Directed Path-Decompositions

2020 ◽  
Vol 34 (1) ◽  
pp. 415-430
Author(s):  
Joshua Erde
Keyword(s):  
Directed Path ◽  
Path Decompositions

Path decompositions of digraphs

1974 ◽  
Vol 10 (3) ◽  
pp. 421-427 ◽  
Author(s):  
Brian R. Alspach ◽  
Norman J. Pullman
Keyword(s):  
Path Decomposition ◽  
The Family ◽  
Path Decompositions ◽  
Simple Paths ◽  
Equality Holding ◽  
Path Number

A path decomposition of a digraph G (having no loops or multiple arcs) is a family of simple paths such that every arc of G lies on precisely one of the paths of the family. The path number, pn(G) is the minimal number of paths necessary to form a path decomposition of G.We show that pn(G) ≥ max{0, od(v)-id(v)} the sum taken over all vertices v of G, with equality holding if G is acyclic. If G is a subgraph of a tournament on n vertices we show that pn(G) ≤ with equality holding if G is transitive.We conjecture that pn(G) ≤ for any digraph G on n vertices if n is sufficiently large, perhaps for all n ≥ 4.

scholarly journals Path decompositions of digraphs and their applications to Weyl algebra

2015 ◽  
Vol 67 ◽  
pp. 36-54 ◽  
Author(s):  
Askar Dzhumadil'daev ◽  
Damir Yeliussizov
Keyword(s):  
Weyl Algebra ◽  
Path Decompositions

scholarly journals Controllability of Weighted and Directed Networks with Nonidentical Node Dynamics

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Linying Xiang ◽  
Jonathan J. H. Zhu ◽  
Fei Chen ◽  
Guanrong Chen
Keyword(s):  
Graph Theory ◽  
Control Theory ◽  
Matrix Theory ◽  
Sufficient Conditions ◽  
Directed Path ◽  
Directed Networks ◽  
Leaders And Followers ◽  
Heterogenous Network

The concept of controllability from control theory is applied to weighted and directed networks with heterogenous linear or linearized node dynamics subject to exogenous inputs, where the nodes are grouped into leaders and followers. Under this framework, the controllability of the controlled network can be decomposed into two independent problems: the controllability of the isolated leader subsystem and the controllability of the extended follower subsystem. Some necessary and/or sufficient conditions for the controllability of the leader-follower network are derived based on matrix theory and graph theory. In particular, it is shown that a single-leader network is controllable if it is a directed path or cycle, but it is uncontrollable for a complete digraph or a star digraph in general. Furthermore, some approaches to improving the controllability of a heterogenous network are presented. Some simulation examples are given for illustration and verification.

scholarly journals p-box: a new graph model

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Mauricio Soto ◽  
Christopher Thraves-Caro
Keyword(s):  
Graph Theory ◽  
Graph Model ◽  
Directed Path ◽  
Outerplanar Graphs ◽  
New Model ◽  
Model Based ◽  
Representative Element ◽  
International Audience ◽  
Graph Families

Graph Theory International audience In this document, we study the scope of the following graph model: each vertex is assigned to a box in ℝd and to a representative element that belongs to that box. Two vertices are connected by an edge if and only if its respective boxes contain the opposite representative element. We focus our study on the case where boxes (and therefore representative elements) associated to vertices are spread in ℝ. We give both, a combinatorial and an intersection characterization of the model. Based on these characterizations, we determine graph families that contain the model (e. g., boxicity 2 graphs) and others that the new model contains (e. g., rooted directed path). We also study the particular case where each representative element is the center of its respective box. In this particular case, we provide constructive representations for interval, block and outerplanar graphs. Finally, we show that the general and the particular model are not equivalent by constructing a graph family that separates the two cases.

scholarly journals Fully Dynamic Transitive Closure in Plane Dags with one Source and one Sink

1994 ◽  
Vol 1 (30) ◽  
Author(s):  
Thore Husfeldt
Keyword(s):  
Transitive Closure ◽  
Upper Bounds ◽  
Directed Acyclic Graphs ◽  
Closure Problem ◽  
Directed Path ◽  
Lower And Upper Bounds ◽  
New Techniques ◽  
Time Dynamic ◽  
Acyclic Graphs ◽  
Closure Algorithm

We give an algorithm for the Dynamic Transitive Closure Problem for planar directed acyclic graphs with one source and one sink. The graph can be updated in logarithmic time under arbitrary edge insertions and deletions that preserve the embedding. Queries of the form `is there a directed path from u to v ?' for arbitrary vertices u and v can be answered in logarithmic time. The size of the data structure and the initialisation time are linear in the number of edges.<br /> <br />The result enlarges the class of graphs for which a logarithmic (or even polylogarithmic) time dynamic transitive closure algorithm exists. Previously, the only algorithms within the stated resource bounds put restrictions on the topology of the graph or on the delete operation. To obtain our result, we use a new characterisation of the transitive closure in plane graphs with one source and one sink and introduce new techniques to exploit this characterisation.<br /> <br />We also give a lower bound of Omega(log n/log log n) on the amortised complexity of the problem in the cell probe model with logarithmic word size. This is the first dynamic directed graph problem with almost matching lower and upper bounds.

scholarly journals Path Decompositions of Tournaments

2021 ◽  
pp. 195-200
Author(s):  
António Girão ◽  
Bertille Granet ◽  
Daniela Kühn ◽  
Allan Lo ◽  
Deryk Osthus
Keyword(s):  
Path Decompositions

scholarly journals On the restrictive channel thickness estimation

2007 ◽  
Vol 20 (3) ◽  
pp. 499-506
Author(s):  
Iskandar Karapetyan
Keyword(s):  
Positive Integer ◽  
Heuristic Algorithms ◽  
Clique Number ◽  
Directed Path ◽  
Channel Routing ◽  
Routing Problem ◽  
Thickness Estimation ◽  
Channel Thickness ◽  
Np Complete ◽  
Routing Method

Channel routing is an important phase of physical design of LSI and VLSI chips. The channel routing method was first proposed by Akihiro Hashimoto and James Stevens [1]. The method was extensively studied by many authors and applied to different technologies. At present there are known many effective heuristic algorithms for channel routing. A. LaPaugh [2] proved that the restrictive routing problem is NP-complete. In this paper we prove that for every positive integer k there is a restrictive channel C for which ?(C)>? (HG)+L(VG)+k, where ? (C) is the thickness of the channel, ?(HG) is clique number of the horizontal constraints graph HG and L(VG) is the length of the longest directed path in the vertical constraints graph VG.

scholarly journals Random minimal directed spanning trees and Dickman-type distributions

2004 ◽  
Vol 36 (03) ◽  
pp. 691-714 ◽  
Author(s):  
Mathew D. Penrose ◽  
Andrew R. Wade
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Dirichlet Distribution ◽  
Maximal Length ◽  
Directed Path ◽  
Limiting Distributions ◽  
Limit Distributions ◽  
Tree Construction ◽  
Directed Spanning Tree ◽  
Westerly Direction

In Bhatt and Roy's minimal directed spanning tree construction fornrandom points in the unit square, all edges must be in a south-westerly direction and there must be a directed path from each vertex to the root placed at the origin. We identify the limiting distributions (for largen) for the total length of rooted edges, and also for the maximal length of all edges in the tree. These limit distributions have been seen previously in analysis of the Poisson-Dirichlet distribution and elsewhere; they are expressed in terms of Dickman's function, and their properties are discussed in some detail.

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