Discrete Morse theory for the collapsibility of supremum sections

The Dushnik-Miller dimension of a poset $\le$ is the minimal number $d$ of linear extensions $\le_1, \ldots , \le_d$ of $\le$ such that $\le$ is the intersection of $\le_1, \ldots , \le_d$. Supremum sections are simplicial complexes introduced by Scarf and are linked to the Dushnik-Miller as follows: the inclusion poset of a simplicial complex is of Dushnik-Miller dimension at most $d$ if and only if it is included in a supremum section coming from a representation of dimension $d$. Collapsibility is a topoligical property of simplicial complexes which has been introduced by Whitehead and which resembles to shellability. While Ossona de Mendez proved in that a particular type of supremum sections are shellable, we show in this article that supremum sections are in general collapsible thanks to the discrete Morse theory developped by Forman

Planar graphs as L-intersection or L-contact graphs

The L-intersection graphs are the graphs that have a representation as intersection graphs of axis parallel shapes in the plane. A subfamily of these graphs are {L, |, --}-contact graphs which are the contact graphs of axis parallel L, |, and -- shapes in the plane. We prove here two results that were conjectured by Chaplick and Ueckerdt in 2013. We show that planar graphs are L-intersection graphs, and that triangle-free planar graphs are {L, |, --}-contact graphs. These results are obtained by a new and simple decomposition technique for 4-connected triangulations. Our results also provide a much simpler proof of the known fact that planar graphs are segment intersection graphs

On the Distance Identifying Set Meta-Problem and Applications to the Complexity of Identifying Problems on Graphs

Numerous problems consisting in identifying vertices in graphs using distances are useful in domains such as network verification and graph isomorphism. Unifying them into a meta-problem may be of main interest. We introduce here a promising solution named Distance Identifying Set. The model contains Identifying Code (IC), Locating Dominating Set (LD) and their generalizations $r$-IC and $r$-LD where the closed neighborhood is considered up to distance $r$. It also contains Metric Dimension (MD) and its refinement $r$-MD in which the distance between two vertices is considered as infinite if the real distance exceeds $r$. Note that while IC = 1-IC and LD = 1-LD, we have MD = $\infty$-MD; we say that MD is not local In this article, we prove computational lower bounds for several problems included in Distance Identifying Set by providing generic reductions from (Planar) Hitting Set to the meta-problem. We mainly focus on two families of problem from the meta-problem: the first one, called bipartite gifted local, contains $r$-IC, $r$-LD and $r$-MD for each positive integer $r$ while the second one, called 1-layered, contains LD, MD and $r$-MD for each positive integer $r$. We have: - the 1-layered problems are NP-hard even in bipartite apex graphs, - the bipartite gifted local problems are NP-hard even in bipartite planar graphs, - assuming ETH, all these problems cannot be solved in $2^{o(\sqrt{n})}$ when restricted to bipartite planar or apex graph, respectively, and they cannot be solved in $2^{o(n)}$ on bipartite graphs, - even restricted to bipartite graphs, they do not admit parameterized algorithms in $2^{O(k)}.n^{O(1)}$ except if W[0] = W[2]. Here $k$ is the solution size of a relevant identifying set. In particular, Metric Dimension cannot be solved in $2^{o(n)}$ under ETH, answering a question of Hartung in 2013

Degreewidth: a New Parameter for Solving Problems on Tournaments

In the paper, we define a new parameter for tournaments called degreewidth which can be seen as a measure of how far is the tournament from being acyclic. The degreewidth of a tournament $T$ denoted by $\Delta(T)$ is the minimum value $k$ for which we can find an ordering $\langle v_1, \dots, v_n \rangle$ of the vertices of $T$ such that every vertex is incident to at most $k$ backward arcs (\textit{i.e.} an arc $(v_i,v_j)$ such that $j<i$). Thus, a tournament is acyclic if and only if its degreewidth is zero. Additionally, the class of sparse tournaments defined by Bessy et al. [ESA 2017] is exactly the class of tournaments with degreewidth one. We first study computational complexity of finding degreewidth. Namely, we show it is NP-hard and complement this result with a $3$-approximation algorithm. We also provide a cubic algorithm to decide if a tournament is sparse. Finally, we study classical graph problems \textsc{Dominating Set} and \textsc{Feedback Vertex Set} parameterized by degreewidth. We show the former is fixed parameter tractable whereas the latter is NP-hard on sparse tournaments. Additionally, we study \textsc{Feedback Arc Set} on sparse tournaments

Discrete Morse theory for the collapsibility of supremum sections

International audienceThe Dushnik-Miller dimension of a poset $\le$ is the minimal number $d$ of linear extensions $\le_1, \ldots , \le_d$ of $\le$ such that $\le$ is the intersection of $\le_1, \ldots , \le_d$. Supremum sections are simplicial complexes introduced by Scarf and are linked to the Dushnik-Miller as follows: the inclusion poset of a simplicial complex is of Dushnik-Miller dimension at most $d$ if and only if it is included in a supremum section coming from a representation of dimension $d$. Collapsibility is a topoligical property of simplicial complexes which has been introduced by Whitehead and which resembles to shellability. While Ossona de Mendez proved in that a particular type of supremum sections are shellable, we show in this article that supremum sections are in general collapsible thanks to the discrete Morse theory developped by Forman

From planar graphs to higher dimension

Dans cette thèse on cherche à généraliser certaines propriétés des graphes planaires aux dimensions supérieures en remplaçant les graphes par des complexes simpliciaux. En particulier on étudie la dimension de Dushnik-Miller qui mesure à quel point un ordre partiel ressemble à un ordre total. Appliquée aux complexes simpliciaux, cette dimension semble capturer des propriétés géométriques. Concernant ce sujet, on infirme une conjecture assurant que n'importe quel complexe simplicial de dimension de Dushnik-Miller au plus d+1 peut être représenté par un complexe de TD-Delaunay dans RR d, qui est une variante des graphes de Delaunay dans le plan. On montre que toute section supremum, qui est un complexe simplicial particulier relié à la dimension de Dushnik-Miller, est ``collapsible'', c'est-à-dire que l'on peut atteindre un point unique en retirant dans le bon ordre les faces du complexe. On introduit la notion d'empilements d'escaliers et on démontre que la dimension de Dushnik-Miller est reliée aux complexes de contacts de tels empilements. On démontre aussi de nouveaux résultats sur les graphes planaires.Les deux résultats suivants sur la représentabilité des graphes planaires sont démontrés : tout graphe planaire est le graphe d'intersection de llcorner et tout graphe planaire sans triangle est le graphe de contact de {llcorner, | , -}. On introduit et étudie une nouvelle notion sur les graphes planaires que l'on appelle ``Möbius stanchion systems'' qui sont reliés à des questions sur les plongements unicellulaires des graphes planaires.In this thesis we look for generalizations of some properties of planar graphs to higher dimensions by replacing graphs by simplicial complexes.In particular we study the Dushnik-Miller dimension which measures how a partial order is far from being a linear order.When applied to simplicial complexes, this dimension seems to capture some geometric properties.In this idea, we disprove a conjecture asserting that any simplicial complex of Dushnik-Miller dimension at most d+1 can be represented as a TD-Delaunay complex in RR d, which is a variant of the well known Delaunay graphs in the plane.We show that any supremum section, particular simplicial complexes related to the Dushnik-Miller dimension, is collapsible, which means that it is possible to reach the single point by removing in a certain order the faces of the complex.We introduce the notion of stair packings and we prove that the Dushnik-Miller dimension is connected to contact complexes of such packings.We also prove new results on planar graphs.The two following theorems about representations of planar graphs are proved: any planar graph is an llcorner-intersection graph and any triangle-free planar graph is an {llcorner, | , -}-contact graph.We introduce and study a new notion on planar graphs called Möbius stanchion systems which is related to questions about unicellular embeddings of planar graphs

Des graphes planaires vers des dimensions supérieures

In this thesis we look for generalizations of some properties of planar graphs to higher dimensions by replacing graphs by simplicial complexes.In particular we study the Dushnik-Miller dimension which measures how a partial order is far from being a linear order.When applied to simplicial complexes, this dimension seems to capture some geometric properties.In this idea, we disprove a conjecture asserting that any simplicial complex of Dushnik-Miller dimension at most d+1 can be represented as a TD-Delaunay complex in RR d, which is a variant of the well known Delaunay graphs in the plane.We show that any supremum section, particular simplicial complexes related to the Dushnik-Miller dimension, is collapsible, which means that it is possible to reach the single point by removing in a certain order the faces of the complex.We introduce the notion of stair packings and we prove that the Dushnik-Miller dimension is connected to contact complexes of such packings.We also prove new results on planar graphs.The two following theorems about representations of planar graphs are proved: any planar graph is an llcorner-intersection graph and any triangle-free planar graph is an {llcorner, | , -}-contact graph.We introduce and study a new notion on planar graphs called Möbius stanchion systems which is related to questions about unicellular embeddings of planar graphs.Dans cette thèse on cherche à généraliser certaines propriétés des graphes planaires aux dimensions supérieures en remplaçant les graphes par des complexes simpliciaux. En particulier on étudie la dimension de Dushnik-Miller qui mesure à quel point un ordre partiel ressemble à un ordre total. Appliquée aux complexes simpliciaux, cette dimension semble capturer des propriétés géométriques. Concernant ce sujet, on infirme une conjecture assurant que n'importe quel complexe simplicial de dimension de Dushnik-Miller au plus d+1 peut être représenté par un complexe de TD-Delaunay dans RR d, qui est une variante des graphes de Delaunay dans le plan. On montre que toute section supremum, qui est un complexe simplicial particulier relié à la dimension de Dushnik-Miller, est ``collapsible'', c'est-à-dire que l'on peut atteindre un point unique en retirant dans le bon ordre les faces du complexe. On introduit la notion d'empilements d'escaliers et on démontre que la dimension de Dushnik-Miller est reliée aux complexes de contacts de tels empilements. On démontre aussi de nouveaux résultats sur les graphes planaires.Les deux résultats suivants sur la représentabilité des graphes planaires sont démontrés : tout graphe planaire est le graphe d'intersection de llcorner et tout graphe planaire sans triangle est le graphe de contact de {llcorner, | , -}. On introduit et étudie une nouvelle notion sur les graphes planaires que l'on appelle ``Möbius stanchion systems'' qui sont reliés à des questions sur les plongements unicellulaires des graphes planaires

Discrete Morse theory for the collapsibility of supremum sections

International audienceThe Dushnik-Miller dimension of a poset $\le$ is the minimal number $d$ of linear extensions $\le_1, \ldots , \le_d$ of $\le$ such that $\le$ is the intersection of $\le_1, \ldots , \le_d$. Supremum sections are simplicial complexes introduced by Scarf and are linked to the Dushnik-Miller as follows: the inclusion poset of a simplicial complex is of Dushnik-Miller dimension at most $d$ if and only if it is included in a supremum section coming from a representation of dimension $d$. Collapsibility is a topoligical property of simplicial complexes which has been introduced by Whitehead and which resembles to shellability. While Ossona de Mendez proved in that a particular type of supremum sections are shellable, we show in this article that supremum sections are in general collapsible thanks to the discrete Morse theory developped by Forman

Möbius Stanchion Systems

International audienceConsider a group of stanchions linked together in a waiting line. In order to paint both sides of every stanchion you will need to lift your paintbrush as many times as the number of faces of the corresponding plane graph. As a lazy graph theorist you want to twist the strips between stanchions in a Möbius fashion such that you do not need to lift up your paintbrush. We call such a twist a MSS and we investigate the space of all MSSs of a planar graph. Our main results are that all the MSSs are connected by a series of two elementary operations, and that the space of MSSs does not depend on the planar embedding of the graph