scholarly journals The Edmonds-Gallai Decomposition for the $k$-Piece Packing Problem

10.37236/1905 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Marek Janata ◽  
Martin Loebl ◽  
Jácint Szabó
Keyword(s):  
Polynomial Algorithm ◽  
Undirected Graph ◽  
Packing Problem ◽  
Matching Problem ◽  
Graph Packing ◽  
Path Packing ◽  
New Type ◽  
Vertex Sets ◽  
Simple Connected Graph ◽  
Type Decomposition

Generalizing Kaneko's long path packing problem, Hartvigsen, Hell and Szabó consider a new type of undirected graph packing problem, called the $k$-piece packing problem. A $k$-piece is a simple, connected graph with highest degree exactly $k$ so in the case $k=1$ we get the classical matching problem. They give a polynomial algorithm, a Tutte-type characterization and a Berge-type minimax formula for the $k$-piece packing problem. However, they leave open the question of an Edmonds-Gallai type decomposition. This paper fills this gap by describing such a decomposition. We also prove that the vertex sets coverable by $k$-piece packings have a certain matroidal structure.

scholarly journals An Edmonds-Gallai-Type Decomposition for the j-Restricted k-Matching Problem

10.37236/7837 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Yanjun Li ◽  
Jácint Szabó
Keyword(s):  
Positive Integer ◽  
Undirected Graph ◽  
Decomposition Theorem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Alternative Proof ◽  
Connected Components ◽  
Matching Problem ◽  
Np Hardness ◽  
Type Decomposition

Given a non-negative integer $j$ and a positive integer $k$, a $j$-restricted $k$-matching in a simple undirected graph is a $k$-matching, so that each of its connected components has at least $j+1$ edges. The maximum non-negative node weighted $j$-restricted $k$-matching problem was recently studied by Li who gave a polynomial-time algorithm and a min-max theorem for $0 \leqslant j < k$, and also proved the NP-hardness of the problem with unit node weights and $2 \leqslant k \leqslant j$. In this paper we derive an Edmonds–Gallai-type decomposition theorem for the $j$-restricted $k$-matching problem with $0 \leqslant j < k$, using the analogous decomposition for $k$-piece packings given by Janata, Loebl and Szabó, and give an alternative proof to the min-max theorem of Li.

scholarly journals A Note on $\mathtt{V}$-free 2-matchings

10.37236/5258 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Kristóf Bérczi ◽  
Attila Bernáth ◽  
Máté Vizer
Keyword(s):  
Bipartite Graph ◽  
Packing Problem ◽  
Bipartite Graphs ◽  
Matching Problem ◽  
Hypergraph Matching ◽  
Path Packing ◽  
Np Complete

Motivated by a conjecture of Liang, we introduce a restricted path packing problem in bipartite graphs that we call a $\mathtt{V}$-free $2$-matching. We verify the conjecture through a weakening of the hypergraph matching problem. We close the paper by showing that it is NP-complete to decide whether one of the color classes of a bipartite graph can be covered by a $\mathtt{V}$-free $2$-matching.

Research about Post Processing for Fagor CNC System Based on Visual C++

2011 ◽  
Vol 80-81 ◽  
pp. 1273-1277
Author(s):  
Chao Wang ◽  
Yuan Yao ◽  
Qian Sheng Zhao ◽  
Qing Xi Hu
Keyword(s):  
File Format ◽  
Post Processing ◽  
Cnc Machine ◽  
Cnc System ◽  
Matching Problem ◽  
Processing Procedure ◽  
Cnc Programming ◽  
New Type ◽  
Machine Systems ◽  
Cam Systems

Post processing is one of the key technologies of CNC programming. Nowadays, the wide application of new type CNC machines is limited by the matching problem between the CAM systems and the developing CNC machine systems. This paper based on XDK9070 CNC engraving machine, researched the file format of cutter location files generated from Pro/Toolmaker and the CNC program format of FAGOR system, expounded the method and process for post processing, designed a special post processing procedure for FAGOR CNC system based on Visual C++, verified the correctness and practicality of the post processing procedure by a example of product processing.

scholarly journals Local improvement algorithms for a path packing problem: A performance analysis based on linear programming

2021 ◽  
Vol 49 (1) ◽  
pp. 62-68
Author(s):  
K.M.J. De Bontridder ◽  
B.V. Halldórsson ◽  
M.M. Halldórsson ◽  
C.A.J. Hurkens ◽  
J.K. Lenstra ◽  
...  
Keyword(s):  
Linear Programming ◽  
Performance Analysis ◽  
Packing Problem ◽  
Local Improvement ◽  
Path Packing ◽  
A Performance

scholarly journals On the bipartite graph packing problem

2017 ◽  
Vol 227 ◽  
pp. 149-155 ◽  
Author(s):  
Bálint Vásárhelyi
Keyword(s):  
Bipartite Graph ◽  
Packing Problem ◽  
Graph Packing

Cut Star of an Undirected Graph

2021 ◽  
pp. 2150006
Author(s):  
Saeid Hanifehnezhad ◽  
Ardeshir Dolati
Keyword(s):  
Data Structure ◽  
Global Minimum ◽  
Undirected Graph ◽  
Minimum Cut ◽  
New Type

Suppose that [Formula: see text] is an undirected graph. An ordered pair [Formula: see text] of the vertices of the graph [Formula: see text] is called a pendant pair for the graph if [Formula: see text] is a minimum cut separating [Formula: see text] and [Formula: see text] Stoer and Wagner obtained a global minimum cut of [Formula: see text] by using pendant pairs of [Formula: see text] and its contractions. A Gomory Hu tree of the graph [Formula: see text] is a very useful data structure which gives us all the minimum s-t cuts of [Formula: see text] for every pair of distinct vertices [Formula: see text] and [Formula: see text] In this paper, we construct a new type of tree for the graph [Formula: see text] called cut star, by using pendant pairs of [Formula: see text] and its contractions. A cut star of the graph [Formula: see text] is constructed more quickly than a Gomory Hu tree of [Formula: see text] We characterize a class of graphs for which a cut star of a graph of this class is also a Gomory Hu tree.

scholarly journals LIST COLORING OF BLOCK GRAPHS AND COMPLETE BIPARTITE GRAPHS

World Science ◽  
2021 ◽  
Author(s):  
Albert Khachik Sahakyan
Keyword(s):  
Bipartite Graph ◽  
Polynomial Algorithm ◽  
Undirected Graph ◽  
Complete Bipartite Graph ◽  
Vertex Coloring ◽  
List Coloring ◽  
Block Graph ◽  
Block Graphs ◽  
Complete Bipartite Graphs ◽  
Np Complete

List coloring is a vertex coloring of a graph where each vertex can be restricted to a list of allowed colors. For a given graph G and a set L(v) of colors for every vertex v, a list coloring is a function that maps every vertex v to a color in the list L(v) such that no two adjacent vertices receive the same color. It was first studied in the 1970s in independent papers by Vizing and by Erdős, Rubin, and Taylor. A block graph is a type of undirected graph in which every biconnected component (block) is a clique. A complete bipartite graph is a bipartite graph with partitions V 1, V 2 such that for every two vertices v_1∈V_1 and v_2∈V_2 there is an edge (v 1, v 2). If |V_1 |=n and |V_2 |=m it is denoted by K_(n,m). In this paper we provide a polynomial algorithm for finding a list coloring of block graphs and prove that the problem of finding a list coloring of K_(n,m) is NP-complete even if for each vertex v the length of the list is not greater than 3 (|L(v)|≤3).

scholarly journals Complete Acyclic Colorings

10.37236/8752 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Stefan Felsner ◽  
Winfried Hochstättler ◽  
Kolja Knauer ◽  
Raphael Steiner
Keyword(s):  
Chromatic Number ◽  
Undirected Graph ◽  
Directed Cycle ◽  
Vertex Arboricity ◽  
Classical Parameters ◽  
Two Parameters ◽  
Vertex Sets ◽  
Dichromatic Number

We study two parameters that arise from the dichromatic number and the vertex-arboricity in the same way that the achromatic number comes from the chromatic number. The adichromatic number of a digraph is the largest number of colors its vertices can be colored with such that every color induces an acyclic subdigraph but merging any two colors yields a monochromatic directed cycle. Similarly, the a-vertex arboricity of an undirected graph is the largest number of colors that can be used such that every color induces a forest but merging any two yields a monochromatic cycle. We study the relation between these parameters and their behavior with respect to other classical parameters such as degeneracy and most importantly feedback vertex sets.

Solving the maximal matching problem with DNA molecules in Adleman–Lipton model

2016 ◽  
Vol 09 (02) ◽  
pp. 1650019 ◽  
Author(s):  
Zhaocai Wang ◽  
Zuwen Ji ◽  
Ziyi Su ◽  
Xiaoming Wang ◽  
Kai Zhao
Keyword(s):  
Undirected Graph ◽  
Applied Mathematics ◽  
Vertex Cover ◽  
Real Life ◽  
Matching Problem ◽  
Maximal Matching ◽  
Dna Strands ◽  
Path Search ◽  
Hard Problems ◽  
Np Complete

The maximal matching problem (MMP) is to find maximal edge subsets in a given undirected graph, that no pair of edges are adjacent in the subsets. It is a vitally important NP-complete problem in graph theory and applied mathematics, having numerous real life applications in optimal combination and linear programming fields. It can be difficultly solved by the electronic computer in exponential level time. Meanwhile in previous studies deoxyribonucleic acid (DNA) molecular operations usually were used to solve NP-complete continuous path search problems, e.g. HPP, traveling salesman problem, rarely for NP-hard problems with discrete vertices or edges solutions, such as the minimum vertex cover problem, graph coloring problem and so on. In this paper, we present a DNA algorithm for solving the MMP with DNA molecular operations. For an undirected graph with [Formula: see text] vertices and [Formula: see text] edges, we reasonably design fixed length DNA strands representing vertices and edges of the graph, take appropriate steps and get the solutions of the MMP in proper length range using [Formula: see text] time. We extend the application of DNA molecular operations and simultaneously simplify the complexity of the computation.

scholarly journals Equitably strong nonsplit equitable domination in graphs

2021 ◽  
Author(s):  
P. Nataraj ◽  
Sundareswaran Raman ◽  
V. Swaminathan
Keyword(s):  
Undirected Graph ◽  
Vertex Set ◽  
New Type ◽  
Domination In Graphs

In a simple, finite and undirected graph G with vertex set V and edge set E, Prof. Sampathkumar defined degree equitability among vertices of G. Two vertices u and v are said to be degree equitable if |deg(u) − deg(v)| ≤ 1. Equitable domination has been defined and studied in [7]. V.R.Kulli and B.Janakiram defined strong non - split domination in a graph [12]. In this paper, the equitable version of this new type of domination is studied

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