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### Local complementation and interlacement graphs

Hubert de Fraysseix
1981 Discrete Mathematics
We denote by C;, l.he graph obtained by a local complement~ztion at x0.  ...  By carrying out local complementations on A(M) in some right order, eke can o5:ain the fundamental interlacement graph associated with an arbitrary partition of E into a base B and a cobase Bi.  ...

### Nullity invariance for pivot and the interlace polynomial

Robert Brijder, Hendrik Jan Hoogeboom
2011 Linear Algebra and its Applications
We consider its consequences for graphs, and in particular generalize the recursive relation of the interlace polynomial and simplify its proof.  ...  Brijder is supported by the Netherlands Organization for Scientific Research (NWO), project "Annotated graph mining".  ...  Acknowledgements We thank the referee for comments and corrections on the paper. In particular, the focus of the paper is improved. R.  ...

### Interlacement in 4-regular graphs: a new approach using nonsymmetric matrices [article]

Lorenzo Traldi
2012 arXiv   pre-print
Let F be a 4-regular graph with an Euler system C.  ...  If C and C' are Euler systems of F then M(C,C') and M(C',C) are inverses, and for any circuit partition P, M(C',P)=M(C',C)M(C,P).  ...  Interlacement and local complements A graph G = (V (G), E(G)) is given by a finite set V (G) of vertices, and a finite set E(G) of edges.  ...

### On the interlace polynomials

Lorenzo Traldi
2013 Journal of combinatorial theory. Series B (Print)
The multivariate polynomial incorporates several different interlace polynomials that have been studied by different authors, and its properties include invariance under a refined version of local complementation  ...  The interlace polynomials of Arratia, Bollobás and Sorkin [R. Arratia, B. Bollobás, G.B. Sorkin, The interlace polynomial of a graph, J. Combin. Theory Ser. B 92 (2004) 199-233; R. Arratia, B.  ...  The idea of using label-switching local complementations was inspired by many conversations with them while we studied the use of interlacement to describe the Jones polynomial and Kauffman bracket of  ...

### A bracket polynomial for graphs, IV. Undirected Euler circuits, graph-links and multiply marked graphs [article]

Lorenzo Traldi
2010 arXiv   pre-print
Here we extend the graph bracket to graphs whose vertices may carry different kinds of marks, and we show how multiply marked graphs encode interlacement with respect to arbitrary (undirected) Euler systems  ...  of the interlacement graph associated to a directed Euler system of the universe graph of D.  ...  Zulli and an anonymous referee for advice, encouragement and inspiration.  ...

### On the interlace polynomials [article]

Lorenzo Traldi
2012 arXiv   pre-print
The multivariate polynomial incorporates several different interlace polynomials that have been studied by different authors, and its properties include invariance under a refined version of local complementation  ...  The interlace polynomials of Arratia, Bollob\'as and Sorkin [J. Combin. Theory Ser.  ...  interlacement, and using labeled transpositions and pivots rather than labeled κ-transformations and local complementations.  ...

### Interlacement of double curves of immersed spheres [article]

Boldizsar Kalmar
2017 arXiv   pre-print
Our proof uses a result of Lippner and we further generalize the ideas of Fraysseix and Ossona de Mendez, which leads us to directed interlacement graphs of paired trees and their local complementation  ...  We characterize those unions of embedded disjoint circles in the 2-sphere which can be the multiple point set of a generic immersion of the 2-sphere into 3-dimensional space in terms of the interlacement  ...  A version of (directed) local complementation of directed graphs will be used later in the paper, which we define here (cf. local complementation in [Bou87] ).  ...

### Interlace polynomials: Enumeration, unimodality and connections to codes

Lars Eirik Danielsen, Matthew G. Parker
2010 Discrete Applied Mathematics
The interlace polynomial q was introduced by Arratia, Bollobas, and Sorkin. It encodes many properties of the orbit of a graph under edge local complementation (ELC).  ...  The interlace polynomial Q, introduced by Aigner and van der Holst, similarly contains information about the orbit of a graph under local complementation (LC).  ...  Given a graph G = (V , E) and an edge {u, v} ∈ E, edge local complementation (ELC) on {u, v} transforms G into G (uv) Fig. 4 . 4 Example of an LC orbit.  ...

### The Interlace Polynomial [article]

2016 arXiv   pre-print
In this paper, we survey results regarding the interlace polynomial of a graph, connections to such graph polynomials as the Martin and Tutte polynomials, and generalizations to the realms of isotropic  ...  systems and delta-matroids.  ...  We define the local complement G ∗ v to be the graph obtained from G by interchanging edges and non-edges in N (v).  ...

### A Two-Variable Interlace Polynomial [article]

Richard Arratia, Bela Bollobas, Gregory B. Sorkin
2004 arXiv   pre-print
The second computation is by a three-term reduction formula involving a graph pivot; the pivot arose previously in the study of interlacement and Euler circuits in four-regular graphs.  ...  We consider a few properties and specializations of the two-variable interlace polynomial.  ...  We are also grateful to Hein van der Holst for providing us with a manuscript copy of [AvdH04] and for helpful discussions on the interlace polynomials.  ...

### Edge-local equivalence of graphs [article]

Maarten Van den Nest, Bart De Moor
2005 arXiv   pre-print
The local complement G*i of a simple graph G at one of its vertices i is obtained by complementing the subgraph induced by the neighborhood of i and leaving the rest of the graph unchanged.  ...  We call two graphs edge-locally equivalent if they are related by a sequence of edge-local complementations.  ...  Second, there is an intimate connection between (edge-)local complementation of graphs and the interlace polynomial [16, 17, 18, 19, 20] , which is a recently introduced graph polynomial motivated by  ...

### On the linear algebra of local complementation [article]

Lorenzo Traldi
2011 arXiv   pre-print
In particular, we show that the equivalence relation on simple graphs generated by local complementation can also be generated by an operation defined using inverse matrices.  ...  We explore the connections between the linear algebra of symmetric matrices over GF(2) and the circuit theory of 4-regular graphs.  ...  Brijder and D. P. Ilyutko for many enlightening conversations. The final version of the paper also profited from the careful attention of two anonymous readers.  ...

### On the linear algebra of local complementation

Lorenzo Traldi
2012 Linear Algebra and its Applications
In particular, we show that the equivalence relation on simple graphs generated by local complementation can also be generated by an operation defined using inverse matrices.  ...  We explore the connections between the linear algebra of symmetric matrices over GF(2) and the circuit theory of 4-regular graphs.  ...  Brijder and D.P. Ilyutko for many enlightening conversations. The final version of the paper also profited from the careful attention of two anonymous readers.  ...

### One and Two-Variable Interlace Polynomials: A Spectral Interpretation [chapter]

Constanza Riera, Matthew G. Parker
2006 Lecture Notes in Computer Science
We relate the one-and two-variable interlace polynomials of a graph to the spectra of a quadratic boolean function with respect to a strategic subset of local unitary transforms.  ...  By so doing we establish links between graph theory, cryptography, coding theory, and quantum entanglement.  ...  We then defined the HN-interlace polynomial, and derived its form for the clique, the line, and the clique-line-clique functions.  ...

### Splitting cubic circle graphs

Lorenzo Traldi
2016 Discussiones Mathematicae Graph Theory
We also deduce that up to isomorphism, K_4 and K_3,3 are the only 3-connected, 3-regular circle graphs.  ...  We show that every 3-regular circle graph has at least two pairs of twin vertices; consequently no such graph is prime with respect to the split decomposition.  ...  Acknowledgment We are grateful to Robert Brijder for the many inspirations provided by our long correspondence and collaboration.  ...
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