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Minimal Triangulations for Graphs with "Few" Minimal Separators [chapter]

Vincent Bouchitté, Ioan Todinca
1998 Lecture Notes in Computer Science  
We give a characterization of minimal triangulation of graphs using the notion of \maximal set of neighbor separators".  ...  R esum e Nous donnons une caract erisation des triangulations minimales des graphes en utilisant la notion de \ensemble maximal de s eparateurs voisins".  ...  Therefore, it is easy to compute the maximalsets of neighbor separators for any AT-free graph with \few" separators.  ... 
doi:10.1007/3-540-68530-8_29 fatcat:vrscgz5zorbuhcjgkg3gpgyhle

On Distance-d Independent Set and other problems in graphs with few minimal separators [article]

Pedro Montealegre, Ioan Todinca
2016 arXiv   pre-print
Let be the class of graphs with at most (n) minimal separators, for some polynomial . We show that the odd powers of a graph G have at most as many minimal separators as G.  ...  classes with polynomially many minimal separators.  ...  We thank Iyad Kanj for fruitful discussions on the subject.  ... 
arXiv:1607.04545v1 fatcat:s3y6oshzzfhhdbfnufiyv43ide

AMP Chain Graphs: Minimal Separators and Structure Learning Algorithms [article]

Mohammad Ali Javidian, Marco Valtorta, Pooyan Jamshidi
2020 arXiv   pre-print
These include finding a minimal separator from a restricted set of nodes, finding a minimal separator for two given disjoint sets, and testing whether a given separator is minimal.  ...  We address the problem of finding a minimal separator in an Andersson-Madigan-Perlman chain graph (AMP CG), namely, finding a set Z of nodes that separates a given nonadjacent pair of nodes such that no  ...  Dag Sonntag for providing us with code that helped in the data generating procedure. This work has been partially supported by AFRL and DARPA (FA8750-16-2-0042).  ... 
arXiv:2002.10870v2 fatcat:fisupuhs2zhmnjberidjlfmigi

AMP Chain Graphs: Minimal Separators and Structure Learning Algorithms

Mohammad Ali Javidian, Marco Valtorta, Pooyan Jamshidi
2020 The Journal of Artificial Intelligence Research  
These include finding a minimal separator from a restricted set of nodes, finding a minimal separator for two given disjoint sets, and testing whether a given separator is minimal.  ...  This paper deals with chain graphs (CGs) under the Andersson–Madigan–Perlman (AMP) interpretation.  ...  Dag Sonntag for providing us with code that helped in the data generating procedure. This work has been partially supported by AFRL and DARPA (FA8750-16-2-0042).  ... 
doi:10.1613/jair.1.12101 fatcat:h6iroeqrjbfb5exu5onsxvn444

Rigidity with few locations [article]

Karim Adiprasito, Eran Nevo
2019 arXiv   pre-print
Graphs triangulating the 2-sphere are generically rigid in 3-space, due to Gluck-Dehn-Alexandrov-Cauchy.  ...  This assertion extends to the triangulations of any fixed compact connected surface, where the upper bound obtained on the size of A increases with the genus.  ...  Acknowledgments We thank Isabella Novik and Orit Raz for helpful discussions, and Bob Connelly, Louis Theran and Walter Whiteley for helpful comments and references.  ... 
arXiv:1806.03322v3 fatcat:urp3r7mn25blbar7kovvhdc3fy

Minimal separators in graph classes defined by small forbidden induced subgraphs [article]

Martin Milanič, Nevena Pivač
2019 arXiv   pre-print
In particular, many problems that are NP-hard for general graphs are known to become polynomial-time solvable for classes of graphs with a polynomially bounded number of minimal separators.  ...  Minimal separators in graphs are an important concept in algorithmic graph theory.  ...  Conclusion In this work we considered graphs with "few" minimal separators.  ... 
arXiv:1903.04534v1 fatcat:2uspjndqovhbfnmzgt7xjrfb6u

Three-dimensional normal pseudomanifolds with relatively few edges [article]

Biplab Basak, Ed Swartz
2018 arXiv   pre-print
Zheng observed that for all d > 0 there are triangulations of S^d ∗R P^2 with g_2=3. She asked if this is the only nonspherical topology possible for g_2(Δ)=3.  ...  For example, whenever the one-skeleton of Δ equals the one-skeleton of the star of a vertex, then Δ has relatively minimal g_2.  ...  Let ∆ be a triangulation of K with graph cone point v. For K = S 3 , ∆ can be ∂∆ 4 . For nonorientable K, ∆ can be appropriate connected sums of the 6-vertex projective plane.  ... 
arXiv:1803.08942v2 fatcat:ecvksplprjdotemt5xbyti5ohi

Drawing Graphs on Few Circles and Few Spheres [article]

Myroslav Kryven, Alexander Ravsky, Alexander Wolff
2019 arXiv   pre-print
Given a drawing of a graph, its visual complexity is defined as the number of geometrical entities in the drawing, for example, the number of segments in a straight-line drawing or the number of arcs in  ...  For complete, complete bipartite, and platonic graphs, we analyze their spherical cover numbers and compare them to their affine cover numbers as well as their segment and arc numbers.  ...  For the platonic solids, the lower bounds (see Table 4 ) that we computed using the MIP turned out to be tight. We conclude with a few open problems.  ... 
arXiv:1709.06965v3 fatcat:qdd6yzhj45d5zkzblrytvwu3wi

Balanced triangulations on few vertices and an implementation of cross-flips [article]

Lorenzo Venturello
2019 arXiv   pre-print
3-manifolds on few vertices.  ...  As a result we exhibit a vertex minimal balanced triangulation of the real projective plane, of the dunce hat and of the real projective space, as well as several balanced triangulations of surfaces and  ...  I would also like to thank Isabella Novik and Hailun Zheng for helpful comments on a preliminary version of this paper and Bruno Benedetti for pointing out the results in [BL13] .  ... 
arXiv:1811.10271v2 fatcat:bgmamlcvwrddnbysblwq2skf4i

Polynomially Bounding the Number of Minimal Separators in Graphs: Reductions, Sufficient Conditions, and a Dichotomy Theorem

Martin Milanič, Nevena Pivač
2021 Electronic Journal of Combinatorics  
We apply these results, combined with constructions of graphs with exponentially many minimal separators, to develop a dichotomy theorem separating tame from non-tame graph classes within the family of  ...  A graph class is said to be tame if graphs in the class have a polynomially bounded number of minimal separators.  ...  Acknowledgements The authors are grateful to the anonymous reviewers for their helpful remarks.  ... 
doi:10.37236/9428 fatcat:af67rib5szfkrl2xso4c2gnnhq

Efficient algorithms for graphs with few P4's

Luitpold Babel, Ton Kloks, Jan Kratochvı́l, Dieter Kratsch, Haiko Müller, Stephan Olariu
2001 Discrete Mathematics  
We show that a large variety of NP-complete problems can be solved e ciently for graphs with 'few' P4's.  ...  We show that all these problems can be solved in linear time for the class of (q; q − 4)-graphs, for every ÿxed q.  ...  Finally, S ⊂ V is a minimal separator of G if S is a minimal a; b-separator for some nonadjacent vertices a and b of G. The following lemma appeared ÿrst in [9] . Lemma 2.  ... 
doi:10.1016/s0012-365x(00)00258-2 fatcat:ac2qalhb4zh2zcyon3qgjpanem

Polyhedra with few 3-cuts are hamiltonian [article]

Gunnar Brinkmann, Carol T. Zamfirescu
2018 arXiv   pre-print
In 2002 Jackson and Yu have shown this result for the subclass of triangulations. We also prove that polyhedra with at most four 3-cuts have a hamiltonian path.  ...  It is well known that for each k > 6 non-hamiltonian polyhedra with k 3-cuts exist.  ...  We want to thank Brendan McKay for helpful discussions on this topic.  ... 
arXiv:1606.01693v2 fatcat:b5ujhg3hfvb67cwgt3iqu27noq

4-Connected Triangulations on Few Lines [article]

Stefan Felsner
2019 arXiv   pre-print
The same holds for all subgraphs of such triangulations. The proof is based on a corresponding result for diagrams of planar lattices which makes use of orthogonal chain and antichain families.  ...  We show that 4-connected plane triangulations can be redrawn such that edges are represented by straight segments and the vertices are covered by a set of at most √(2n) lines each of them horizontal or  ...  Work on this problem began at the 2018 Bertinoro Workshop of Graph Drawing. I thank the organizers of the event for making this possible.  ... 
arXiv:1908.04524v2 fatcat:ymi5izq5mvatjnpuqjxmmjgw3i

Triangulations with few ears: symmetry classes and disjointness [article]

Andrei Asinowski, Alon Regev
2014 arXiv   pre-print
Certain enumerational and structural problems become easier when one considers only triangulations with few ears. We demonstrate this in two ways.  ...  First, for k=2, 3, we find the number of symmetry classes of triangulations with k ears.  ...  Triangulations having few ears (equivalently, with few internal triangles) are generally easier to understand than triangulations with an arbitrary number of ears.  ... 
arXiv:1309.0743v3 fatcat:2zz7elynevetfatnnx6cu7snau

Polyhedra with Few 3-Cuts are Hamiltonian

G. Brinkmann, C. T. Zamfirescu
2019 Electronic Journal of Combinatorics  
In 2002 Jackson and Yu have shown this result for the subclass of triangulations. We also prove that polyhedra with at most four $3$-cuts have a hamiltonian path.  ...  It is well known that for each $k\ge 6$ non-hamiltonian polyhedra with $k$ $3$-cuts exist.  ...  We want to thank Brendan McKay for helpful discussions on this topic.  ... 
doi:10.37236/7771 fatcat:aicvurbls5gb5gq4thdac7dtrm
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