arXiv subjects

Wen Chean Teh

Publications and source records attributed to Wen Chean Teh.

At least 19 recordsLinked to original sources

arXiv2025

Extremal Trees With Prescribed Burning Numbers

Graph burning is motivated by the spread of social influence, and the burning number measures the speed of the spread. Given that the smallest burning number among the spanning trees of a graph determines the burning number of a connected graph, trees are the main objects of investigation in graph burning. Given a prescribed burning number, our study focuses on identifying the corresponding extremal trees with respect to order up to graph homeomorphism. In this work, we propose the concept of admissible sequences over a homeomorphically irreducible tree in addition to developing a general framework. We then determine whether an admissible sequence induces an extremal tree with a specified burning number. Additionally, we obtain some results on the smallest attainable diameter for extremal $n$-spiders with a prescribed burning number.

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arXiv2024

Ternary is Still Good for Parikh Matrices

The focus of this work is the study of Parikh matrices with emphasis on two concrete problems. In the first part of our presentation we show that a conjecture by Dick at al. in 2021 only stands in the case of ternary alphabets, while providing counterexamples for larger alphabets. In particular, we show that the only type of distinguishability in the case of 3-letter alphabets is the trivial one. The second part of the paper builds on the notion of Parikh matrices for projections of words, discussed initially in this work, and answers, once more in the case of a ternary alphabet, a question posed by Atanasiu et al. in 2022 with regards to the minimal Hamming distance in between words sharing a congruency class.

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arXiv2024

A Note on Graph Burning of Path Forests

Graph burning is a natural discrete graph algorithm inspired by the spread of social contagion. Despite its simplicity, some open problems remain steadfastly unsolved, notably the burning number conjecture, which says that every connected graph of order $m^2$ has burning number at most $m$. Earlier, we showed that the conjecture also holds for a path forest, which is disconnected, provided each of its paths is sufficiently long. However, finding the least sufficient length for this to hold turns out to be nontrivial. In this note, we present our initial findings and conjectures that associate the problem to some naturally impossibly burnable path forests. It is noteworthy that our problem can be reformulated as a topic concerning sumset partition of integers.

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arXiv2022

Burnability of Double Spiders and Path Forests

The burning number of a graph can be used to measure the spreading speed of contagion in a network. The burning number conjecture is arguably the main unresolved conjecture related to this graph parameter, which can be settled by showing that every tree of order $m^2$ has burning number at most $m$. This is known to hold for many classes of trees, including spiders - trees with exactly one vertex of degree greater than two. In fact, it has been verified that certain spiders of order slightly larger than $m^2$ also have burning numbers at most $m$, a result that has then been conjectured to be true for all trees. The first focus of this paper is to verify this slightly stronger conjecture for double spiders - trees with two vertices of degrees at least three and they are adjacent. Our other focus concerns the burning numbers of path forests, a class of graphs in which their burning numbers are naturally related to that of spiders and double spiders. Here, our main result shows that a path forest of order $m^2$ with a sufficiently long shortest path has burning number exactly $m$, the smallest possible for any path forest of the same order.

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arXiv2021

Counting Subwords in Circular Words and Their Parikh Matrices

The word inference problem is to determine languages such that the information on the number of occurrences of those subwords in the language can uniquely identify a word. A considerable amount of work has been done on this problem, but the same cannot be said for circular words despite growing interests on the latter due to their applications -- for example, in splicing systems. Meanwhile, Parikh matrices are useful tools and well established in the study of subword occurrences. In this work, we propose two ways of counting subword occurrences in circular words. We then extend the idea of Parikh matrices to the context of circular words and investigate this extension. Motivated by the word inference problem, we study ambiguity in the identification of a circular word by its Parikh matrix. Accordingly, two rewriting rules are developed to generate ternary circular words which share the same Parikh matrix.

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arXiv2020

Ranks of Strictly Minimal Reaction Systems Induced by Permutations and Cartesian Product

Reaction system is a computing model inspired by the biochemical interaction taking place within the living cells. Various extended or modified frameworks motivated by biological, physical, or purely mathematically considerations have been proposed and received significant amount of attention, notably in the recent years. This study, however, takes after particular early works that concentrated on the mathematical nature of minimal reaction systems in the context-free basic framework and motivated by a recent result on the sufficiency of strictly minimal reaction systems to simulate every reaction system. This paper focuses on the largest reaction system rank attainable by strictly minimal reaction systems, where the rank pertains to the minimum size of a functionally equivalent reaction system. Precisely, we provide a very detailed study for specific strictly minimal reaction system induced by permutations, up to the quaternary alphabet. Along the way, we obtain a general result about reaction system rank for Cartesian product of functions specified by reaction systems.

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arXiv2020

Simulation Of Reaction Systems By The Strictly Minimal Ones

Reaction systems, introduced by Ehrenfeucht and Rozenberg, are elementary computational models based on biochemical reactions transpiring within the living cells. Numerous studies focus on mathematical aspects of minimal reaction systems due to their simplicity and rich generative power. In 2014 Manzoni, Pocas, and Porreca showed that every reaction system can be simulated by some minimal reaction system over an extended background set. Motivated by their work, we introduce the concepts of strictly minimal and hybrid reaction systems. Using our new concepts, the result of Manzoni et al. is revisited and strengthened. We also show that extension of the background set by polynomially bounded many elements is not sufficient to guarantee the aforementioned simulation. Finally, an analogous result for strong simulation is obtained.

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arXiv2019

Graph Burning: Tight Bounds on the Burning Numbers of Path Forests and Spiders

In 2016, Bonato, Janssen, and Roshanbin introduced graph burning as a discrete process that models the spread of social contagion. Although the burning process is a simple algorithm, the problem of determining the least number of rounds needed to completely burn a graph, called the burning number of the graph, is NP-complete even for elementary graph structures like spiders. An early conjecture that every connected graph of order square of m can be burned in at most m rounds is the main motivator of this study. Attempts to prove the conjecture have resulted in various upper bounds for the burning number and validation of the conjecture for certain elementary classes of graphs. In this work, we find a tight upper bound for the order of a spider for it to be burned within a given number of rounds. Our result shows that the tight bound depends on the structure of the spider under consideration, namely the number of arms. This strengthens the previously known results on spiders in relation to the conjecture. More importantly, this opens up potential enquiry into the connection between burning numbers and certain characteristics of graphs. Finally, a tight upper bound for the order of a path forest for it to be burned within a given number of rounds is obtained, thus completing previously known partial corresponding results.

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arXiv2019

M-Ambiguity Sequences for Parikh Matrices and Their Periodicity Revisited

The introduction of Parikh matrices by Mateescu et al. in 2001 has sparked numerous new investigations in the theory of formal languages by various researchers, among whom is Serbanuta. Recently, a decade-old conjecture by Serbanuta on the M-ambiguity of words was disproved, leading to new possibilities in the study of such words. In this paper, we investigate how selective repeated duplications of letters in a word affect the M-ambiguity of the resulting words. The corresponding M-ambiguity of those words are then presented in sequences, which we term as M-ambiguity sequences. We show that nearly all patterns of M-ambiguity sequences are attainable. Finally, by employing certain algebraic approach and some underlying theory in integer programming, we show that repeated periodic duplications of letters of the same type in a word results in an M-ambiguity sequence that is eventually periodic.

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arXiv2018

Parikh Word Representability of Bipartite Permutation Graphs

The class of Parikh word representable graphs were recently introduced. In this work, we further develop its general theory beyond the binary alphabet. Our main result shows that this class is equivalent to the class of bipartite permutation graphs. Furthermore, we study certain graph theoretic properties of these graphs in terms of the arity of the representing word.

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arXiv2018

Parikh Matrices for Powers of Words

Certain upper triangular matrices, termed as Parikh matrices, are often used in the combinatorial study of words. Given a word, the Parikh matrix of that word elegantly computes the number of occurrences of certain predefined subwords in that word. In this paper, we compute the Parikh matrix of any word raised to an arbitrary power. Furthermore, we propose canonical decompositions of both Parikh matrices and words into normal forms. Finally, given a Parikh matrix, the relation between its normal form and the normal forms of words in the corresponding M-equivalence class is established.

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arXiv2018

Parikh Motivated Study on Repetitions in Words

We introduce the notion of general prints of a word, which is substantialized by certain canonical decompositions, to study repetition in words. These associated decompositions, when applied recursively on a word, result in what we term as core prints of the word. The length of the path to attain a core print of a general word is scrutinized. This paper also studies the class of square-free ternary words with respect to the Parikh matrix mapping, which is an extension of the classical Parikh mapping. It is shown that there are only finitely many matrix-equivalence classes of ternary words such that all words in each class are square-free. Finally, we employ square-free morphisms to generate infinitely many pairs of square-free ternary words that share the same Parikh matrix.

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arXiv2017

Compositions of Functions and Permutations Specified by Minimal Reaction Systems

This paper studies mathematical properties of reaction systems that was introduced by Enrenfeucht and Rozenberg as computational models inspired by biochemical reaction in the living cells. In particular, we continue the study on the generative power of functions specified by minimal reaction systems under composition initiated by Salomaa. Allowing degenerate reaction systems, functions specified by minimal reaction systems over a quarternary alphabet that are permutations generate the alternating group on the power set of the background set.

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arXiv2017

A Ramsey Algebraic Study of Matrices

The notion of a topological Ramsey space was introduced by Carlson some 30 years ago. Studying the topological Ramsey space of variable words, Carlson was able to derive many classical combinatorial results in a unifying manner. For the class of spaces generated by algebras, Carlson had suggested that one should attempt a purely combinatorial approach to the study. This approach was later formulated and named Ramsey algebra. In this paper, we continue to look at heterogeneous Ramsey algebras, mainly characterizing various Ramsey algebras involving matrices.

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arXiv2017

Ramsey Orderly Algebras as a New Approach to Ramsey Algebras

Ramsey algebras are algebras that induce Ramsey spaces, which are generalizations of the Ellentuck space and Milliken's space. Previous work suggests a possible local version of Ramsey algebras induced by infinite sequences. Hence, we introduce a new structure called orderly algebra. Under our canonical setup, an algebra is Ramsey if and only if every of its induced orderly algebra is Ramsey. In this paper, we present justifications for this novel notion as a sound approach for further study on Ramsey algebras.

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arXiv2017

Heterogeneous Ramsey Algebras and Classification of Ramsey Vector Spaces

Carlson introduced the notion of a Ramsey space as a generalization to the Ellentuck space. When a Ramsey space is induced by an algebra, Carlson suggested a study of its purely combinatorial version now called Ramsey algebra. Some basic results for homogeneous algebras have been obtained. In this paper, we introduce the notion of a Ramsey algebra for heterogeneous algebras and derive some basic results. Then, we study the Ramsey-algebraic properties of vector spaces.

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arXiv2017

Are Ramsey Algebras Essentially Semigroups

It is known that semigroups are Ramsey algebras. This paper is an attempt to understand the role associativity plays in a binary system being a Ramsey algebra. Specifically, we show that the nonassociative Moufang loop of octonions is not a Ramsey algebra.

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arXiv2017

Strong 2.t and Strong 3.t Transformations for Strong M-equivalence

Parikh matrices have been extensively investigated due to their usefulness in studying subword occurrences in words. Due to the dependency of Parikh matrices on the ordering of the alphabet, strong M-equivalence was proposed as an order-independent alternative to M-equivalence in studying words possessing the same Parikh matrix. This paper introduces and studies the notions of strong 2.t and strong 3.t transformations in determining when two ternary words are strongly M-equivalent. The irreducibility of strong 2.t transformations are then scrutinized, exemplified by a structural characterization of irreducible strong 2.2 transformations. The common limitation of these transformations in characterizing strong M-equivalence is then addressed.

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