scholarly journals A Maximum Cardinality Cut Algorithm for Co-bipartite and Split Graphs Using Bimodular Decomposition

2021 ◽  
Author(s):  
Arman BOYACI ◽  
Mordechai SHALOM
Keyword(s):  
Maximum Cardinality ◽  
Split Graphs

scholarly journals OPEN PACKING NUMBER FOR SOME CLASSES OF PERFECT GRAPHS

2020 ◽  
Vol 6 (2) ◽  
pp. 38
Author(s):  
K. Raja Chandrasekar ◽  
S. Saravanakumar
Keyword(s):  
Bipartite Graph ◽  
Perfect Graphs ◽  
Perfect Graph ◽  
Maximum Cardinality ◽  
Common Neighbor ◽  
Packing Number ◽  
Vertex Set ◽  
Open Packing ◽  
Split Graphs

Let \(G\) be a graph with the vertex set \(V(G)\).  A subset \(S\) of \(V(G)\) is an open packing set of \(G\) if every pair of vertices in \(S\) has no common neighbor in \(G.\)  The maximum cardinality of an open packing set of \(G\) is the open packing number of \(G\) and it is denoted by \(\rho^o(G)\).  In this paper, the exact values of the open packing numbers for some classes of perfect graphs, such as split graphs, \(\{P_4, C_4\}\)-free graphs, the complement of a bipartite graph, the trestled graph of a perfect graph are obtained.

Graphic sequences and split graphs

2016 ◽  
Vol 32 (4) ◽  
pp. 1005-1014
Author(s):  
Jian-hua Yin ◽  
Lei Meng ◽  
Meng-Xiao Yin
Keyword(s):  
Split Graphs

Split graphs

2021 ◽  
pp. 189-206
Author(s):  
Karen L. Collins ◽  
Ann N. Trenk
Keyword(s):  
Split Graphs

scholarly journals The Evolution of Networks and Local Public Good Provision: A Potential Approach

Games ◽  
2021 ◽  
Vol 12 (3) ◽  
pp. 55
Author(s):  
Markus Kinateder ◽  
Luca Paolo Merlino
Keyword(s):  
Public Good ◽  
Nash Equilibria ◽  
Potential Game ◽  
Public Good Provision ◽  
Local Public Good ◽  
Steady State Distribution ◽  
State Distribution ◽  
Long Run ◽  
Split Graphs ◽  
Good Provision

In this paper, we propose a game in which each player decides with whom to establish a costly connection and how much local public good is provided when benefits are shared among neighbors. We show that, when agents are homogeneous, Nash equilibrium networks are nested split graphs. Additionally, we show that the game is a potential game, even when we introduce heterogeneity along several dimensions. Using this result, we introduce stochastic best reply dynamics and show that this admits a unique and stationary steady state distribution expressed in terms of the potential function of the game. Hence, even if the set of Nash equilibria is potentially very large, the long run predictions are sharp.

scholarly journals Benchmarking Quantum Annealing Against “Hard” Instances of the Bipartite Matching Problem

2021 ◽  
Vol 2 (2) ◽  
Author(s):  
Daniel Vert ◽  
Renaud Sirdey ◽  
Stéphane Louise
Keyword(s):  
Simulated Annealing ◽  
Polynomial Time ◽  
Optimal Solution ◽  
Quantum Computers ◽  
Quantum Annealing ◽  
Maximum Cardinality ◽  
Matching Problem ◽  
Bipartite Matching ◽  
Wave Device ◽  
D Wave

AbstractThis paper experimentally investigates the behavior of analog quantum computers as commercialized by D-Wave when confronted to instances of the maximum cardinality matching problem which is specifically designed to be hard to solve by means of simulated annealing. We benchmark a D-Wave “Washington” (2X) with 1098 operational qubits on various sizes of such instances and observe that for all but the most trivially small of these it fails to obtain an optimal solution. Thus, our results suggest that quantum annealing, at least as implemented in a D-Wave device, falls in the same pitfalls as simulated annealing and hence provides additional evidences suggesting that there exist polynomial-time problems that such a machine cannot solve efficiently to optimality. Additionally, we investigate the extent to which the qubits interconnection topologies explains these latter experimental results. In particular, we provide evidences that the sparsity of these topologies which, as such, lead to QUBO problems of artificially inflated sizes can partly explain the aforementioned disappointing observations. Therefore, this paper hints that denser interconnection topologies are necessary to unleash the potential of the quantum annealing approach.

Persistency in maximum cardinality bipartite matchings

1994 ◽  
Vol 15 (3) ◽  
pp. 143-149 ◽  
Author(s):  
Marie-Christine Costa
Keyword(s):  
Maximum Cardinality

Roman domination and 2-independence in trees

2017 ◽  
Vol 09 (02) ◽  
pp. 1750023 ◽  
Author(s):  
Nacéra Meddah ◽  
Mustapha Chellali
Keyword(s):  
Independence Number ◽  
Minimum Weight ◽  
Domination Number ◽  
Independent Set ◽  
Maximum Cardinality ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Number Formula ◽  
Sum Formula ◽  
Roman Dominating Function

A Roman dominating function (RDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying the condition that every vertex [Formula: see text] with [Formula: see text] is adjacent to at least one vertex [Formula: see text] of [Formula: see text] for which [Formula: see text]. The weight of a RDF is the sum [Formula: see text], and the minimum weight of a RDF [Formula: see text] is the Roman domination number [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is a [Formula: see text]-independent set of [Formula: see text] if every vertex of [Formula: see text] has at most one neighbor in [Formula: see text] The maximum cardinality of a [Formula: see text]-independent set of [Formula: see text] is the [Formula: see text]-independence number [Formula: see text] Both parameters are incomparable in general, however, we show that if [Formula: see text] is a tree, then [Formula: see text]. Moreover, all extremal trees attaining equality are characterized.

On class 2 split graphs

2010 ◽  
Vol 39 (17) ◽  
Author(s):  
Sheila de Almeida ◽  
Celia de Mello ◽  
Aurora Morgana
Keyword(s):  
Split Graphs

scholarly journals Finding Points in General Position

2017 ◽  
Vol 27 (04) ◽  
pp. 277-296 ◽  
Author(s):  
Vincent Froese ◽  
Iyad Kanj ◽  
André Nichterlein ◽  
Rolf Niedermeier
Keyword(s):  
Lower Bound ◽  
General Position ◽  
Subset Selection ◽  
Selection Problem ◽  
Fixed Parameter Tractability ◽  
Maximum Cardinality ◽  
Exponential Time ◽  
Running Time ◽  
Fixed Parameter ◽  
Set Of Points

We study the General Position Subset Selection problem: Given a set of points in the plane, find a maximum-cardinality subset of points in general position. We prove that General Position Subset Selection is NP-hard, APX-hard, and present several fixed-parameter tractability results for the problem as well as a subexponential running time lower bound based on the Exponential Time Hypothesis.

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