Related papers: Conjugacy in Patience Sorting monoids
In many matching markets--such as athlete recruitment or academic admissions--participants on one side are evaluated by attribute vectors known to the other side, which in turn applies individual \emph{salience vectors} to assign relative…
We present a solution to the conjugacy decision problem and the conjugacy search problem in Garside groups, which is theoretically simpler than the usual one, with no loss of efficiency. This is done by replacing the well known cycling and…
Convexity in a network (graph) has been recently defined as a property of each of its subgraphs to include all shortest paths between the nodes of that subgraph. It can be measured on the scale [0, 1] with 1 being assigned to fully convex…
The associahedron is a polytope whose graph is the graph of flips on triangulations of a convex polygon. Pseudotriangulations and multitriangulations generalize triangulations in two different ways, which have been unified by Pilaud and…
The monoids of simplicial endomorphisms, i.e. the monoids of endomorphisms in the simplicial category, are submonoids of monoids one finds in Temperley-Lieb algebras, and as the monoids of Temperley-Lieb algebras are linked to situations…
The topic of stable matchings (marriages) in a bipartite graph has become widely popular, starting with the appearance of the classical work by Gale and Shapley. We give a detailed survey on selected known results in this field that…
Graph products of monoids provide a common framework for direct and free products, and graph monoids (also known as free partially commutative monoids). If the monoids in question are groups, then any graph product is, of course, a group.…
In this paper we consider endomorphisms of an undirected cycle graph from Semigroup Theory perspective. Our main aim is to present a process to determine sets of generators with minimal cardinality for the monoids $wEnd(C_n)$ and $End(C_n)$…
Mixed order phase transitions (MOT), which display discontinuous order parameter and diverging correlation length, appear in several seemingly unrelated settings ranging from equilibrium models with long-range interactions to models far…
A recent line of research has concentrated on exploring the links between analytic and combinatorial theories of submodularity, uncovering several key connections between them. In this context, Lov\'asz initiated the study of matroids from…
Let $X$ be a non-empty set of positive integers and $X^*=X\setminus \{1\}$. The divisibility graph $D(X)$ has $X^*$ as the vertex set and there is an edge connecting $a$ and $b$ with $a, b\in X^*$ whenever $a$ divides $b$ or $b$ divides…
A circle graph is the intersection graph of a set of chords in a circle. Keil [Discrete Applied Mathematics, 42(1):51-63, 1993] proved that Dominating Set, Connected Dominating Set, and Total Dominating Set are NP-complete in circle graphs.…
The diameter of a graph measures the maximal distance between any pair of vertices. The diameters of many small-world networks, as well as a variety of other random graph models, grow logarithmically in the number of nodes. In contrast, the…
In mathematics and computer science, connectivity is one of the basic concepts of matroid theory: it asks for the minimum number of elements which need to be removed to disconnect the remaining nodes from each other. It is closely related…
In this article, we consider the monoid of all monotone and order-decreasing partial transformations denoted as $\mathcal{DORP}_{n}$ on an $n$ ordered chain $[n]=\{1, \ldots,n\}$, its two-sided ideal $I(n,p)= \{\rho \in \mathcal{DORP}_{n} :…
The exchange graph of a cluster algebra encodes the combinatorics of mutations of clusters. Through the recent "categorifications" of cluster algebras using representation theory one obtains a whole variety of exchange graphs associated…
In this article we establish relationships between Leavitt path algebras, talented monoids and the adjacency matrices of the underlying graphs. We show that indeed the adjacency matrix generates in some sense the group action on the…
We classify the $\mathscr{R}$-cross-sections of the monoid of order-preserving transformations on the $n$-element chain in terms of certain binary trees.
Patience Sorting is a combinatorial algorithm that can be viewed as an iterated, non-recursive form of the Schensted Insertion Algorithm. In recent work the authors have shown that Patience Sorting provides an algorithmic description for…
The random reversal graph offers new perspectives, allowing to study the connectivity of genomes as well as their most likely distance as a function of the reversal rate. Our main result shows that the structure of the random reversal graph…