Related papers: Chains of binary paths and shifted tableaux

We present a bijection between some quadrangular dissections of an hexagon and unrooted binary trees, with interesting consequences for enumeration, mesh compression and graph sampling. Our bijection yields an efficient uniform random…

Let $P,Q$ be longest paths in a simple graph. We analyze the possible connections between the components of $P\cup Q\setminus (V(P)\cap V(Q))$ and introduce the notion of a bi-traceable graph. We use the results for all the possible…

We consider problems to make a given bidirected graph strongly connected with minimum cardinality of additional signs or additional arcs. For the former problem, we show the minimum number of additional signs and give a linear-time…

It is known that for the 2n-step symmetric simple random walk on Z, two events have the same probability if and only if their sets of paths have the same cardinality. In this article, we construct two kinds of bijections between sets of…

When considering the number of subtrees of trees, the extremal structures which maximize this number among binary trees and trees with a given maximum degree lead to some interesting facts that correlate to other graphical indices in…

We described a simple algorithm running in linear time for each fixed constant $k$, that either establishes that the pathwidth of a graph $G$ is greater than $k$, or finds a path-decomposition of $G$ of width at most $O(2^{k})$. This…

This article focuses on the identification of the number of paths with different lengths between pairs of nodes in complex networks and how, by providing comprehensive information about the network topology, such an information can be…

In this work we introduce and study tree-like tableaux, which are certain fillings of Ferrers diagrams in simple bijection with permutation tableaux and alternative tableaux. We exhibit an elementary insertion procedure on our tableaux…

A path graph is the intersection graph of paths in a tree. A directed path graph is the intersection graph of paths in a directed tree. Even if path graphs and directed path graphs are characterized very similarly, their recognition…

Let $P(G)=(P_{0}(G),P_{1}(G),\cdots, P_{\rho}(G))$ be the path sequence of a graph $G$, where $P_{i}(G)$ is the number of paths with length $i$ and $\rho$ is the length of a longest path in $G$. In this paper, we first give the path…

Numerous problems of both theoretical and practical interest are related to finding shortest (or otherwise optimal) paths in networks, frequently in the presence of some obstacles or constraints. A somewhat related class of problems focuses…

In this paper, we provide a novel enumeration algorithm for the set of all walks of a given length within a directed graph. Our algorithm has worst-case constant delay between outputting succinct representations of such walks, after a…

It is known that both the number of Dyck paths with $2n$ steps and $k$ peaks, and the number of Dyck paths with $2n$ steps and $k$ steps at odd height follow the Narayana distribution. In this paper we present a bijection which explicitly…

We provide enumerating results for partial knight's paths of a given size. We prove algebraically that zigzag knight's paths of a given size ending on the $x$-axis are enumerated by the generalized Catalan numbers, and we give a…

We study augmenting a plane Euclidean network with a segment, called a shortcut, to minimize the largest distance between any two points along the edges of the resulting network. Problems of this type have received considerable attention…

We study problems of reconfiguration of shortest paths in graphs. We prove that the shortest reconfiguration sequence can be exponential in the size of the graph and that it is NP-hard to compute the shortest reconfiguration sequence even…

For sufficiently tame paths in $\mathbb{R}^n$, Euclidean length provides a canonical parametrization of a path by length. In this paper we provide such a parametrization for all continuous paths. This parametrization is based on an…

A binary modified de Bruijn sequence is an infinite and periodic binary sequence derived by removing a zero from the longest run of zeros in a binary de Bruijn sequence. The minimal polynomial of the modified sequence is its unique…

We give a bijective proof of a conjecture of Regev and Vershik on the equality of two multisets of hook numbers of certain skew-Young diagrams. The bijection proves a result that is stronger and more symmetric than the original conjecture,…

A problem studied in Systems Biology is how to find shortest paths in metabolic networks. Unfortunately, simple (i.e., graph theoretic) shortest paths do not properly reflect biochemical facts. An approach to overcome this issue is to use…