A Concise Survey of Matching Preclusion in Interconnection Networks

2019 ◽  
Vol 19 (03) ◽  
pp. 1940006 ◽  
Author(s):  
YAPING MAO ◽  
EDDIE CHENG
Keyword(s):  
Interconnection Networks ◽  
Perfect Matchings ◽  
Minimum Number ◽  
General Topic

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. There are other related parameters and generalization including the strong matching preclusion number, the conditional matching preclusion number, the fractional matching preclusion number, and so on. In this survey, we give an introduction on the general topic of matching preclusion.

MATCHING PRECLUSION AND CONDITIONAL MATCHING PRECLUSION FOR CROSSED CUBES

2012 ◽  
Vol 22 (02) ◽  
pp. 1250005 ◽  
Author(s):  
EDDIE CHENG ◽  
SACHIN PADMANABHAN
Keyword(s):  
Interconnection Networks ◽  
Perfect Matchings ◽  
Single Vertex ◽  
Minimum Number ◽  
Important Variant

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. Recently, the conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond those induced by a single vertex. It is defined to be the minimum number of edges whose deletion results in a graph with no isolated vertices that has neither perfect matchings nor almost-perfect matchings. In this paper, we find the matching preclusion number and the conditional matching preclusion number with the classification of the optimal sets for the class of crossed cubes, an important variant of the class of hypercubes. Indeed, we will establish more general results on the matching preclusion and the conditional matching preclusion problems for a larger class of interconnection networks.

MATCHING PRECLUSION AND CONDITIONAL MATCHING PRECLUSION FOR AUGMENTED CUBES

2010 ◽  
Vol 11 (01n02) ◽  
pp. 35-60 ◽  
Author(s):  
EDDIE CHENG ◽  
RANDY JIA ◽  
DAVID LU
Keyword(s):  
Interconnection Networks ◽  
Perfect Matchings ◽  
Single Vertex ◽  
Minimum Number

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. For many interconnection networks, the optimal sets are precisely those incident to a single vertex. Recently, the conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond those incident to a single vertex. It is defined to be the minimum number of edges whose deletion results in a graph with no isolated vertices that has neither perfect matchings nor almost-perfect matchings. In this paper, we find this number and classify all optimal sets for the augmented cubes, a class of networks designed as an improvement of the hypercubes.

Strong Matching Preclusion of Arrangement Graphs

2016 ◽  
Vol 16 (02) ◽  
pp. 1650004 ◽  
Author(s):  
EDDIE CHENG ◽  
OMER SIDDIQUI
Keyword(s):  
Interconnection Networks ◽  
Perfect Matchings ◽  
Alternating Group ◽  
Star Graphs ◽  
Common Generalization ◽  
Minimum Number

The strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion results in a graph with neither perfect matchings nor almost-perfect matchings. This is an extension of the matching preclusion problem that was introduced by Park and Ihm. The class of arrangement graphs was introduced as a common generalization of the star graphs and alternating group graphs, and to provide an even richer class of interconnection networks. In this paper, the goal is to find the strong matching preclusion number of arrangement graphs and to categorize all optimal strong matching preclusion sets of these graphs.

CONDITIONAL MATCHING PRECLUSION FOR (n,k)-STAR GRAPHS

2013 ◽  
Vol 23 (01) ◽  
pp. 1350004 ◽  
Author(s):  
EDDIE CHENG ◽  
LÁSZLÓ LIPTÁK
Keyword(s):  
Interconnection Networks ◽  
Perfect Matchings ◽  
Optimal Solutions ◽  
Alternate Proof ◽  
Star Graphs ◽  
Minimum Number

The matching preclusion number of an even graph G, denoted by mp (G), is the minimum number of edges whose deletion leaves the resulting graph without perfect matchings. The conditional matching preclusion number of an even graph G, denoted by mp 1(G), is the minimum number of edges whose deletion leaves the resulting graph with neither perfect matchings nor isolated vertices. The class of (n,k)-star graphs is a popular class of interconnection networks for which the matching preclusion number and the classification of the corresponding optimal solutions were known. However, the conditional version of this problem was open. In this paper, we determine the conditional matching preclusion for (n,k)-star graphs as well as classify the corresponding optimal solutions via several new results. In addition, an alternate proof of the results on the matching preclusion problem will also be given.

scholarly journals Fractional Matching Preclusion for Restricted Hypercube-Like Graphs

2019 ◽  
Vol 19 (03) ◽  
pp. 1940010
Author(s):  
HUAZHONG LÜ ◽  
TINGZENG WU
Keyword(s):  
Interconnection Networks ◽  
Perfect Matching ◽  
Parallel Systems ◽  
Perfect Matchings ◽  
Minimum Number ◽  
Fractional Perfect Matching

The restricted hypercube-like graphs, variants of the hypercube, were proposed as desired interconnection networks of parallel systems. The matching preclusion number of a graph is the minimum number of edges whose deletion results in the graph with neither perfect matchings nor almost perfect matchings. The fractional perfect matching preclusion and fractional strong perfect matching preclusion are generalizations of the matching preclusion. In this paper, we obtain fractional matching preclusion number and fractional strong matching preclusion number of restricted hypercube-like graphs, which extend some known results.

Matching Preclusion for Enhanced Pyramid Networks

2019 ◽  
Vol 19 (03) ◽  
pp. 1940009
Author(s):  
XIAQI WEI ◽  
SHURONG ZHANG ◽  
WEIHUA YANG
Keyword(s):  
Interconnection Networks ◽  
Computer Systems ◽  
Perfect Matchings ◽  
Minimum Number ◽  
Edge Failure ◽  
Distributed Computer Systems

The matching preclusion number of a graph is the minimum number of edges whose deletion leaves the resulting graph that has neither perfect matchings nor almost perfect matchings. This concept was introduced as a measure of robustness in the event of edge failure in interconnection networks. The pyramid network is one of the important networks applied in parallel and distributed computer systems. Chen et al. in 2004 proposed a new hierarchy structure, called the enhanced pyramid network, by replacing each mesh in a pyramid network with a torus. An enhanced pyramid network of n layers is denoted by EPM(n). In this paper, we prove that the matching preclusion number of EPM(n) is 9 where n ≥ 4.

Component Edge Connectivity of Hypercubes

2018 ◽  
Vol 29 (06) ◽  
pp. 995-1001 ◽  
Author(s):  
Shuli Zhao ◽  
Weihua Yang ◽  
Shurong Zhang ◽  
Liqiong Xu
Keyword(s):  
Fault Tolerance ◽  
Complete Graph ◽  
Interconnection Networks ◽  
Interconnection Network ◽  
Optimal Solutions ◽  
Edge Connectivity ◽  
Important Measure ◽  
Minimum Number ◽  
Text Component

Fault tolerance is an important issue in interconnection networks, and the traditional edge connectivity is an important measure to evaluate the robustness of an interconnection network. The component edge connectivity is a generalization of the traditional edge connectivity. The [Formula: see text]-component edge connectivity [Formula: see text] of a non-complete graph [Formula: see text] is the minimum number of edges whose deletion results in a graph with at least [Formula: see text] components. Let [Formula: see text] be an integer and [Formula: see text] be the decomposition of [Formula: see text] such that [Formula: see text] and [Formula: see text] for [Formula: see text]. In this note, we determine the [Formula: see text]-component edge connectivity of the hypercube [Formula: see text], [Formula: see text] for [Formula: see text]. Moreover, we classify the corresponding optimal solutions.

A Short Note of Strong Matching Preclusion for a Class of Arrangement Graphs

2020 ◽  
Vol 30 (01) ◽  
pp. 2050001
Author(s):  
Shuangshuang Zhang ◽  
Yuzhi Xiao ◽  
Xia Liu ◽  
Jun Yin
Keyword(s):  
Short Note ◽  
Perfect Matchings ◽  
Minimum Number ◽  
Edge Failure

The strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings. The strong matching preclusion is a well-studied measure for the network invulnerability in the event of edge failure. In this paper, we obtain the strong matching preclusion number for a class of arrangement graphs and categorize their the strong matching preclusion set, which are a supplement of known results.

Analysis on the Component Connectivity of Enhanced Hypercubes

2020 ◽  
Author(s):  
Liqiong Xu ◽  
Litao Guo
Keyword(s):  
Connected Graph ◽  
Interconnection Networks ◽  
Unsolved Problem ◽  
Reliability Evaluation ◽  
Minimum Number ◽  
Appl Math

Abstract Reliability evaluation of interconnection networks is of significant importance to the design and maintenance of interconnection networks. The component connectivity is an important parameter for the reliability evaluation of interconnection networks and is a generalization of the traditional connectivity. The $g$-component connectivity $c\kappa _g (G)$ of a non-complete connected graph $G$ is the minimum number of vertices whose deletion results in a graph with at least $g$ components. Determining the $g$-component connectivity is still an unsolved problem in many interconnection networks. Let $Q_{n,k}$ ($1\leq k\leq n-1$) denote the $(n, k)$-enhanced hypercube. In this paper, let $n\geq 7$ and $1\leq k \leq n-5$, we determine $c\kappa _{g}(Q_{n,k}) = g(n + 1) - \frac{1}{2}g(g + 1) + 1$ for $2 \leq g \leq n$. The previous result in Zhao and Yang (2019, Conditional connectivity of folded hypercubes. Discret. Appl. Math., 257, 388–392) is extended.

Fractional Matching Preclusion for (n, k)-Star Graphs

2018 ◽  
Vol 28 (04) ◽  
pp. 1850017 ◽  
Author(s):  
Tianlong Ma ◽  
Yaping Mao ◽  
Eddie Cheng ◽  
Jinling Wang
Keyword(s):  
Perfect Matching ◽  
Perfect Matchings ◽  
Star Graphs ◽  
Minimum Number ◽  
Fractional Perfect Matching

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings. As a generalization, Liu and Liu introduced the concept of fractional matching preclusion number in 2017. The Fractional Matching Preclusion Number (FMP number) of G is the minimum number of edges whose deletion leaves the resulting graph without a fractional perfect matching. The Fractional Strong Matching Preclusion Number (FSMP number) of G is the minimum number of vertices and/or edges whose deletion leaves the resulting graph without a fractional perfect matching. In this paper, we obtain the FMP number and the FSMP number for (n, k)-star graphs. In addition, all the optimal fractional strong matching preclusion sets of these graphs are categorized.

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