning graph classes, given a concrete presentation of a formal language. We also present a systematic study of graph classes that can be represented by languages in which each letter occurs at most twice. Here, we find graph classes like interval, permutation, circle, bipartite chain, convex, and threshold graphs.

cs.DM↗
arXiv2024

Rollercoasters with Plateaus

In this paper we investigate the problem of detecting, counting, and enumerating (generating) all maximum length plateau-$k$-rollercoasters appearing as a subsequence of some given word (sequence, string), while allowing for plateaus. We define a plateau-$k$-rollercoaster as a word consisting of an alternating sequence of (weakly) increasing and decreasing \emph{runs}, with each run containing at least $k$ \emph{distinct} elements, allowing the run to contain multiple copies of the same symbol consecutively. This differs from previous work, where runs within rollercoasters have been defined only as sequences of distinct values. Here, we are concerned with rollercoasters of \emph{maximum} length embedded in a given word $w$, that is, the longest rollercoasters that are a subsequence of $w$. We present algorithms allowing us to determine the longest plateau-$k$-roller\-coasters appearing as a subsequence in any given word $w$ of length $n$ over an alphabet of size $σ$ in $O(n σk)$ time, to count the number of plateau-$k$-rollercoasters in $w$ of maximum length in $O(n σk)$ time, and to output all of them with $O(n)$ delay after $O(n σk)$ preprocessing. Furthermore, we present an algorithm to determine the longest common plateau-$k$-rollercoaster within a set of words in $O(N k σ)$ where $N$ is the product of all word lengths within the set.

cs.DS↗
arXiv2024

Graph and String Parameters: Connections Between Pathwidth, Cutwidth and the Locality Number

We investigate the locality number, a recently introduced structural parameter for strings (with applications in pattern matching with variables), and its connection to two important graph-parameters, cutwidth and pathwidth. These connections allow us to show that computing the locality number is NP-hard, but fixed-parameter tractable, if parameterised by the locality number or by the alphabet size, which has been formulated as open problems in the literature. Moreover, the locality number can be approximated with ratio O(sqrt(log(opt)) log(n)). An important aspect of our work -- that is relevant in its own right and of independent interest -- is that we identify connections between the string parameter of the locality number on the one hand, and the famous graph parameters of cutwidth and pathwidth, on the other hand. These two parameters have been jointly investigated in the literature and are arguably among the most central graph parameters that are based on "linearisations" of graphs. In this way, we also identify a direct approximation preserving reduction from cutwidth to pathwidth, which shows that any polynomial f(opt,|V|)-approximation algorithm for pathwidth yields a polynomial 2f(2 opt,h)-approximation algorithm for cutwidth on multigraphs (where h is the number of edges). In particular, this translates known approximation ratios for pathwidth into new approximation ratios for cutwidth, namely O(sqrt(log(opt)) log(h)) and O(sqrt(log(opt)) opt) for (multi) graphs with h edges.

cs.DS↗
arXiv2023

$k$-Universality of Regular Languages

A subsequence of a word $w$ is a word $u$ such that $u = w[i_1] w[i_2] \dots w[i_{k}]$, for some set of indices $1 \leq i_1 < i_2 < \dots < i_k \leq \lvert w\rvert$. A word $w$ is $k$-subsequence universal over an alphabet $Σ$ if every word in $Σ^k$ appears in $w$ as a subsequence. In this paper, we study the intersection between the set of $k$-subsequence universal words over some alphabet $Σ$ and regular languages over $Σ$. We call a regular language $L$ \emph{$k$-$\exists$-subsequence universal} if there exists a $k$-subsequence universal word in $L$, and \emph{$k$-$\forall$-subsequence universal} if every word of $L$ is $k$-subsequence universal. We give algorithms solving the problems of deciding if a given regular language, represented by a finite automaton recognising it, is \emph{$k$-$\exists$-subsequence universal} and, respectively, if it is \emph{$k$-$\forall$-subsequence universal}, for a given $k$. The algorithms are FPT w.r.t.~the size of the input alphabet, and their run-time does not depend on $k$; they run in polynomial time in the number $n$ of states of the input automaton when the size of the input alphabet is $O(\log n)$. Moreover, we show that the problem of deciding if a given regular language is \emph{$k$-$\exists$-subsequence universal} is NP-complete, when the language is over a large alphabet. Further, we provide algorithms for counting the number of $k$-subsequence universal words (paths) accepted by a given deterministic (respectively, nondeterministic) finite automaton, and ranking an input word (path) within the set of $k$-subsequence universal words accepted by a given finite automaton.

cs.FL↗
arXiv2023

$α$-$β$-Factorization and the Binary Case of Simon's Congruence

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.

math.CO↗
arXiv2022

MaxSAT with Absolute Value Functions: A Parameterized Perspective

The natural generalization of the Boolean satisfiability problem to optimization problems is the task of determining the maximum number of clauses that can simultaneously be satisfied in a propositional formula in conjunctive normal form. In the weighted maximum satisfiability problem each clause has a positive weight and one seeks an assignment of maximum weight. The literature almost solely considers the case of positive weights. While the general case of the problem is only restricted slightly by this constraint, many special cases become trivial in the absence of negative weights. In this work we study the problem with negative weights and observe that the problem becomes computationally harder - which we formalize from a parameterized perspective in the sense that various variations of the problem become W[1]-hard if negative weights are present. Allowing negative weights also introduces new variants of the problem: Instead of maximizing the sum of weights of satisfied clauses, we can maximize the absolute value of that sum. This turns out to be surprisingly expressive even restricted to monotone formulas in disjunctive normal form with at most two literals per clause. In contrast to the versions without the absolute value, however, we prove that these variants are fixed-parameter tractable. As technical contribution we present a kernelization for an auxiliary problem on hypergraphs in which we seek, given an edge-weighted hypergraph, an induced subgraph that maximizes the absolute value of the sum of edge-weights.

cs.CC↗