• arXiv.cs.DM Pub Date : 2020-07-01
Flavia Bonomo-Braberman; Guillermo A. Durán; Nina Pardal; Martín D. Safe

A graph is circle if its vertices are in correspondence with a family of chords in a circle in such a way that every two distinct vertices are adjacent if and only if the corresponding chords have nonempty intersection. Even though there are diverse characterizations of circle graphs, a structural characterization by minimal forbidden induced subgraphs for the entire class of circle graphs is not known

更新日期：2020-07-02
• arXiv.cs.DM Pub Date : 2020-06-30
Ben Cameron; Chính T. Hoàng; Joe Sawada

For $k \geq 3$, we prove (i) there is a finite number of $k$-vertex-critical $(P_2+\ell P_1)$-free graphs and (ii) $k$-vertex-critical $(P_3+P_1)$-free graphs have at most $2k-1$ vertices. Together with previous research, these results imply the following characterization where $H$ is a graph of order four: There is a finite number of $k$-vertex-critical $H$-free graphs for fixed $k \geq 5$ if and

更新日期：2020-07-02
• arXiv.cs.DM Pub Date : 2020-06-30
Flavia Bonomo-Braberman; Fabiano S. Oliveira; Moysés S. Sampaio Jr.; Jayme L. Szwarcfiter

Interval and proper interval graphs are very well-known graph classes, for which there is a wide literature. As a consequence, some generalizations of interval graphs have been proposed, in which graphs in general are expressed in terms of $k$ interval graphs, by splitting the graph in some special way. As a recent example of such an approach, the classes of $k$-thin and proper $k$-thin graphs have

更新日期：2020-07-01
• arXiv.cs.DM Pub Date : 2020-06-30
Flavia Bonomo-Braberman; Carolina L. Gonzalez; Fabiano S. Oliveira; Moysés S. Sampaio Jr.; Jayme L. Szwarcfiter

The thinness of a graph is a width parameter that generalizes some properties of interval graphs, which are exactly the graphs of thinness one. Many NP-complete problems can be solved in polynomial time for graphs with bounded thinness, given a suitable representation of the graph. In this paper we study the thinness and its variations of graph products. We show that the thinness behaves "well" in

更新日期：2020-07-01
• arXiv.cs.DM Pub Date : 2020-06-27
Rishabh Iyer; Jeff Bilmes

Submodular Functions are a special class of set functions, which generalize several information-theoretic quantities such as entropy and mutual information [1]. Submodular functions have subgradients and subdifferentials [2] and admit polynomial-time algorithms for minimization, both of which are fundamental characteristics of convex functions. Submodular functions also show signs similar to concavity

更新日期：2020-07-01
• arXiv.cs.DM Pub Date : 2020-06-30
Nicolas Bousquet; Alice Joffard; Paul Ouvrard

Given a graph $G$ and an integer $k$, a token addition and removal ({\sf TAR} for short) reconfiguration sequence between two dominating sets $D_{\sf s}$ and $D_{\sf t}$ of size at most $k$ is a sequence $S= \langle D_0 = D_{\sf s}, D_1 \ldots, D_\ell = D_{\sf t} \rangle$ of dominating sets of $G$ such that any two consecutive dominating sets differ by the addition or deletion of one vertex, and no

更新日期：2020-07-01
• arXiv.cs.DM Pub Date : 2020-06-30
C. A. Jothishwaran; Anton Tkachenko; Sugata Gangopadhyay; Constanza Riera; Pantelimon Stanica

We propose a quantum algorithm to estimate the Gowers $U_2$ norm of a Boolean function, and extend it into a second algorithm to distinguish between linear Boolean functions and Boolean functions that are $\epsilon$-far from the set of linear Boolean functions, which seems to perform better than the classical BLR algorithm. Finally, we outline an algorithm to estimate Gowers $U_3$ norms of Boolean

更新日期：2020-07-01
• arXiv.cs.DM Pub Date : 2020-06-30
Dibyayan Chakraborty; Sandip Das; Florent Foucaud; Harmender Gahlawat; Dimitri Lajou; Bodhayan Roy

We study the complexity of finding the \emph{geodetic number} on subclasses of planar graphs and chordal graphs. A set $S$ of vertices of a graph $G$ is a \emph{geodetic set} if every vertex of $G$ lies in a shortest path between some pair of vertices of $S$. The \textsc{Minimum Geodetic Set (MGS)} problem is to find a geodetic set with minimum cardinality of a given graph. The problem is known to

更新日期：2020-07-01
• arXiv.cs.DM Pub Date : 2020-06-30
David Schaller; Peter F. Stadler; Marc Hellmuth

Best match graphs (BMGs) are vertex-colored directed graphs that were introduced to model the relationships of genes (vertices) from different species (colors) given an underlying evolutionary tree that is assumed to be unknown. In real-life applications, BMGs are estimated from sequence similarity data. Measurement noise and approximation errors therefore result in empirically determined graphs that

更新日期：2020-07-01
• arXiv.cs.DM Pub Date : 2020-06-30
Jacob Focke; Leslie Ann Goldberg; Marc Roth; Stanislav Živný

We study the problem of computing the parity of the number of homomorphisms from an input graph $G$ to a fixed graph $H$. Faben and Jerrum [ToC'15] introduced an explicit criterion on the graph $H$ and conjectured that, if satisfied, the problem is solvable in polynomial time and, otherwise, the problem is complete for the complexity class $\oplus\mathrm{P}$ of parity problems. We verify their conjecture

更新日期：2020-07-01
• arXiv.cs.DM Pub Date : 2020-06-27
Khaled Elbassioni

Given a partially order set (poset) $P$, and a pair of families of ideals $\cI$ and filters $\cF$ in $P$ such that each pair $(I,F)\in \cI\times\cF$ has a non-empty intersection, the dualization problem over $P$ is to check whether there is an ideal $X$ in $P$ which intersects every member of $\cF$ and does not contain any member of $\cI$. Equivalently, the problem is to check for a distributive lattice

更新日期：2020-06-30
• arXiv.cs.DM Pub Date : 2020-06-29
Panagiotis Charalampopoulos; Jakub Radoszewski; Wojciech Rytter; Tomasz Waleń; Wiktor Zuba

The notions of periodicity and repetitions in strings, and hence these of runs and squares, naturally extend to two-dimensional strings. We consider two types of repetitions in 2D-strings: 2D-runs and quartics (quartics are a 2D-version of squares in standard strings). Amir et al. introduced 2D-runs, showed that there are $O(n^3)$ of them in an $n \times n$ 2D-string and presented a simple construction

更新日期：2020-06-30
• arXiv.cs.DM Pub Date : 2020-06-29

For a given $\varepsilon > 0$, we say that a graph $G$ is $\epsilon$-flexibly $k$-choosable if the following holds: for any assignment $L$ of lists of size $k$ on $V(G)$, if a preferred color is requested at any set $R$ of vertices, then at least $\epsilon |R|$ of these requests may be satisfied by some $L$-coloring. We consider flexible list colorings in several graph classes with certain special

更新日期：2020-06-30
• arXiv.cs.DM Pub Date : 2020-06-29
Anita Liebenau; Nick Wormald

We provide asymptotic formulae for the numbers of bipartite graphs with given degree sequence, and of loopless digraphs with given in- and out-degree sequences, for a wide range of parameters. Our results cover medium range densities and close the gaps between the results known for the sparse and dense ranges. In the case of bipartite graphs, these results were proved by Greenhill, McKay and Wang in

更新日期：2020-06-30
• arXiv.cs.DM Pub Date : 2020-06-28
Akbar Rafiey; Yuichi Yoshida

The problem of maximizing nonnegative monotone submodular functions under a certain constraint has been intensively studied in the last decade, and a wide range of efficient approximation algorithms have been developed for this problem. Many machine learning problems, including data summarization and influence maximization, can be naturally modeled as the problem of maximizing monotone submodular functions

更新日期：2020-06-30
• arXiv.cs.DM Pub Date : 2020-06-28
Waïss Azizian; Marc Lelarge

Various classes of Graph Neural Networks (GNN) have been proposed and shown to be successful in a wide range of applications with graph structured data. In this paper, we propose a theoretical framework able to compare the expressive power of these GNN architectures. The current universality theorems only apply to intractable classes of GNNs. Here, we prove the first approximation guarantees for practical

更新日期：2020-06-30
• arXiv.cs.DM Pub Date : 2020-06-27
Ramtin Afshar; Michael T. Goodrich; Pedro Matias; Martha C. Osegueda

In this paper, we study the parallel query complexity of reconstructing biological and digital phylogenetic trees from simple queries involving their nodes. This is motivated from computational biology, data protection, and computer security settings, which can be abstracted in terms of two parties, a \emph{responder}, Alice, who must correctly answer queries of a given type regarding a degree-$d$

更新日期：2020-06-30
• arXiv.cs.DM Pub Date : 2020-06-26
Josef Rukavicka

Let $\exp^{k,\alpha}$ denote a tetration function defined as follows: $\exp^{1,\alpha}=2^{\alpha}$ and $\exp^{k+1,\alpha}=2^{\exp^{k,\alpha}}$, where $k,\alpha$ are positive integers. Let $\Delta_n$ denote an alphabet with $n$ letters. If $L\subseteq\Delta_n^*$ is an infinite language such that for each $u\in L$ there is $v\in L$ with $\vert v\vert\leq \exp^{k,\alpha}\vert u\vert$ then we call $L$

更新日期：2020-06-30
• arXiv.cs.DM Pub Date : 2020-06-26
Esther M. Arkin; Rathish Das; Jie Gao; Mayank Goswami; Joseph S. B. Mitchell; Valentin Polishchuk; Csaba D. Toth

Motivated by indoor localization by tripwire lasers, we study the problem of cutting a polygon into small-size pieces, using the chords of the polygon. Several versions are considered, depending on the definition of the "size" of a piece. In particular, we consider the area, the diameter, and the radius of the largest inscribed circle as a measure of the size of a piece. We also consider different

更新日期：2020-06-29
• arXiv.cs.DM Pub Date : 2020-06-25
Luke Postle

In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\ge 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log t})$ and hence is $O(t\sqrt{\log t})$-colorable. Recently, Norin, Song and the author showed that every graph with no $K_t$ minor is $O(t(\log t)^{\beta})$-colorable

更新日期：2020-06-29
• arXiv.cs.DM Pub Date : 2020-06-25
Jan Dreier; Philipp Kuinke; Peter Rossmanith

Complex networks are everywhere. They appear for example in the form of biological networks, social networks, or computer networks and have been studied extensively. Efficient algorithms to solve problems on complex networks play a central role in today's society. Algorithmic meta-theorems show that many problems can be solved efficiently. Since logic is a powerful tool to model problems, it has been

更新日期：2020-06-26
• arXiv.cs.DM Pub Date : 2020-06-25
Thomas Fernique

We consider the packings in the plane of discs of radius $1$ and $\sqrt{2}-1$ when the proportions of each type of disc are fixed. The maximal density is determined and the densest packings are described. In particular, it is shown that a phase separation phenomenon appears when there is an excess of small discs.

更新日期：2020-06-26
• arXiv.cs.DM Pub Date : 2020-06-25
Andrea Hornakova; Roberto Henschel; Bodo Rosenhahn; Paul Swoboda

We present an extension to the disjoint paths problem in which additional \emph{lifted} edges are introduced to provide path connectivity priors. We call the resulting optimization problem the lifted disjoint paths problem. We show that this problem is NP-hard by reduction from integer multicommodity flow and 3-SAT. To enable practical global optimization, we propose several classes of linear inequalities

更新日期：2020-06-26
• arXiv.cs.DM Pub Date : 2020-06-24
Ramon Ferrer-i-Cancho; Carlos Gómez-Rodríguez; Juan Luis Esteban

A fundamental problem in network science is the normalization of the topological or physical distance between vertices, that requires understanding the range of variation of the unnormalized distances. Here we investigate the limits of the variation of the physical distance in linear arrangements of the vertices of trees. In particular, we investigate various problems on the sum of edge lengths in

更新日期：2020-06-26
• arXiv.cs.DM Pub Date : 2020-06-25
Nicolas Bousquet; Takehiro Ito; Yusuke Kobayashi; Haruka Mizuta; Paul Ouvrard; Akira Suzuki; Kunihiro Wasa

Let $G$ be a graph and $T_1,T_2$ be two spanning trees of $G$. We say that $T_1$ can be transformed into $T_2$ via an edge flip if there exist two edges $e \in T_1$ and $f$ in $T_2$ such that $T_2= (T_1 \setminus e) \cup f$. Since spanning trees form a matroid, one can indeed transform a spanning tree into any other via a sequence of edge flips, as observed by Ito et al. We investigate the problem

更新日期：2020-06-26
• arXiv.cs.DM Pub Date : 2020-06-25
Wenjun Cai; Aaron Potechin

Graph matrices are a type of matrix which appears when analyzing the sum of squares hierarchy and other methods using higher moments. However, except for rough norm bounds, little is known about graph matrices. In this paper, we take a step towards better understanding graph matrices by determining the spectrum of the singular values of Z-shaped graph matrices.

更新日期：2020-06-26
• arXiv.cs.DM Pub Date : 2020-06-24
Ryan Alweiss; Yang P. Liu; Mehtaab Sawhney

We study discrepancy minimization for vectors in $\mathbb{R}^n$ under various settings. The main result is the analysis of a new simple random process in multiple dimensions through a comparison argument. As corollaries, we obtain bounds which are tight up to logarithmic factors for several problems in online vector balancing posed by Bansal, Jiang, Singla, and Sinha (STOC 2020), as well as linear

更新日期：2020-06-26
• arXiv.cs.DM Pub Date : 2020-06-24
Paul Jungeblut; Torsten Ueckerdt

Let $G = (V,E)$ be a plane graph. A face $f$ of $G$ is guarded by an edge $vw \in E$ if at least one vertex from $\{v,w\}$ is on the boundary of $f$. For a planar graph class $\mathcal{G}$ we ask for the minimal number of edges needed to guard all faces of any $n$-vertex graph in $\mathcal{G}$. We prove that $\lfloor n/3 \rfloor$ edges are always sufficient for quadrangulations and give a construction

更新日期：2020-06-25
• arXiv.cs.DM Pub Date : 2020-06-24
Hideo Bannai; Takuya Mieno; Yuto Nakashima

We revisit the so-called "Three Squares Lemma" by Crochemore and Rytter [Algorithmica 1995] and, using arguments based on Lyndon words, derive a more general variant which considers three overlapping squares which do not necessarily share a common prefix. We also give an improved upper bound of $n\log_2 n$ on the maximum number of (occurrences of) primitively rooted squares in a string of length $n$

更新日期：2020-06-25
• arXiv.cs.DM Pub Date : 2020-06-23

Submodular function maximization has found a wealth of new applications in machine learning models during the past years. The related supermodular maximization models (submodular minimization) also offer an abundance of applications, but they appeared to be highly intractable even under simple cardinality constraints. Hence, while there are well-developed tools for maximizing a submodular function

更新日期：2020-06-25
• arXiv.cs.DM Pub Date : 2020-06-24
Sreeya Ghosh; Sudhakar Sahoo; Sk. Sarif Hassan; Jayanta Kumar Das; Pabitra Pal Choudhury

Cellular Automaton (CA) and an Integral Value Transformation (IVT) are two well established mathematical models which evolve in discrete time steps. Theoretically, studies on CA suggest that CA is capable of producing a great variety of evolution patterns. However computation of non-linear CA or higher dimensional CA maybe complex, whereas IVTs can be manipulated easily. The main purpose of this paper

更新日期：2020-06-25
• arXiv.cs.DM Pub Date : 2020-06-23
Irène Marcovici; Thomas Stoll; Pierre-Adrien Tahay

We introduce a family of block-additive automatic sequences, that are obtained by allocating a weight to each couple of digits, and defining the $n$th term of the sequence as being the total weight of the integer $n$ written in base $k$. Under an additional difference condition on the weight function, these sequences can be interpreted as generalised Rudin-Shapiro sequences, and we prove that they

更新日期：2020-06-24
• arXiv.cs.DM Pub Date : 2020-06-23
Sebastian M. Cioabă; Gordon Royle; Zhao Kuang Tan

In this paper, we study the flip graph on the perfect matchings of a complete graph of even order. We investigate its combinatorial and spectral properties including connections to the signed reversal graph and we improve a previous upper bound on its chromatic number.

更新日期：2020-06-24
• arXiv.cs.DM Pub Date : 2020-06-23
Srinibas Swain; C. Paul Bonnington; Graham Farr; Kerri Morgan

Since the pioneering work of R. M. Foster in the 1930s, many graph repositories have been created to support research in graph theory. This survey reviews many of these graph repositories and summarises the scope and contents of each repository. We identify opportunities for the development of repositories that can be queried in more flexible ways.

更新日期：2020-06-24
• arXiv.cs.DM Pub Date : 2020-06-22

Given a connected vertex-weighted graph $G$, the maximum weight internal spanning tree (MaxwIST) problem asks for a spanning tree of $G$ that maximizes the total weight of internal nodes. This problem is NP-hard and APX-hard, with the currently best known approximation factor $1/2$ (Chen et al., Algorithmica 2019). For the case of claw-free graphs, Chen et al. present an involved approximation algorithm

更新日期：2020-06-24
• arXiv.cs.DM Pub Date : 2020-06-22
Martin Dyer; Catherine Greenhill; Pieter Kleer; James Ross; Leen Stougie

There is a well-known connection between hypergraphs and bipartite graphs, obtained by treating the incidence matrix of the hypergraph as the biadjacency matrix of a bipartite graph. We use this connection to describe and analyse a rejection sampling algorithm for sampling simple uniform hypergraphs with a given degree sequence. Our algorithm uses, as a black box, an algorithm $\mathcal{A}$ for sampling

更新日期：2020-06-23
• arXiv.cs.DM Pub Date : 2020-06-21
Lhouari Nourine; Simon Vilmin

In this paper we are interested in decomposing a dihypergraph $\mathcal{H} = (V, \mathcal{E})$ into simpler dihypergraphs, that can be handled more efficiently. We study the properties of dihypergraphs that can be hierarchically decomposed into trivial dihypergraphs, \ie vertex hypergraph. The hierarchical decomposition is represented by a full labelled binary tree called $\mathcal{H}$-tree, in the

更新日期：2020-06-23
• arXiv.cs.DM Pub Date : 2020-06-22

In the Metric Capacitated Covering (MCC) problem, given a set of balls $\mathcal{B}$ in a metric space $P$ with metric $d$ and a capacity parameter $U$, the goal is to find a minimum sized subset $\mathcal{B}'\subseteq \mathcal{B}$ and an assignment of the points in $P$ to the balls in $\mathcal{B}'$ such that each point is assigned to a ball that contains it and each ball is assigned with at most

更新日期：2020-06-23
• arXiv.cs.DM Pub Date : 2020-06-22
Francesco Dolce; Edita Pelantová

Droubay, Justin and Pirillo that each word of length $n$ contains at most $n+1$ distinct palindromes. A finite "rich word" is a word with maximal number of palindromic factors. The definition of palindromic richness can be naturally extended to infinite words. Sturmian words and Rote complementary symmetric sequences form two classes of binary rich words, while episturmian words and words coding $d$-interval

更新日期：2020-06-23
• arXiv.cs.DM Pub Date : 2020-06-21
Luke Postle

In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\ge 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log t})$ and hence is $O(t\sqrt{\log t})$-colorable. Recently, Norin, Song and the author showed that every graph with no $K_t$ minor is $O(t(\log t)^{\beta})$-colorable

更新日期：2020-06-23
• arXiv.cs.DM Pub Date : 2020-06-20
Calvin Beideman; Karthekeyan Chandrasekaran; Chao Xu

We address counting and optimization variants of multicriteria global min-cut and size-constrained min-$k$-cut in hypergraphs. 1. For an $r$-rank $n$-vertex hypergraph endowed with $t$ hyperedge-cost functions, we show that the number of multiobjective min-cuts is $O(r2^{tr}n^{3t-1})$. In particular, this shows that the number of parametric min-cuts in constant rank hypergraphs for a constant number

更新日期：2020-06-23
• arXiv.cs.DM Pub Date : 2020-06-19
Gwenaël Joret; Piotr Micek; Bruce Reed; Michiel Smid

The clique chromatic number of a graph is the minimum number of colours needed to colour its vertices so that no inclusion-wise maximal clique which is not an isolated vertex is monochromatic. We show that every graph of maximum degree $\Delta$ has clique chromatic number $O\left(\frac{\Delta}{\log~\Delta}\right)$. We obtain as a corollary that every $n$-vertex graph has clique chromatic number $O 更新日期：2020-06-23 • arXiv.cs.DM Pub Date : 2020-06-18 Emmanuel Abbe; Peter Ralli The r-th power of a graph modifies a graph by connecting every vertex pair within distance r. This paper gives a generalization of the Alon-Boppana Theorem for the r-th power of graphs, including irregular graphs. This leads to a generalized notion of Ramanujan graphs, those for which the powered graph has a spectral gap matching the derived Alon-Boppana bound. In particular, we show that certain graphs 更新日期：2020-06-22 • arXiv.cs.DM Pub Date : 2020-06-19 Émilie Charlier; Célia Cisternino; Manon Stipulanti We consider numeration systems based on a$d$-tuple$\mathbf{U}=(U_1,\ldots,U_d)$of sequences of integers and we define$(\mathbf{U},\mathbb{K})$-regular sequences through$\mathbb{K}$-recognizable formal series, where$\mathbb{K}$is any semiring. We show that, for any$d$-tuple$\mathbf{U}$of Pisot numeration systems and any commutative semiring$\mathbb{K}$, this definition does not depend on 更新日期：2020-06-22 • arXiv.cs.DM Pub Date : 2020-06-18 Nina Otter; Mason A. Porter A key concern in network analysis is the study of social positions and roles of actors in a network. The notion of "position" refers to an equivalence class of nodes that have similar ties to other nodes, whereas a "role" is an equivalence class of compound relations that connect the same pairs of nodes. An open question in network science is whether it is possible to simultaneously perform role and 更新日期：2020-06-19 • arXiv.cs.DM Pub Date : 2020-06-18 Leran Cai; Thomas Sauerwald; Luca Zanetti A random walk is a basic stochastic process on graphs and a key primitive in the design of distributed algorithms. One of the most important features of random walks is that, under mild conditions, they converge to a stationary distribution in time that is at most polynomial in the size of the graph. This fundamental property, however, only holds if the graph does not change over time, while on the 更新日期：2020-06-19 • arXiv.cs.DM Pub Date : 2020-06-18 Noga Alon; Sepehr Assadi A recent palette sparsification theorem of Assadi, Chen, and Khanna [SODA'19] states that in every$n$-vertex graph$G$with maximum degree$\Delta$, sampling$O(\log{n})$colors per each vertex independently from$\Delta+1$colors almost certainly allows for proper coloring of$G$from the sampled colors. Besides being a combinatorial statement of its own independent interest, this theorem was shown 更新日期：2020-06-19 • arXiv.cs.DM Pub Date : 2020-06-18 Jesse MulderijDelft University of Technology; Bob HuismanNederlandse Spoorwegen; Denise TönissenVrije Universiteit Amsterdam; Koos van der LindenDelft University of Technology; Mathijs de WeerdtDelft University of Technology In between transportation services, trains are parked and maintained at shunting yards. The conflict-free routing of trains to and on these yards and the scheduling of service and maintenance tasks is known as the train unit shunting and service problem. Efficient use of the capacity of these yards is becoming increasingly important, because of increasing numbers of trains without proportional extensions 更新日期：2020-06-19 • arXiv.cs.DM Pub Date : 2020-06-18 Adil Erzin; Stepan Nazarenko; Gregory Melidi; Roman Plotnikov We consider a Two-Bar Charts Packing Problem (2-BCPP), in which it is necessary to pack two-bar charts (2-BCs) in a unit-height strip of minimum length. The problem is a generalization of the Bin Packing Problem (BPP). Earlier, we proposed an$O(n^2)$-time algorithm that constructs the packing which length at most$2\cdot OPT+1$, where$OPT$is the minimum length of the packing of$n$2-BCs. In this 更新日期：2020-06-19 • arXiv.cs.DM Pub Date : 2020-06-17 Lucas Mol; Narad Rampersad; Jeffrey Shallit An overlap-free (or$\beta$-free) word$w$over a fixed alphabet$\Sigma$is extremal if every word obtained from$w$by inserting a single letter from$\Sigma$at any position contains an overlap (or a factor of exponent at least$\beta$, respectively). We find all lengths which admit an extremal overlap-free binary word. For every extended real number$\beta$such that$2^+\leq\beta\leq 8/3$, we 更新日期：2020-06-19 • arXiv.cs.DM Pub Date : 2020-06-17 Édouard Bonnet; Colin Geniet; Eun Jung Kim; Stéphan Thomassé; Rémi Watrigant The twin-width of a graph$G$is the minimum integer$d$such that$G$has a$d$-contraction sequence, that is, a sequence of$|V(G)|-1$iterated vertex identifications for which the overall maximum number of red edges incident to a single vertex is at most$d$, where a red edge appears between two sets of identified vertices if they are not homogeneous in$G$. We show that if a graph admits a$d$-contraction 更新日期：2020-06-18 • arXiv.cs.DM Pub Date : 2020-06-17 James Trimble This note introduces the exact solver Bute for the exact treedepth problem, along with two variants of the solver. Each of these solvers computes an elimination tree in a bottom-up fashion by finding sets of vertices that induce subgraphs of small treedepth, then combining sets of vertices together with a root vertex to produce larger sets. The algorithms make use of a trie data structure to reduce 更新日期：2020-06-18 • arXiv.cs.DM Pub Date : 2020-06-17 Christoph Hertrich; Christian Weiß; Heiner Ackermann; Sandy Heydrich; Sven O. Krumke In this paper we study a proportionate flow shop of batching machines with release dates and a fixed number$m \geq 2$of machines. The scheduling problem has so far barely received any attention in the literature, but recently its importance has increased significantly, due to applications in the industrial scaling of modern bio-medicine production processes. We show that for any fixed number of machines 更新日期：2020-06-18 • arXiv.cs.DM Pub Date : 2020-06-17 Isha Agarwal; Matvey Borodin; Aidan Duncan; Kaylee Ji; Tanya Khovanova; Shane Lee; Boyan Litchev; Anshul Rastogi; Garima Rastogi; Andrew Zhao We analyze a coin-based game with two players where, before starting the game, each player selects a string of length$n$comprised of coin tosses. They alternate turns, choosing the outcome of a coin toss according to specific rules. As a result, the game is deterministic. The player whose string appears first wins. If neither player's string occurs, then the game must be infinite. We study several 更新日期：2020-06-18 • arXiv.cs.DM Pub Date : 2020-06-16 Matthieu Rosenfeld We propose a new proof technique that aims to be applied to the same problems as the Lov\'asz Local Lemma or the entropy-compression method. We present this approach in the context of non-repetitive colorings and we use it to improve upper-bounds relating different non-repetitive numbers to the maximal degree of a graph. It seems that there should be other more interesting applications to the presented 更新日期：2020-06-16 • arXiv.cs.DM Pub Date : 2020-06-16 Masood Masjoody In this paper the notions of {\em trapping} and {\em confining} the robber on a graph are introduced. We present some necessary conditions for graphs$G$not containing the path on$k$vertices (for some$k\ge 4$) as an induced subgraph so that$k-3$cops do not have a winning strategy on$G$. Utilizing the latter conditions together with a characterization of cographs, we show that on cograph the 更新日期：2020-06-16 • arXiv.cs.DM Pub Date : 2020-06-16 Matthew Kwan; Lisa Sauermann; Yufei Zhao Sometimes, it is possible to represent a complicated polytope as a projection of a much simpler polytope. To quantify this phenomenon, the extension complexity of a polytope$P$is defined to be the minimum number of facets in a (possibly higher-dimensional) polytope from which$P\$ can be obtained as a (linear) projection. This notion has been studied for several decades, motivated by its relevance

更新日期：2020-06-16
• arXiv.cs.DM Pub Date : 2020-06-15
Sebastian Buß; Hendrik Molter; Rolf Niedermeier; Maciej Rymar

The betweenness centrality of a graph vertex measures how often this vertex is visited on shortest paths between other vertices of the graph. In the analysis of many real-world graphs or networks, betweenness centrality of a vertex is used as an indicator for its relative importance in the network. In particular, it is among the most popular tools in social network analysis. In recent years, a growing

更新日期：2020-06-15
• arXiv.cs.DM Pub Date : 2020-06-15
Vidya Sagar Sharma; Piyush Srivastava

In neuroscience, an important aspect of understanding the function of a neural circuit is to determine which, if any, of the neurons in the circuit are vital for the biological behavior governed by the neural circuit. A similar problem is to determine whether a given small set of neurons may be enough for the behavior to be displayed, even if all other neurons in the circuit are deactivated. Such a

更新日期：2020-06-15
• arXiv.cs.DM Pub Date : 2020-06-13
Prajnanaswaroopa S; Geetha J; Somasundaram K

The Total coloring conjecture states that any simple graph G with maximum degree D can be totally colored with at most D+2 colors. In this paper, we have obtained the total chromatic number for some classes of Cayley graphs.

更新日期：2020-06-13
Contents have been reproduced by permission of the publishers.

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