arXiv subjects

Thomas Magnard

Publications and source records attributed to Thomas Magnard.

3 recordsLinked to original sources

arXiv2021

An FPT algorithm for the embeddability of graphs into two-dimensional simplicial complexes

We consider the embeddability problem of a graph G into a two-dimensional simplicial complex C: Given G and C, decide whether G admits a topological embedding into C. The problem is NP-hard, even in the restricted case where C is homeomorphic to a surface. We prove that the problem is fixed-parameter tractable in the size of the two-dimensional complex, by providing an O(2^{poly(c)}.n^2)-time algorithm. If G embeds into C, we can compute a representation of an embedding in the same amount of time. Moreover, we show that several known problems reduce to this one, such as the crossing number and the planarity number problems, and, under some conditions, the embedding extension problem. Our approach is to reduce to the case where G has bounded branchwidth via an irrelevant vertex method, and to apply dynamic programming. We do not rely on any component of the existing linear-time algorithms for embedding graphs on a fixed surface, but only on algorithms from graph minor theory. However, by combining our results with a linear-time algorithm for embedding graphs on surfaces and with a very recent result for the irrelevant vertex method, we can decide whether G embeds into C in f(c).O(n) time, for some function f.

cs.CG↗

arXiv2019

Shellings and sheddings induced by collapses

We say that a pure simplicial complex ${\mathbf K}$ of dimension $d$ satisfies the removal-collapsibility condition if ${\mathbf K}$ is either empty or ${\mathbf K}$ becomes collapsible after removing $\tilde \beta_d ({\mathbf K}; {\mathbb Z}_2)$ facets, where $\tilde \beta_d ({\mathbf K}; {\mathbb Z}_2)$ denotes the $d$th reduced Betti number. In this paper, we show that if the link of each face of a pure simplicial complex ${\mathbf K}$ (including the link of the empty face which is the whole ${\mathbf K}$) satisfy the removal-collapsibility condition, then the second barycentric subdivision of ${\mathbf K}$ is vertex decomposable and in particular shellable. This is a higher dimensional generalization of a result of Hachimori, who proved that that if the link of each vertex of a pure 2-dimensional simplicial complex ${\mathbf K}$ is connected, and ${\mathbf K}$ becomes simplicially collapsible after removing $\tilde{\chi}({\mathbf K})$ facets, where $\tilde \chi ({\mathbf K})$ denotes the reduced Euler characteristic, then the second barycentric subdivision of ${\mathbf K}$ is shellable. For the proof, we introduce a new variant of decomposability of a simplicial complex, stronger than vertex decomposability, which we call star decomposability. This notion may be of independent interest.

math.CO↗

arXiv2018

Embedding graphs into two-dimensional simplicial complexes

We consider the problem of deciding whether an input graph G admits a topological embedding into a two-dimensional simplicial complex C. This problem includes, among others, the embeddability problem of a graph on a surface and the topological crossing number of a graph, but is more general. The problem is NP-complete when C is part of the input, and we give a polynomial-time algorithm if the complex C is fixed. Our strategy is to reduce the problem to an embedding extension problem on a surface, which has the following form: Given a subgraph H' of a graph G', and an embedding of H' on a surface S, can that embedding be extended to an embedding of G' on S? Such problems can be solved, in turn, using a key component in Mohar's algorithm to decide the embeddability of a graph on a fixed surface (STOC 1996, SIAM J. Discr. Math. 1999).

cs.CG↗