scholarly journals Bounded-Degree Graphs have Arbitrarily Large Geometric Thickness

10.37236/1029 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
János Barát ◽  
Jiří Matoušek ◽  
David R. Wood
Keyword(s):  
Graph Drawing ◽  
Line Drawing ◽  
Geometric Graphs ◽  
Regular Graphs ◽  
Open Problems ◽  
Bounded Degree ◽  
Large N ◽  
Straight Line ◽  
Minimum Integer ◽  
Geometric Thickness

The geometric thickness of a graph $G$ is the minimum integer $k$ such that there is a straight line drawing of $G$ with its edge set partitioned into $k$ plane subgraphs. Eppstein [Separating thickness from geometric thickness. In Towards a Theory of Geometric Graphs, vol. 342 of Contemp. Math., AMS, 2004] asked whether every graph of bounded maximum degree has bounded geometric thickness. We answer this question in the negative, by proving that there exists $\Delta$-regular graphs with arbitrarily large geometric thickness. In particular, for all $\Delta\geq9$ and for all large $n$, there exists a $\Delta$-regular graph with geometric thickness at least $c\sqrt{\Delta}\,n^{1/2-4/\Delta-\epsilon}$. Analogous results concerning graph drawings with few edge slopes are also presented, thus solving open problems by Dujmović et al. [Really straight graph drawings. In Proc. 12th International Symp. on Graph Drawing (GD '04), vol. 3383 of Lecture Notes in Comput. Sci., Springer, 2004] and Ambrus et al. [The slope parameter of graphs. Tech. Rep. MAT-2005-07, Department of Mathematics, Technical University of Denmark, 2005].

ON EXTENDING A PARTIAL STRAIGHT-LINE DRAWING

2006 ◽  
Vol 17 (05) ◽  
pp. 1061-1069 ◽  
Author(s):  
MAURIZIO PATRIGNANI
Keyword(s):  
Computational Complexity ◽  
Planar Graph ◽  
Graph Drawing ◽  
Line Drawing ◽  
Np Hard ◽  
Open Problems ◽  
Straight Line

We investigate the computational complexity of the following problem. Given a planar graph in which some vertices have already been placed in the plane, place the remaining vertices to form a planar straight-line drawing of the whole graph. We show that this extensibility problem, proposed in the 2003 "Selected Open Problems in Graph Drawing" [1], is NP-hard.

scholarly journals Bounded-Degree Graphs can have Arbitrarily Large Slope Numbers

10.37236/1139 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
János Pach ◽  
Dömötör Pálvölgyi
Keyword(s):  
Maximum Degree ◽  
Line Drawing ◽  
Bounded Degree ◽  
Straight Line ◽  
Bounded Degree Graphs

We construct graphs with $n$ vertices of maximum degree $5$ whose every straight-line drawing in the plane uses edges of at least $n^{1/6-o(1)}$ distinct slopes.

scholarly journals Partitioning Graph Drawings and Triangulated Simple Polygons into Greedily Routable Regions

2017 ◽  
Vol 27 (01n02) ◽  
pp. 121-158 ◽  
Author(s):  
Martin Nöllenburg ◽  
Roman Prutkin ◽  
Ignaz Rutter
Keyword(s):  
Routing Protocol ◽  
Graph Drawing ◽  
Geographic Routing ◽  
Np Hard ◽  
Line Drawings ◽  
Simple Polygons ◽  
Straight Line ◽  
Starting Point ◽  
Minimum Decomposition ◽  
Geographic Routing Protocol

A greedily routable region (GRR) is a closed subset of [Formula: see text], in which any destination point can be reached from any starting point by always moving in the direction with maximum reduction of the distance to the destination in each point of the path. Recently, Tan and Kermarrec proposed a geographic routing protocol for dense wireless sensor networks based on decomposing the network area into a small number of interior-disjoint GRRs. They showed that minimum decomposition is NP-hard for polygonal regions with holes. We consider minimum GRR decomposition for plane straight-line drawings of graphs. Here, GRRs coincide with self-approaching drawings of trees, a drawing style which has become a popular research topic in graph drawing. We show that minimum decomposition is still NP-hard for graphs with cycles and even for trees, but can be solved optimally for trees in polynomial time, if we allow only certain types of GRR contacts. Additionally, we give a 2-approximation for simple polygons, if a given triangulation has to be respected.

scholarly journals Non-perturbative aspects of Sp(2N) gauge theories

2021 ◽  
Author(s):  
◽  
Jack Holligan
Keyword(s):  
Gauge Theories ◽  
Fine Tuning ◽  
Global Symmetry ◽  
Dirac Fermions ◽  
Open Problems ◽  
Abelian Gauge ◽  
General Behaviour ◽  
Yang Mills ◽  
Large N ◽  
Glueball Spectrum

Yang-Mills theories based on the symplectic groups – denoted by Sp(2N) – are inter-esting for both theoretical and phenomenological reasons. Sp(2N) theories with two fundamental Dirac fermions give rise to pseudo-Nambu-Goldstone bosons which can be interpreted as a composite Higgs particle. This framework can describe the existing Higgs boson without the need for unnatural fine-tuning. This justifies a programme of wider investigations of Sp(2N) gauge theories aimed at understanding their general behaviour. In this work, we study the glueball mass spectrum for Sp(2N) Yang-Mills theories using the variational method applied to Monte-Carlo generated gauge config-urations. This is carried out both for finite N and in the limit N → ∞. The results are compared to existing results for SU(N) Yang-Mills theories, again, for finite- and large-N. Our glueball analysis is then used to investigate some conjectures related to the behaviour of the spectrum in Yang-Mills theories based on a generic non-Abeliangauge group G. We also find numerical evidence that Sp(2N) groups confine both for finite and large N. As well as studying the glueball spectrum, we examine the quenched-meson spectrum for fermions in the fundamental, antisymmetric and sym-metric representations for N = 2 and N = 3. This study enables us to provide a first account of how the related observables vary with N. The investigations presented in this work contribute to our understanding of the non-perturbative dynamics of Sp(2N) gauge theories in connection with Higgs compositeness and, more in general, with fun-damental open problems in non-Abelian gauge theories such as confinement and global symmetry breaking.

scholarly journals THE POINT-SET EMBEDDABILITY PROBLEM FOR PLANE GRAPHS

2013 ◽  
Vol 23 (04n05) ◽  
pp. 357-395 ◽  
Author(s):  
THERESE BIEDL ◽  
MARTIN VATSHELLE
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Line Drawing ◽  
Np Hard ◽  
Bounded Treewidth ◽  
Straight Line ◽  
Triangulated Graphs ◽  
Point Set ◽  
Fixed Embedding ◽  
Set Of Points

In this paper, we study the point-set embeddability problem, i.e., given a planar graph and a set of points, is there a mapping of the vertices to the points such that the resulting straight-line drawing is planar? It was known that this problem is NP-hard if the embedding can be chosen, but becomes polynomial for triangulated graphs of treewidth 3. We show here that in fact it can be answered for all planar graphs with a fixed combinatorial embedding that have constant treewidth and constant face-degree. We prove that as soon as one of the conditions is dropped (i.e., either the treewidth is unbounded or some faces have large degrees), point-set embeddability with a fixed embedding becomes NP-hard. The NP-hardness holds even for a 3-connected planar graph with constant treewidth, triangulated planar graphs, or 2-connected outer-planar graphs. These results also show that the convex point-set embeddability problem (where faces must be convex) is NP-hard, but we prove that it becomes polynomial if the graph has bounded treewidth and bounded maximum degree.

scholarly journals Distinct Distances in Graph Drawings

10.37236/831 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Paz Carmi ◽  
Vida Dujmović ◽  
Pat Morin ◽  
David R. Wood
Keyword(s):  
Maximum Degree ◽  
Combinatorial Geometry ◽  
Complete Graphs ◽  
Bounded Degree ◽  
Cartesian Products ◽  
Bounded Treewidth ◽  
Line Drawings ◽  
Straight Line ◽  
Minimum Number ◽  
Complete Bipartite Graphs

The distance-number of a graph $G$ is the minimum number of distinct edge-lengths over all straight-line drawings of $G$ in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the distance-number of trees, graphs with no $K^-_4$-minor, complete bipartite graphs, complete graphs, and cartesian products. Our main results concern the distance-number of graphs with bounded degree. We prove that $n$-vertex graphs with bounded maximum degree and bounded treewidth have distance-number in ${\cal O}(\log n)$. To conclude such a logarithmic upper bound, both the degree and the treewidth need to be bounded. In particular, we construct graphs with treewidth $2$ and polynomial distance-number. Similarly, we prove that there exist graphs with maximum degree $5$ and arbitrarily large distance-number. Moreover, as $\Delta$ increases the existential lower bound on the distance-number of $\Delta$-regular graphs tends to $\Omega(n^{0.864138})$.

scholarly journals Edge-Removal and Non-Crossing Configurations in Geometric Graphs

2010 ◽  
Vol Vol. 12 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Oswin Aichholzer ◽  
Sergio Cabello ◽  
Ruy Fabila-Monroy ◽  
David Flores-Peñaloza ◽  
Thomas Hackl ◽  
...  
Keyword(s):  
Extremal Problem ◽  
General Position ◽  
Geometric Graph ◽  
Perfect Matchings ◽  
Geometric Graphs ◽  
Line Segments ◽  
Straight Line ◽  
Edge Removal ◽  
International Audience ◽  
Point Set

Graphs and Algorithms International audience A geometric graph is a graph G = (V, E) drawn in the plane, such that V is a point set in general position and E is a set of straight-line segments whose endpoints belong to V. We study the following extremal problem for geometric graphs: How many arbitrary edges can be removed from a complete geometric graph with n vertices such that the remaining graph still contains a certain non-crossing subgraph. The non-crossing subgraphs that we consider are perfect matchings, subtrees of a given size, and triangulations. In each case, we obtain tight bounds on the maximum number of removable edges.

Frozen (Δ + 1)-colourings of bounded degree graphs

2020 ◽  
pp. 1-14
Author(s):  
Marthe Bonamy ◽  
Nicolas Bousquet ◽  
Guillem Perarnau
Keyword(s):  
Graph Theory ◽  
Maximum Degree ◽  
Mixing Time ◽  
Glauber Dynamics ◽  
Regular Graphs ◽  
Connected Component ◽  
Bounded Degree ◽  
Large Girth ◽  
Bounded Degree Graphs

Abstract Let G be a graph on n vertices and with maximum degree Δ, and let k be an integer. The k-recolouring graph of G is the graph whose vertices are k-colourings of G and where two k-colourings are adjacent if they differ at exactly one vertex. It is well known that the k-recolouring graph is connected for $k\geq \Delta+2$ . Feghali, Johnson and Paulusma (J. Graph Theory83 (2016) 340–358) showed that the (Δ + 1)-recolouring graph is composed by a unique connected component of size at least 2 and (possibly many) isolated vertices. In this paper, we study the proportion of isolated vertices (also called frozen colourings) in the (Δ + 1)-recolouring graph. Our first contribution is to show that if G is connected, the proportion of frozen colourings of G is exponentially smaller in n than the total number of colourings. This motivates the use of the Glauber dynamics to approximate the number of (Δ + 1)-colourings of a graph. In contrast to the conjectured mixing time of O(nlog n) for $k\geq \Delta+2$ colours, we show that the mixing time of the Glauber dynamics for (Δ + 1)-colourings restricted to non-frozen colourings can be Ω(n2). Finally, we prove some results about the existence of graphs with large girth and frozen colourings, and study frozen colourings in random regular graphs.

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