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

Power domination on triangular grids

The concept of power domination emerged from the problem of monitoring electrical systems. Given a graph G and a set S $\subseteq$ V (G), a set M of monitored vertices is built as follows: at first, M contains only the vertices of S and their direct neighbors, and then each time a vertex in M has exactly one neighbor not in M, this neighbor is added to M. The power domination number of a graph G is the minimum size of a set S such that this process ends up with the set M containing every vertex of G. We here show that the power domination number of a triangular grid T\_k with hexagonal-shape border of length k -- 1 is exactly $\lceil k/3 \rceil.Comment: Canadian Conference on Computational Geometry, Jul 2017, Ottawa, Canad

Power domination in maximal planar graphs

Power domination in graphs emerged from the problem of monitoring an electrical system by placing as few measurement devices in the system as possible. It corresponds to a variant of domination that includes the possibility of propagation. For measurement devices placed on a set S of vertices of a graph G, the set of monitored vertices is initially the set S together with all its neighbors. Then iteratively, whenever some monitored vertex v has a single neighbor u not yet monitored, u gets monitored. A set S is said to be a power dominating set of the graph G if all vertices of G eventually are monitored. The power domination number of a graph is the minimum size of a power dominating set. In this paper, we prove that any maximal planar graph of order n $\ge$ 6 admits a power dominating set of size at most (n--2)/4

Convex polygons in Cartesian products

We study several problems concerning convex polygons whose vertices lie in aCartesian product of two sets of n real numbers (for short, grid). First, we prove that everysuch grid contains Ω(log n) points in convex position and that this bound is tight up to aconstant factor. We generalize this result to d dimensions (for a fixed d ∈ N), and obtaina tight lower bound of Ω(logd−1 n) for the maximum number of points in convex positionin a d-dimensional grid. Second, we present polynomial-time algorithms for computing thelongest x- or y-monotone convex polygonal chain in a grid that contains no two points withthe same x- or y-coordinate. We show that the maximum size of a convex polygon with suchunique coordinates can be efficiently approximated up to a factor of 2. Finally, we presentexponential bounds on the maximum number of point sets in convex position in such grids,and for some restricted variants. These bounds are tight up to polynomial factors

Convex Polygons in Cartesian Products

We study several problems concerning convex polygons whose vertices lie in a Cartesian product of two sets of n real numbers (for short, grid). First, we prove that every such grid contains a convex polygon with Omega(log n) vertices and that this bound is tight up to a constant factor. We generalize this result to d dimensions (for a fixed d in N), and obtain a tight lower bound of Omega(log^{d-1}n) for the maximum number of points in convex position in a d-dimensional grid. Second, we present polynomial-time algorithms for computing the longest convex polygonal chain in a grid that contains no two points with the same x- or y-coordinate. We show that the maximum size of such a convex polygon can be efficiently approximated up to a factor of 2. Finally, we present exponential bounds on the maximum number of convex polygons in these grids, and for some restricted variants. These bounds are tight up to polynomial factors

Mutations in ZMYND10, a Gene Essential for Proper Axonemal Assembly of Inner and Outer Dynein Arms in Humans and Flies, Cause Primary Ciliary Dyskinesia

Primary ciliary dyskinesia (PCD) is a ciliopathy characterized by airway disease, infertility, and laterality defects, often caused by dual loss of the inner dynein arms (IDAs) and outer dynein arms (ODAs), which power cilia and flagella beating. Using whole-exome and candidate-gene Sanger resequencing in PCD-affected families afflicted with combined IDA and ODA defects, we found that 6/38 (16%) carried biallelic mutations in the conserved zinc-finger gene BLU (ZMYND10). ZMYND10 mutations conferred dynein-arm loss seen at the ultrastructural and immunofluorescence level and complete cilia immotility, except in hypomorphic p.Val16Gly (c.47T>G) homozygote individuals, whose cilia retained a stiff and slowed beat. In mice, Zmynd10 mRNA is restricted to regions containing motile cilia. In a Drosophila model of PCD, Zmynd10 is exclusively expressed in cells with motile cilia: chordotonal sensory neurons and sperm. In these cells, P-element-mediated gene silencing caused IDA and ODA defects, proprioception deficits, and sterility due to immotile sperm. Drosophila Zmynd10 with an equivalent c.47T>G (p.Val16Gly) missense change rescued mutant male sterility less than the wild-type did. Tagged Drosophila ZMYND10 is localized primarily to the cytoplasm, and human ZMYND10 interacts with LRRC6, another cytoplasmically localized protein altered in PCD. Using a fly model of PCD, we conclude that ZMYND10 is a cytoplasmic protein required for IDA and ODA assembly and that its variants cause ciliary dysmotility and PCD with laterality defects

Graphes planaires : dessins non-alignés, domination de puissance et énumération d'orientations Eulériennes

In this thesis, we present results on three different problems concerning planar graphs. We first give some new results on planar non-aligned drawings, i.e., planar grid drawings where vertices are all on different rows and columns. We show that not every planar graph has a non-aligned drawing on a grid with n rows and columns, but we present two algorithms generating a non-aligned polyline drawings on such a grid requiring either n − 3 or min( (2n−5)/3 , #{separating triangles} + 1) bends in total. Concerning non-minimal grids, we give two algorithms drawing a planar non-aligned drawing on grids with area O(n^4). We also give specific results for 4-connected graphs and nested-triangle graphs.The second topic is power domination in planar graphs. We present a family of graphs with power dominating number γ_P at least n/6 . We then prove that for every maximal planar graph G of order n, γ_P (G) ≤ (n−2)/4 , and we give a constructive algorithm. We also prove that for triangular grids Tk of dimension k with hexagonal-shape border, γ_P (Tk) = \lceil k/3 \rceil.Finally, we focus on the enumeration of planar Eulerian orientations. After proposing a new decomposition for these maps, we define subsets and supersets of planar Eulerian orientations with parameter k, generated by looking at the orientations of the last 2k − 1 edges around the root vertex. For each set, we give a system of functional equations defining its generating function, and we prove that it is always algebraic. This way, we show that the growth rate of planar Eulerian orientations is between 11.56 and 13.005.Dans cette thèse, nous étudions trois problèmes concernant les graphes planaires.Nous travaillons tout d’abord sur les dessins planaires non-alignés, c’est-à-dire des dessins planaires de graphes sur une grille sans que deux sommets se trouvent sur la même ligne ou la même colonne. Nous caractérisons les graphes planaires possédant un tel dessin sur une grille à n lignes et n colonnes, et nous présentons deux algorithmes générant un dessin planaire non-aligné avec arêtes brisées sur cette grille pour tout graphe planaire, avec n − 3 ou min( (2n−5)/3 , #{triangles séparateurs} + 1) brisures au total. Nous proposons également deux algorithmes dessinant un dessin planaire non-aligné sur des grilles d’aire O(n^4). Nous donnons des résultats spécifiques concernant les graphes 4-connexes et de typetriangle-emboîté.Le second sujet de cette thèse est la domination de puissance dans les graphes planaires. Nous exhibons une famille de graphes ayant un nombre de domination de puissance γ_P au moins égal à n/6 . Nous montrons aussi que pour tout graphe planaire maximal G à n ≥ 6 sommets, γ_P (G) ≤ (n−2)/4 . Enfin, nous étudions les grilles triangulaires Tk à bord hexagonal de dimension k et nous montrons que γ_P (Tk ) = \lceil k/3 \rceil.Nous étudions également l’énumération des orientations planaires Eulériennes. Nous proposons tout d’abord une nouvelle décomposition de ces cartes. Puis, en considérant les orientations des dernières 2k − 1 arêtes autour de la racine, nous définissons des sous- et sur-ensembles des orientations planaires Eulériennes paramétrés par k. Pour chaque classe, nous proposons un système d’équations fonctionnelles définissant leur série génératrice, et nous prouvons que celle-ci est toujours algébrique. Nous montrons ainsi que la constante de croissance des orientations planaires Eulériennes est comprise entre 11.56 et 13.005

Planar graphs : non-aligned drawings, power domination and enumeration of Eulerian orientations

Dans cette thèse, nous présentons trois problèmes concernant les graphes planaires.Nous travaillons tout d'abord sur les dessins planaires non-alignés, c'est-à-dire des dessins planaires de graphes sur une grille sans que deux sommets se trouvent sur la même ligne ou la même colonne.Nous caractérisons les graphes planaires possédant un tel dessin sur une grille de taille n x n, et nous présentons deux algorithmes générant un dessin planaire non-aligné avec arêtes brisées sur cette grille pour tout graphe planaire, avec n-3 ou min((2n-3)/(5),#{triangles séparateurs}+1) brisures au total.Nous proposons également deux algorithmes dessinant un dessin planaire non-aligné sur des grilles d'aire O(n⁴). Nous donnons des résultats spécifiques concernant les graphes 4-connexes et de type triangle-emboîté.Le second sujet de cette thèse est la domination de puissance dans les graphes planaires. Nous exhibons une famille de graphes ayant un nombre de domination de puissance γP au moins égal à n/6. Nous montrons aussi que pour tout graphe planaire maximal G à n≥6 sommets, γP(G) ≤ (n-2)/4. Enfin, nous étudions les grilles triangulaires Tk à bord hexagonal de dimension k et nous montrons que γP(Tk) = [k/3]. Nous étudions également l'énumération des orientations planaires Eulériennes. Nous proposons une nouvelle décomposition de ces cartes. En considérant les orientations des dernières 2k-1 arêtes autour de la racine, nous définissons des sous- et sur-ensembles des orientations planaires Eulériennes paramétrés par k.Pour chaque classe, nous proposons un système d'équations fonctionnelles définissant leur série génératrice, et nous prouvons que celle-ci est toujours algébrique. Nous montrons ainsi que la constance de croissance des orientations planaires Eulériennes est entre 11.56 et 13.005.In this thesis, we present results on three different problems concerning planar graphs.We first give some new results on planar non-aligned drawings, i.e. planar grid drawings where vertices are all on different rows and columns.We show that not every planar graph has a non-aligned drawing on an n x n-grid, but we present two algorithms generating a non-aligned polyline drawings on such a grid requiring either n-3 or min((2n-3)/(5),#{separating triangles}+1) bends in total.Concerning non-minimal grids, we give two algorithms drawing a planar non-aligned drawing on grids with area of order n⁴. We also give specific results for 4-connected graphs and nested-triangle graphs.The second topic is power domination in planar graphs. We present a family of graphs with power dominating number γP at least n/6. We then prove that for every maximal planar graph G of order n, γP(G) ≤ (n-2)/4, and we give a constructive algorithm. We also prove that for triangular grids Tk of dimension k with hexagonal-shape border, γP(Tk) = [k/3].Finally, we focus on the enumeration of planar Eulerian orientations. After proposing a new decomposition for these maps, we define subsets and supersets of planar Eulerian orientations with parameter k, generated by looking at the orientations of the last 2k-1 edges around the root vertex.For each set, we give a system of functional equations defining its generating function, and we prove that it is always algebraic.This way, we show that the growth rate of planar Eulerian orientations is between 11.56 and 13.005

Power domination on triangular grids with triangular and hexagonal shape

The concept of power domination emerged from the problem of monitoring electrical systems. Given a graph G and a set S ⊆ V (G), a set M of monitored vertices is built as follows: at first, M contains only the vertices of S and their direct neighbors, and then each time a vertex in M has exactly one neighbor not in M , this neighbor is added to M. The power domination number of a graph G is the minimum size of a set S such that this process ends up with the set M containing every vertex of G. We show that the power domination number of a triangular grid H_k with hexagonal-shaped border of length k − 1 is the ceiling of k/3 , and the one of a triangular grid T_k with triangular-shaped border of length k − 1 is the ceiling of k/4

Rook-Drawing for Plane Graphs

Motivated by visualization of large graphs, we introduce a new type of graph drawing called “rook-drawing”. A rook-drawing of a graph G is obtained by placing the n nodes of G on the intersections of a regular grid, such that each row and column of the grid supports exactly one node. This paper focuses on rook-drawings of planar graphs. We first give a linear algorithm to compute a planar straight-line rook-drawing for outerplanar graphs. We then characterize the maximal planar graphs admitting a planar straight-line rook-drawing, which are unique for a given order. Finally, we give a linear time algorithm to compute a polyline planar rook-drawing for plane graphs with at most n−3 bent edges