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### A Characterisation of the Minimal Triangulations of Permutation Graphs [chapter]

Daniel Meister
Graph-Theoretic Concepts in Computer Science
We give a non-trivial characterisation of the class of interval graphs that are minimal triangulations of permutation graphs and obtain as a surprising result that only "a few" interval graphs are minimal  ...  We (negatively) answer the question whether every interval graph is a minimal triangulation of a permutation graph.  ...  Ø Ö ÓØ Ð Ø Ò ÓÖ ÓØ Ö Ø Ò Ú ÖØ × ÓÖ ÓÒ Ú ÖØ Ü × Ð Ø ÓÖ Ö Ø Ò Ú ÖØ Ü Ò Ø ÓØ Ö × ÙÒÐ ×× ¬ º ÁÒ ÐÐ Ø × × ×¸× ÒØ Ö× Ø× Û Ø Ø ÓÖ Ö Ó Ö Ø Ò¹ Ð º À Ò ¸Ø ÓÖ Ö× Ó Ø Ö Ø Ò Ð × ¬Ò Ñ Ü Ñ Ð × Ø Ó Ô ÖÛ × ÒÓÒ¹ ÒØ Ö× Ø  ...

### Treewidth and minimum fill-in on permutation graphs in linear time

Daniel Meister
2010 Theoretical Computer Science
The main theoretical result of the paper is a characterisation of the minimal triangulations of a permutation graph, that also leads to a succinct and linear-time computable representation of the set of  ...  minimum fill-in on permutation graphs. • let H be a minimal triangulation of G of smallest clique number among the minimal triangulations of G; the treewidth of G, denoted as tw(G), is equal to the clique  ...  Minimal triangulations from permutation diagrams We will give a characterisation of the minimal triangulations of a permutation graph.  ...

### A Construction Principle for Tight and Minimal Triangulations of Manifolds

Benjamin A. Burton, Basudeb Datta, Nitin Singh, Jonathan Spreer
2016 Experimental Mathematics
A triangulation is tight if all its piecewise linear embeddings into a Euclidean space are as convex as allowed by the topology of the underlying manifold.  ...  Tight triangulations are conjectured to be strongly minimal, and proven to be so for dimensions ≤ 3.  ...  Acknowledgements The authors thank the anonymous referees for useful comments.  ...

### Treewidth computations I. Upper bounds

Hans L. Bodlaender, Arie M.C.A. Koster
2010 Information and Computation
This paper gives an overview of several upper bound heuristics that have been proposed and tested for the problem of determining the treewidth of a graph and finding tree decompositions.  ...  For more and more applications, it is important to be able to compute the treewidth of a given graph and to find tree decompositions of small width reasonably fast.  ...  We introduce the characterisation by using equivalent characterisations of chordal graphs, also known as triangulated graphs. Lemma 2 (Gavril Definition 3.  ...

### The Branch-Width of Circular-Arc Graphs [chapter]

Frédéric Mazoit
2006 Lecture Notes in Computer Science
We prove that the branch-width of circular-arc graphs can be computed in polynomial time.  ...  A triangulation of a graph G is a chordal super-graph of G. A triangulation H of G is minimal if no strict sub-graph of H is a triangulation of G. Definition 1 (Efficient triangulation).  ...  Since blocks of a graph are characterised by the minimal separators that border them, we can realise blocks with scan-lines.  ...

### Which graphs are rigid in $$\ell _p^d$$?

Sean Dewar, Derek Kitson, Anthony Nixon
2021 Journal of Global Optimization
A catalogue of rigidity preserving graph moves is also provided for the more general class of strictly convex and smooth normed spaces and we show that every triangulation of the sphere is independent  ...  Finally, we prove that every triangulation of the projective plane is minimally rigid in $$\ell _p^3$$ ℓ p 3 .  ...  Our final main result concerns the rigidity of triangulated surfaces in dimension 3. It is well-known that the graph of a triangulated sphere is minimally rigid in the Euclidean space eg).  ...

### The fibres of the Scott map on polygon tilings are the flip equivalence classes

Karin Baur, Paul P. Martin
2018 Monatshefte für Mathematik (Print)
We thus obtain a map from tilings of surfaces to permutations of the marked points on boundary components, the Scott map.  ...  We define a map from tilings of surfaces with marked points to strand diagrams, generalising Scott's construction for the case of triangulations of polygons.  ...  , and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.  ...

### Matching fields and lattice points of simplices [article]

Georg Loho, Ben Smith
2019 arXiv   pre-print
As additional results, we show a cryptomorphic description of linkage matching fields and characterise the flip graph of a linkage matching field in terms of its prodsimplicial flag complex.  ...  Given a triangulation of a product of two simplices encoded by a set of spanning trees on a bipartite node set, we similarly prove that the bijection from left to right degree vectors of the trees is enough  ...  We thank Pavel Galashin, Gaku Liu and Alex Postnikov for communicating the related work [10] . We would like to thank the referee for helpful comments.  ...

### Which graphs are rigid in ℓ_p^d? [article]

Sean Dewar, Derek Kitson, Anthony Nixon
2020 arXiv   pre-print
Finally, we prove that every triangulation of the projective plane is minimally rigid in ℓ_p^3.  ...  A catalogue of rigidity preserving graph moves is also provided for the more general class of strictly convex and smooth normed spaces and we show that every triangulation of the sphere is independent  ...  It is wellknown that the graph of a triangulated sphere is minimally rigid in the Euclidean space 3 2 and that, in general, triangulations of closed surfaces are generically rigid in 3 2 (see [9,  ...

### The positive tropical Grassmannian, the hypersimplex, and the m=2 amplituhedron [article]

Tomasz Lukowski and Matteo Parisi and Lauren K. Williams
2021 arXiv   pre-print
We consider the positroid dissections of both objects, which informally, are subdivisions of Δ_k+1,n (respectively, 𝒜_n,k,2(Z)) into a disjoint union of images of positroid cells of the positive Grassmannian  ...  At first glance, Δ_k+1,n and 𝒜_n,k,2(Z) seem very different - the former is an (n-1)-dimensional polytope, while the latter is a 2k-dimensional non-polytopal subset of Gr_k,k+2.  ...  We start with a dissection of A 4,2,2 coming from the plabic graph shown below (corresponding to the decorated permutation (3, 4, 1, 2)), and a dissection of A 4,1,2 (corresponding to the permutations  ...

### An O(n2) time algorithm for the minimal permutation completion problem

Christophe Crespelle, Anthony Perez, Ioan Todinca
2018 Discrete Applied Mathematics
Such a graph is called a minimal triangulation of G, and the MINIMAL TRIANGULATION 21 problem has been known to be polynomial since 1976 [15, 17].  ...  Such a representation is called 31 a permutation model of the graph. We give an O(n 2 ) time algorithm computing a minimal permutation completion 32 of an arbitrary graph.  ...  into modules, defined as follows: Combinatorial characterisation 126 Given a permutation graph G, a vertex x to insert in G and a permutation completion H of G + x, we want to obtain 127 a permutation  ...

### Page 3531 of Mathematical Reviews Vol. , Issue 93g [page]

1993 Mathematical Reviews
It surveys and extends earlier COMBINATORICS 93g:05004 work of W. Voss on the characterisation of groups by triangulations of orientable surfaces. This approach was introduced by H.  ...  Numbers and graphs; 12. On a special class of infinite graphs; 13. Building of the theory of minimal graphs. Chapter 7 continues the topic of partial orderings started in the first volume.  ...

### Odd Cycle Transversals and Independent Sets in Fullerene Graphs

Luerbio Faria, Sulamita Klein, Matěj Stehlík
2012 SIAM Journal on Discrete Mathematics
We deduce two further conjectures on the independence number of fullerene graphs, as well as a new upper bound on the smallest eigenvalue of a fullerene graph.  ...  Both bounds are sharp, and we characterise the extremal graphs. This proves conjectures of Doslic and Vukicevic, and of Daugherty.  ...  out a gap in the proof of Theorem 5.1.  ...

### Brauer graph algebras [article]

Sibylle Schroll
2017 arXiv   pre-print
In the last section, after defining Green's walk around the Brauer graph, a complete description of the Auslander Reiten quiver of a Brauer graph algebra is given.  ...  The interpretation of flips of diagonals in triangulations of marked oriented surfaces as derived equivalences of Brauer graph algebras and the comparison of derived equivalences of Brauer graph algebras  ...  In [46] for every Brauer graph algebra a minimal set of relations is determined: The relations of type I and III are always minimal and the only relations of type II appearing in a minimal generating  ...

### Brauer Graph Algebras [chapter]

Sibylle Schroll
2018 CRM Short Courses
In the last section, after defining Green's walk around the Brauer graph, a complete description of the Auslander Reiten quiver of a Brauer graph algebra is given.  ...  The interpretation of flips of diagonals in triangulations of marked oriented surfaces as derived equivalences of Brauer graph algebras and the comparison of derived equivalences of Brauer graph algebras  ...  In [46] for every Brauer graph algebra a minimal set of relations is determined: The relations of type I and III are always minimal and the only relations of type II appearing in a minimal generating  ...
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