Planar Graphs Research Articles

The maximum number of odd cycles in a planar graph

AbstractHow many copies of a fixed odd cycle, , can a planar graph contain? We answer this question asymptotically for and prove a bound which is tight up to a factor of 3/2 for all other values of . This extends the prior results of Cox and Martin and of Lv, Győri, He, Salia, Tompkins, and Zhu on the analogous question for even cycles. Our bounds result from a reduction to the following maximum likelihood question: which probability mass on the edges of some clique maximizes the probability that edges sampled independently from form either a cycle or a path?

Unit-length Rectangular Drawings of Graphs

A rectangular drawing of a planar graph $G$ is a planar drawing of $G$ in which vertices are mapped to grid points, edges are mapped to horizontal and vertical straight-line segments, and faces are drawn as rectangles. Sometimes this latter constraint is relaxed for the outer face. In this paper, we study rectangular drawings in which the edges have unit length. We show a complexity dichotomy for the problem of deciding the existence of a unit-length rectangular drawing, depending on whether the outer face must also be drawn as a rectangle or not. Specifically, we prove that the problem is NP-complete for biconnected graphs when the drawing of the outer face is not required to be a rectangle, even if the sought drawing must respect a given combinatorial embedding, whereas it is polynomial-time solvable, both in the fixed and the variable embedding settings, if the outer face is required to be drawn as a rectangle. Furthermore, we provide a linear-time algorithm for deciding whether a plane graph admits an embedding-preserving unit-length rectangular drawing if the drawing of the outer face is prescribed. As a by-product of our research, we provide the first polynomial-time algorithm to test whether a planar graph $G$ admits a rectangular drawing, for general instances of maximum degree $4$.

The scaling limit of random cubic planar graphs

AbstractWe study the random cubic planar graph with an even number of vertices. We show that the Brownian map arises as Gromov–Hausdorff–Prokhorov scaling limit of as tends to infinity, after rescaling distances by for a specific constant . This is the first time a model of random graphs that are not embedded into the plane is shown to converge to the Brownian map. Our approach features a new method that allows us to relate distances on random graphs to first‐passage percolation distances on their 3‐connected core.

ZONAL LABELING OF EDGE COMB PRODUCT OF GRAPHS

Given a plane graph $G=(V,E)$. A zonal labeling of graph $G$ is defined as an assignment of the two nonzero elements of the ring $\mathbb{Z}_3$, which are $1$ and $2$, to the vertices of $G$ such that the sum of the labels of the vertices on the border of each region of the graph is $0\in\mathbb{Z}_3$. A graph $G$ that possess such a labeling is termed as zonal graph. This paper will characterize edge comb product graphs that are zonal. The results show that $P_m\trianglerighteq_eC_n$, $C_n\trianglerighteq_e C_r$, $S_p\trianglerighteq_e C_n$, and $S_p\trianglerighteq_e F_t$ are zonal in some cases, but not in others.

On the Biplanarity of Blowups

The 2-blowup of a graph is obtained by replacing each vertex with two non-adjacent copies; a graph is biplanar if it is the union of two planar graphs. We disprove a conjecture of Gethner that 2-blowups of planar graphs are biplanar: iterated Kleetopes are counterexamples. Additionally, we construct biplanar drawings of 2-blowups of planar graphs whose duals have two-path induced path partitions, and drawings with split thickness two of 2-blowups of 3-chromatic planar graphs, and of graphs that can be decomposed into a Hamiltonian path and a dual Hamiltonian path.

Light 3-faces in 3-polytopes without adjacent triangles

Over the last decades, a lot of research has been devoted to structural and coloring problems on plane graphs that are sparse in this or that sense.In this note we deal with the densest among sparse 3-polytopes, namely those having no adjacent 3-cycles. Borodin (1996) proved that such 3-polytopes have a vertex of degree at most 4 and, moreover, an edge with the degree-sum of its end-vertices at most 9, where both bounds are sharp.By d(v) denote the degree of a vertex v. An edge e=xy in a 3-polytope is an (i,j)-edge if d(x)≤i and d(y)≤j. The well-known (3,5;4,4)-Archimedean solid corresponds to a plane quadrangulation in which every edge joins a 3-vertex with a 5-vertex.We prove that every 3-polytope with neither adjacent 3-cycles nor (3,5)-edges has a 3-face with the degree-sum of its incident vertices (weight) at most 16, which bound is sharp.

Error and attack vulnerability of Apollonian networks

Abstract This article examines the resilience of different Apollonian network (AN) types—deterministic, random, and evolutionary—to systematic attacks. ANs, members of the family of maximal planar graphs, possess unique properties such as high clustering coefficients, small-world properties, scale-free behavior, Euclidean and space-filling properties, and modularity. These peculiarities require a thorough investigation of their robustness. This work presents a novel approach to studying ANs by implementing evolutionary Apollonian networks (EANs). These EANs include various probabilities distribution functions, including exponential, degenerate, logistic, Pareto, and stable (Cauchy, Lévy, Normal) distributions. To improve the robustness of these networks, we propose a novel edge rewiring mechanism using a genetic algorithm (GA). The GA aims to optimize a combined metric that includes the Flow Robustness of Degree (SFRD), Betweenness (SFRB), and Dangalchev's closeness (SFRC) centralities while preserving the original degree distribution and structural properties of the network. To evaluate the effectiveness of this approach, we use various robustness measures to assess the resilience of different AN types. The results show that SFRB, SFRD, and SFRC effectively rank ANs based on their robustness.

Polyhedra with Hexagonal and Triangular Faces and Three Faces Around Each Vertex

AbstractWe analyze polyhedra composed of hexagons and triangles with three faces around each vertex, and their 3-regular planar graphs of edges and vertices, which we call “trihexes”. Trihexes are analogous to fullerenes, which are 3-regular planar graphs whose faces are all hexagons and pentagons. Every trihex can be represented as the quotient of a hexagonal tiling of the plane under a group of isometries generated by $$180^\circ $$ 180 ∘ rotations. Every trihex can also be described with either one or three “signatures”: triples of numbers that describe the arrangement of the rotocenters of these rotations. Simple arithmetic rules relate the three signatures that describe the same trihex. We obtain a bijection between trihexes and equivalence classes of signatures as defined by these rules. Labeling trihexes with signatures allows us to put bounds on the number of trihexes for a given number of vertices v in terms of the prime factorization of v and to prove a conjecture concerning trihexes that have no “belts” of hexagons.

The complexity of the perfect matching‐cut problem

AbstractPERFECT MATCHING‐CUT is the problem of deciding whether a graph has a perfect matching that contains an edge‐cut. We show that this problem is NP‐complete for planar graphs with maximum degree four, for planar graphs with girth five, for bipartite five‐regular graphs, for graphs of diameter three, and for bipartite graphs of diameter four. We show that there exist polynomial‐time algorithms for the following classes of graphs: claw‐free, ‐free, diameter two, bipartite with diameter three, and graphs with bounded treewidth.

Counting circuit double covers

AbstractWe study a counting version of Cycle Double Cover Conjecture. We discuss why it is more interesting to count circuits (i.e., graphs isomorphic to for some ) instead of cycles (graphs with all degrees even). We give an almost‐exponential lower bound for graphs with a surface embedding of representativity at least 4. We also prove an exponential lower bound for planar graphs. We conjecture that any bridgeless cubic graph has at least circuit double covers and we show an infinite class of graphs for which this bound is tight.

Redicolouring digraphs: Directed treewidth and cycle-degeneracy

Given a digraph D=(V,A) on n vertices and a vertex v∈V, the cycle-degree of v is the minimum size of a set S⊆V(D)∖{v} intersecting every directed cycle of D containing v. From this definition of cycle-degree, we define the c-degeneracy (or cycle-degeneracy) of D, which we denote by δc∗(D). It appears to be a nice generalisation of the undirected degeneracy. For instance, the dichromatic number χ→(D) of D is bounded above by δc∗(D)+1, where χ→(D) is the minimum integer k such that D admits a k-dicolouring, i.e., a partition of its vertices into k acyclic subdigraphs.In this work, using this new definition of cycle-degeneracy, we extend several evidences for Cereceda’s conjecture (Cereceda, 2007) to digraphs. The k-dicolouring graph of D, denoted by Dk(D), is the undirected graph whose vertices are the k-dicolourings of D and in which two k-dicolourings are adjacent if they differ on the colour of exactly one vertex. This is a generalisation of the k-colouring graph of an undirected graph G, in which the vertices are the proper k-colourings of G.We show that Dk(D) has diameter at most Oδc∗(D)(nδc∗(D)+1) (respectively O(n2) and (δc∗(D)+1)n) when k is at least δc∗(D)+2 (respectively 32(δc∗(D)+1) and 2(δc∗(D)+1)). This improves known results on digraph redicolouring (Bousquet et al., 2023). Next, we extend a result due to Feghali (2021) to digraphs, showing that Dd+1(D) has diameter at most Od,ϵ(n(logn)d−1) when D has maximum average cycle-degree at most d−ϵ.We then show that two proofs of Bonamy and Bousquet (2018) for undirected graphs can be extended to digraphs. The first one uses the digrundy number of a digraph χ→g(D), which is the worst number of colours used in a greedy dicolouring. If k≥χ→g(D)+1, we show that Dk(D) has diameter at most 4⋅χ→(D)⋅n. The second one uses the D-width of a digraph, denoted by Dw(D), which is a generalisation of the treewidth to digraphs. If k≥Dw(D)+2, we show that Dk(D) has diameter at most 2(n2+n).Finally, we give a general theorem which makes a connection between the recolourability of a digraph D and the recolourability of its underlying graph UG(D). Assume that G is a class of undirected graphs, closed under edge-deletion and with bounded chromatic number, and let k≥χ(G) (i.e., k≥χ(G) for every G∈G) be such that, for every n-vertex graph G∈G, the diameter of the k-colouring graph of G is bounded by f(n) for some function f. We show that, for every n-vertex digraph D such that UG(D)∈G, the diameter of Dk(D) is bounded by 2f(n). For instance, this result directly extends a number of results on planar graph recolouring to planar digraph redicolouring.

Construction of floorplans for plane graphs over polygonal boundaries

Construction of floorplans for plane graphs over polygonal boundaries

Application of drag reduction agents as a method for reducing heat loss during pumping through a pipeline

Relevance. The need to maintain the temperature regime of product pumping under increasingly complex conditions of pipeline routes by reducing heat losses of oil and petroleum product pipelines, as well as expanding the scope of application of anti-turbulence additives. Aim. To determine the effect of anti-turbulence additives on heat and hydraulic losses in the pipeline and to propose cost-effective ways to use additives when pumping through pipelines. Objects. Heat and hydraulic losses in pipelines pumping oil and oil products. Methods. Mathematical analysis of the influence of anti-turbulence additives on the thermohydraulic properties of the flow to assess the prospects for increasing the energy efficiency of pumping liquid through pipelines by introducing polymer additives. Results. The flow temperature was calculated using anti-turbulence additive depending on its concentration and efficiency, taking into account the dependence of the properties of the pumped product on temperature. The authors have constructed the graphic dependences of the economic components of heat and hydraulic losses on the concentration of anti-turbulence additives. The economic feasibility of the decision was estimated in terms of calculating the difference in pumping costs with and without anti-turbulence additives. The authors identified the relationship between the impact of losses from friction heat and heat exchange with the environment on heat losses, and analyzed the change in these parameters after the introduction of anti-turbulence additives. For the pipeline under consideration, the parameters of which correspond to standard pumping ones, the сonclusions were drawn on the predominance of the contribution of the additive hydraulic efficiency over the thermal one. This indicates the advisability of using an anti-turbulence additive as an agent for reducing heat losses at high values of the efficiency of the polymer additive, however, the overall economic efficiency is maximum at lower concentrations of the agent. The authors constructed a planar graph reflecting the dependence of the temperature at the end of the pipeline section on two coordinates: the length of the section and the hydraulic efficiency of the introduced additive.

Графовые алгоритмы для расчёта распределения следов амурского тигра на территории Приморского края

The paper analyzes the distribution of the Amur tiger (Panthera tigris altaica) tracks in the Primorsky Krai municipalities according to observations obtained during the one-time winter count organized by the Pacific Institute of Geography of the Far Eastern Branch of the Russian Academy of Sciences in 2005. This task is related to the need to calculate the number of animals based on the tracks of their vital activity. Based on the initial data we form a sample to analyze a set of tracks to identify intersection areas of various tiger habitats in Primorsky Krai. The map of Primorsky Krai municipalities was presented in the form of a planar graph to identify their intersections. A recurrent algorithm for identifying tiger clusters was developed using graph theory methods. According to them, the territory of Primorsky Krai is divided into northern, central, southern and southwestern zones with intersections. The districts of Primorsky Krai included in the designated zones are listed. The total number of tracks in intersecting adjacent zones was counted to identify possible tiger migrations in Primorsky Krai. A quantitative assessment of the degree of intersection of the selected zones by the number of tiger tracks is given. The proposed calculation scheme can be used to clarify and verify the results of modeling the spatial distribution and total population size of the Amur tiger.

Metric basis and metric dimension of some infinite planar graphs

Let [Formula: see text] be a nontrivial connected simple graph and [Formula: see text] be the distance between the vertices [Formula: see text] and [Formula: see text] in [Formula: see text]. The metric dimension of a graph [Formula: see text], denoted by dim([Formula: see text]), refers to the smallest set of vertices required to uniquely identify every vertex in the graph [Formula: see text]. A family of simple connected graphs say [Formula: see text], where [Formula: see text] has a constant metric dimension, if dim[Formula: see text] is finite and does not depend on the choice of [Formula: see text] in [Formula: see text]. In this paper, we consider two infinite families of planar graphs, say [Formula: see text], where [Formula: see text] and [Formula: see text], where [Formula: see text], and investigate their metric basis as well as the metric dimension. Additionally, we prove that the metric basis for these two graphs are independent.

Computing the Edge Irregularity Strength of Some Classes of Grid Graphs

Let G be a simple graph. A function ϕ:V(G)→{1,2,…,k} a vertex k-labeling which assigns labels to the vertices of G. For any edge xy in G, we define the weight of this edge as wϕ(xy)=ϕ(x)+ϕ(y). If all the edge weights are distinct, then ϕ is termed as an edge irregular k -labeling of G. The smallest possible value of k for which the graph G possesses an edge irregular k-labeling is denoted as the edge irregularity strength of G and is represented as es(G). In this paper, we investigate the edge irregular k-labeling of some classes of grid graphs, namely rhombic graph Rmn, triangular graph Lmn and octagonal graph Omn. As by-product, we obtain their precise value of edge irregularity strength.

Edge open packing: Complexity, algorithmic aspects, and bounds

Given a graph G, two edges e1,e2∈E(G) are said to have a common edge e≠e1,e2 if e joins an endvertex of e1 to an endvertex of e2. A subset B⊆E(G) is an edge open packing set in G if no two edges of B have a common edge in G, and the maximum cardinality of such a set in G is called the edge open packing number, ρeo(G), of G. In this paper, we prove that the decision version of the edge open packing number is NP-complete even when restricted to graphs with universal vertices, Eulerian bipartite graphs, and planar graphs with maximum degree 4, respectively. In contrast, we present a linear-time algorithm that computes the edge open packing number of a tree. We also resolve two problems posed in the seminal paper (Chelladurai et al. (2022) [5]). Notably, we characterize the graphs G that attain the upper bound ρeo(G)≤|E(G)|/δ(G), and provide lower and upper bounds for the edge-deleted subgraph of a graph and establish the corresponding realization result.

Dimer piling problems and interacting field theory

The dimer tiling problem asks in how many ways can the edges of a graph be covered by dimers so that each site is covered once. In the special case of a planar graph, this problem has a solution in terms of a free fermionic field theory. We rediscover and explore an expression for the number of coverings of an arbitrary graph by arbitrary objects in terms of an interacting fermionic field theory first proposed by Samuel. Generalizations of the dimer tiling problem, which we call “dimer piling problems,” demand that each site be covered N times by indistinguishable dimers. Our field theory provides a solution of these problems in the large-N limit. We give a similar path integral representation for certain lattice coloring problems. Published by the American Physical Society 2024

L(3,2,1)-labeling of certain planar graphs

Given a graph G=(V,E) of maximum degree Δ, denoting by d(x,y) the distance in G between nodes x,y∈V, an L(3,2,1)-labeling of G is an assignment l from V to the set of non-negative integers such that |l(x)−l(y)|≥3 if x and y are adjacent, |l(x)−l(y)|≥2 if d(x,y)=2, and |l(x)−l(y)|≥1 if d(x,y)=3, for all x and y in V. The L(3,2,1)-number λ(G) is the smallest positive integer such that G admits an L(3,2,1)-labeling with labels from {0,1,…,λ(G)}.In this paper, the L(3,2,1)-number of certain planar graphs is determined, proving that it is linear in Δ, although the general upper bound for the L(3,2,1)-number of planar graphs is quadratic in Δ.

Approximation schemes for Min-Sum [formula omitted]-Clustering

We consider the Min-Sum k-Clustering (k-MSC) problem. Given a set of points in a metric which is represented by an edge-weighted graph G=(V,E) and a parameter k, the goal is to partition the points V into k clusters such that the sum of distances between all pairs of the points within the same cluster is minimized.The k-MSC problem is known to be APX-hard on general metrics. The best known approximation algorithms for the problem obtained by Behsaz et al. (2019) achieve an approximation ratio of O(log|V|) in polynomial time for general metrics and an approximation ratio 2+ϵ in quasi-polynomial time for metrics with bounded doubling dimension. No approximation schemes for k-MSC (when k is part of the input) is known for any non-trivial metrics prior to our work. In fact, most of the previous works rely on the simple fact that there is a 2-approximate reduction from k-MSC to the balanced k-median problem and design approximation algorithms for the latter to obtain an approximation for k-MSC.In this paper, we obtain the first Quasi-Polynomial Time Approximation Schemes (QPTAS) for the problem on metrics induced by graphs of bounded treewidth, graphs of bounded highway dimension, graphs of bounded doubling dimensions (including fixed dimensional Euclidean metrics), and planar and minor-free graphs. We bypass the barrier of 2 for k-MSC by introducing a new clustering problem, which we call min-hub clustering, which is a generalization of balanced k-median and is a trade off between center-based clustering problems (such as balanced k-median) and pair-wise clustering (such as Min-Sum k-clustering). We then show how one can find approximation schemes for Min-hub clustering on certain classes of metrics.