The interchange graph of a finite graph

Rooij, A. C. M. van; Wilf, Herbert S.

doi:10.1007/bf01904834

Cited by 97 publications

(46 citation statements)

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“…The node-equivalent network is the line graph [23][24][25] of the original network. The line graph is a network where the links of the original network are represented by a node and are connected to those nodes that represent links that were first neighbors in the original network.…”

Section: Summary and Discussionmentioning

confidence: 99%

Dynamics of link states in complex networks: The case of a majority rule

et al. 2012

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Motivated by the idea that some characteristics are specific to the relations between individuals and not to the individuals themselves, we study a prototype model for the dynamics of the states of the links in a fixed network of interacting units. Each link in the network can be in one of two equivalent states. A majority link-dynamics rule is implemented, so that in each dynamical step the state of a randomly chosen link is updated to the state of the majority of neighboring links. Nodes can be characterized by a link heterogeneity index, giving a measure of the likelihood of a node to have a link in one of the two states. We consider this link-dynamics model in fully connected networks, square lattices, and Erdös-Renyi random networks. In each case we find and characterize a number of nontrivial asymptotic configurations, as well as some of the mechanisms leading to them and the time evolution of the link heterogeneity index distribution. For a fully connected network and random networks there is a broad distribution of possible asymptotic configurations. Most asymptotic configurations that result from link dynamics have no counterpart under traditional node dynamics in the same topologies.

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Section: Summary and Discussionmentioning

confidence: 99%

Dynamics of link states in complex networks: The case of a majority rule

et al. 2012

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“…(3) G does not have K , , , as an induced subgraph, and if two odd triangles have a common edge, then the subgraph induced by their vertices is K4 (Van Rooij and Wilf [4]). (4) None of a set of nine graphs is an induced subgraph of G (Beineke [l]).…”

Section: Characterization Of Ps-graphsmentioning

confidence: 99%

Path graphs

Broersma¹,

Hoede

1989

Journal of Graph Theory

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The concept of a line graph is generalized to that of a path graph. The path graph f,(G) of a graph G is obtained by representing the paths Pk in G by vertices and joining two vertices whenever the corresponding paths f k in G form a path f k + , or a cycle C,. f,-graphs are characterized and investigated on isomorphism and traversability. Trees and unicyclic graphs with hamiltonian /?,-graphs are characterized.

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“…Lehot [7] and Roussopoulos [8] have independently given lineartime algorithms for detecting whether a graph is a line graph and outputting its root graph. Van Rooij and Wilf [9] gave a characterization of line graphs in terms of nine forbidden subgraphs. The characterization can be stated as follows; A triangle in a graph is even if every other node is adjacent to 0 or 2 nodes in the triangle; it is odd otherwise.…”

Section: Line Graph and CC Graphsmentioning

confidence: 99%

Estimating Population Size via Line Graph Reconstruction

Halldórsson

Blokh

Sharan

2012

Lecture Notes in Computer Science

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Background: We propose a novel graph theoretic method to estimate haplotype population size from genotype data. The method considers only the potential sharing of haplotypes between individuals and is based on transforming the graph of potential haplotype sharing into a line graph using a minimum number of edge and vertex deletions. Results: We show that the resulting line graph deletion problems are NP complete and provide exact integer programming solutions for them. We test our approach using extensive simulations of multiple population evolution and genotypes sampling scenarios. Our results also indicate that the method may be useful in comparing populations and it may be used as a first step in a method for haplotype phasing. Conclusions: Our computational experiments show that when most of the sharings are true sharings the problem can be solved very fast and the estimated size is very close to the true size; when many of the potential sharings do not stem from true haplotype sharing, our method gives reasonable lower bounds on the underlying number of haplotypes. In comparison, a naive approach of phasing the input genotypes provides trivial upper bounds of twice the number of genotypes.

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The interchange graph of a finite graph

Cited by 97 publications

References 1 publication

Dynamics of link states in complex networks: The case of a majority rule

Dynamics of link states in complex networks: The case of a majority rule

Path graphs

Estimating Population Size via Line Graph Reconstruction

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