Rabinovitch showed in 1978 that the interval orders having a representation consisting of only closed unit intervals have order dimension at most 3. This article shows that the same dimension bound applies to two other classes of posets: those having a representation consisting of unit intervals (but with a mixture of open and closed intervals allowed) and those having a representation consisting of closed intervals with lengths in $\{0,1\}$.
We study restricted homomorphism dualities in the context of classes with bounded expansion. This presents a generalization of restricted dualities obtained earlier for bounded degree graphs and also for proper minor closed classes. This is related to distance coloring of graphs and to the ''approximative version'' of Hadwiger conjecture.
We introduce and characterise grid classes, which are natural generalisations of other well-studied permutation classes. This characterisation allows us to give a new, short proof of the Fibonacci dichotomy: the number of permutations of length n in a permutation class is either at least as large as the nth Fibonacci number or is eventually polynomial.
It is well known that Universal Cycles of $k$-letter words on an $n$-letter alphabet exist for all $k$ and $n$. In this paper, we prove that Universal Cycles exist for restricted classes of words, including: non-bijections, equitable words (under suitable restrictions), ranked permutations, and "passwords".
A matching $M$ in a graph $G$ is said to be uniquely restricted if there is no other matching in $G$ that matches the same set of vertices as $M$. We describe a polynomial-time algorithm to compute a maximum cardinality uniquely restricted matching in an interval graph, thereby answering a question of Golumbic et al. ("Uniquely restricted matchings", M. C. Golumbic, T. Hirst and M. Lewenstein, Algorithmica, 31:139--154, 2001). Our algorithm actually solves the more general problem of computing a maximum cardinality "strong independent set" in an interval nest digraph, which may be of independe…
This work considers lattice walks restricted to the quarter plane, with steps taken from a set of cardinality three. We present a complete classification of the generating functions of these walks with respect to the classes algebraic, transcendental holonomic and non-holonomic. The principal results are a new algebraic class related to Kreweras' walks; two new non-holonomic classes; and enumerative data on some other classes. These results provide strong evidence for conjectures which use combinatorial criteria to classify the generating functions all nearest neighbour walks in the quarter pl…
Let A be a finite set with at least two elements. The composition of two classes I and J of operations on A, is defined as the set of all compositions of functions in I with functions in J. This binary operation gives a monoid structure to the set E_A of all equational classes of operations on A. The set E_A of equational classes of operations on A also constitutes a complete distributive lattice under intersection and union. Clones of operations, i.e. classes containing all projections and idempotent under class composition, also form a lattice which is strictly contained in E_A. In the Boole…
Zeilberger's enumeration schemes can be used to completely automate the enumeration of many permutation classes. We extend his enumeration schemes so that they apply to many more permutation classes and describe the Maple package WILFPLUS, which implements this process. We also compare enumeration schemes to three other systematic enumeration techniques: generating trees, substitution decompositions, and the insertion encoding.
We generalize Monge property of real matrices for interval matrices. We define two classes of interval matrices with Monge property - in a strong and in a weak sense. We study fundamental properties of both classes. We show several different characterizations of the strong Monge property. For weak Monge property we give a polynomial characterization and several sufficient and necessary conditions. For both classes we study closure properties. We further propose a generalization of an algorithm by Deineko \& Filonenko which for a given matrix returns row and column permutations such that the pe…
We investigate a generalization of stacks that we call $\mathcal{C}$-machines. We show how this viewpoint rapidly leads to functional equations for the classes of permutations that $\mathcal{C}$-machines generate, and how these systems of functional equations can frequently be solved by either the kernel method or, much more easily, by guessing and checking. General results about the rationality, algebraicity, and the existence of Wilfian formulas for some classes generated by $\mathcal{C}$-machines are given. We also draw attention to some relatively small permutation classes which, although …
Lonesum matrices are matrices that are uniquely reconstructible from their row and column sum vectors. These matrices are enumerated by the poly-Bernoulli numbers that are related to the multiple zeta values and have a rich literature in number theory. Combinatorially, lonesum matrices are in bijection with many other combinatorial objects: several permutation classes, other matrix classes, acyclic orientations in graphs etc. Motivated of these facts, we study in this paper lonesum matrices with restriction on the number of columns and rows of the same type.
An interval $k$-graph is the intersection graph of a family $\mathcal{I}$ of intervals of the real line partitioned into at most $k$ classes with vertices adjacent if and only if their corresponding intervals intersect and belong to different classes. In this paper we discuss the interval $k$-graphs that are the incomparability graphs of orders; i.e., cocomparability interval $k$-graphs or interval $k$-orders. Interval $2$-orders have been characterized in many ways, but we show that analogous characterizations do not carry over to interval $k$-orders, for $k > 2$. We describe the structure of…
A set partition avoids a pattern if no subdivision of that partition standardizes to the pattern. There exists a bijection between set partitions and restricted growth functions (RGFs) on which Wachs and White defined four statistics of interest to this work. We first characterize the restricted growth functions of several avoidance classes based on partitions of size four, enumerate these avoidance classes, and consider the distribution of the Wachs and White statistics across these avoidance classes. We also investigate the equidistribution of statistics between avoidance classes based on mu…
Bernshteyn and Lee defined a new notion, weak degeneracy, which is slightly weaker than the ordinary degeneracy. It is proved that strictly $f$-degenerate transversal is a common generalization of list coloring, $L$-forested-coloring and DP-coloring. In this paper, we consider three classes of graphs, including planar graphs without any configuration in Fig. 2, toroidal graphs without any configuration in Fig. 5, and planar graphs without intersecting $5$-cycles. We give structural results for each class of graphs, and prove each structure is reducible for weakly $3$-degenerate and the existen…
We collect some general results on graph limits associated to hereditary classes of graphs. As examples, we consider some classes defined by forbidden subgraphs and some classes of intersection graphs, including triangle-free graphs, chordal graphs, cographs, interval graphs, unit interval graphs, threshold graphs, and line graphs.
Let $E_n^r=\{[τ]_a=(τ_1^{(a_1)},...,τ_n^{(a_n)})| τ\in S_n,\ 1\leq a_i\leq r\}$ be the set of all signed permutations on the symbols 1,2,...,n with signs 1,2,...,r. We prove, for every 2-letter signed pattern $[τ]_a$, that the number of $[τ]_a$-avoiding signed permutations in $E_n^r$ is given by the formula $\sum\limits_{j=0}^n j!(r-1)^j{n\choose j}^2$. Also we prove that there are only one Wilf class for r=1, four Wilf classes for r=2, and six Wilf classes for $r\geq 3$.
In the mid-1990s, Stanley and Stembridge conjectured that the chromatic symmetric functions of claw-free co-comparability (also called incomparability) graphs were e-positive. The quest for the proof of this conjecture has led to an examination of other, related graph classes. In 2013 Guay-Paquet proved that if unit interval graphs are e-positive, that implies claw-free incomparability graphs are as well. Inspired by this approach, we consider a related case and prove that unit interval graphs whose complement is also a unit interval graph are e-positive. We introduce the concept of strongly $…
In 1969, Roberts introduced proper and unit interval graphs and proved that these classes are equal. Natural generalizations of unit interval graphs called $k$-length interval graphs were considered in which the number of different lengths of intervals is limited by $k$. Even after decades of research, no insight into their structure is known and the complexity of recognition is open even for $k=2$. We propose generalizations of proper interval graphs called $k$-nested interval graphs in which there are no chains of $k+1$ intervals nested in each other. It is easy to see that $k$-nested interv…
This paper unifies problems and results related to (embedding) universal and homomorphism universal structures. On the one side we give a new combinatorial proof of the existence of universal objects for homomorphism defined classes of structures (thus reproving a result of Cherlin, Shelah and Shi) and on the other side this leads to the new proof of the existence of dual objects (established by Nešetřil and Tardif). Our explicite approach has further applications to special structures such as variants of the rational Urysohn space. We also solve a related extremal problem which shows the opti…
In this work, we introduce the \emph{interval permutation segment (IP-SEG)} model that naturally generalizes the geometric intersection models of interval and permutation graphs. We study properties of two graph classes that arise from the IP-SEG model and present a family of forbidden subgraphs for these classes. In addition, we present polynomial algorithms for the clique and independent set problems on these classes, when the model is given as part of the input.
To each finite subset of $\mathbb{Z}^2$ (a diagram), one can associate a subvariety of a complex Grassmannian (a diagram variety), and a representation of a symmetric group (a Specht module). Liu has conjectured that the cohomology class of a diagram variety is represented by the Frobenius characteristic of the corresponding Specht module. We give a counterexample to this conjecture. However, we show that for the diagram variety of a permutation diagram, Liu's conjectured cohomology class $σ$ is at least an upper bound on the actual class $τ$, in the sense that $σ- τ$ is a nonnegative linear c…
An odd diagram class is a set of permutations with the same odd diagram. Brenti, Carnevale and Tenner showed that each odd diagram class is an interval in the Bruhat order. They conjectured that such intervals are rank-symmetric. In this paper, we present an algorithm to partition an odd diagram class in a uniform manner. As an application, we obtain that the Poincaré polynomial of an odd diagram class factors into polynomials of the form $1+t+\cdots+t^m$. This in particular resolves the conjecture of Brenti, Carnevale and Tenner.
In this paper, we provide algorithms to rank, unrank, and randomly generate certain degree-restricted classes of Cayley trees. Specifically, we consider classes of trees that have a given degree sequence or a given multiset of degrees. If the underlying set of trees have n vertices, then the largest ranks involved in each case are of order n! so that it takes O(nlog(n)) bits just to write down the ranks. Our ranking and unranking algorithms for these degree-restricted classes are as efficient as can be expected since we show that they require O(n^2log(n)) bit operations if the underlying trees…
In this paper, we define and characterize signed interval graphs and bigraphs introducing the concept of negative interval. Also we have shown that these classes of graphs are respectively a generalization of well known classes of interval graphs and interval bigraphs. In this context we have observed that signed interval graphs coincide with the complement of Threshold tolerance graphs(co-TT graphs) introduced by Monma, Reed and Trotter \cite{22}. Finally, we have solved the open problem of forbidden induced subgraph characterization of co-TT graphs posed by them in the same paper.
Baker devised a powerful technique to obtain approximation schemes for various problems restricted to planar graphs. Her technique can be directly extended to various other graph classes, among the most general ones the graphs avoiding a fixed apex graph as a minor. Further generalizations (e.g., to all proper minor closed graph classes) are known, but they use a combination of techniques and usually focus on somewhat restricted classes of problems. We present a new type of graph decompositions (thin systems of overlays) generalizing Baker's technique and leading to straightforward polynomial-…
Baker devised a technique to obtain approximation schemes for many optimization problems restricted to planar graphs; her technique was later extended to more general graph classes. In particular, using the Baker's technique and the minor structure theorem, Dawar et al. gave Polynomial-Time Approximation Schemes (PTAS) for all monotone optimization problems expressible in the first-order logic when restricted to a proper minor-closed class of graphs. We define a Baker game formalizing the notion of repeated application of Baker's technique interspersed with vertex removal, prove that monotone …
Klavik et al. [arXiv:1207.6960] recently introduced a generalization of recognition called the bounded representation problem which we study for the classes of interval and proper interval graphs. The input gives a graph G and in addition for each vertex v two intervals L_v and R_v called bounds. We ask whether there exists a bounded representation in which each interval I_v has its left endpoint in L_v and its right endpoint in R_v. We show that the problem can be solved in linear time for interval graphs and in quadratic time for proper interval graphs. Robert's Theorem states that the class…
In social networks the {\sc Strong Triadic Closure} is an assignment of the edges with strong or weak labels such that any two vertices that have a common neighbor with a strong edge are adjacent. The problem of maximizing the number of strong edges that satisfy the strong triadic closure was recently shown to be NP-complete for general graphs. Here we initiate the study of graph classes for which the problem is solvable. We show that the problem admits a polynomial-time algorithm for two unrelated classes of graphs: proper interval graphs and trivially-perfect graphs. To complement our result…
A set of vertices in a graph is c-colorable if the subgraph induced by the set has a proper c-coloring. In this paper, we study the problem of finding a step-by-step transformation (reconfiguration) between two c-colorable sets in the same graph. This problem generalizes the well-studied Independent Set Reconfiguration problem. As the first step toward a systematic understanding of the complexity of this general problem, we study the problem on classes of perfect graphs. We first focus on interval graphs and give a combinatorial characterization of the distance between two c-colorable sets. Th…
