Planar Graphs Research Articles

Last car decomposition of planar maps

We give new equations which characterize the generating functions of planar quadrangulations and planar triangulations, with zero, one, or two boundaries. The proof is inspired by the Lackner–Panholzer last car decomposition of parking trees and consists in applying a similar decomposition to the peeling trees of planar maps.

Kramers-Wannier Duality and Random-Bond Ising Model.

We present a new combinatorial approach to the Ising model incorporating arbitrary bond weights on planar graphs. In contrast to existing methodologies, the exact free energy is expressed as the determinant of a set of ordered and disordered operators defined on a planar graph and the corresponding dual graph, respectively, thereby explicitly demonstrating the Kramers-Wannier duality. The implications of our derived formula for the Random-Bond Ising Model are further elucidated.

Planar Turán Number of the Θ6

Let F be a nonempty family of graphs. A graph 𝐺 is called F -free if it contains no graph from F as a subgraph. For a positive integer 𝑛, the planar Turán number of F, denoted by exp (𝑛, F), is the maximum number of edges in an 𝑛-vertex F -free planar graph.Let Θ𝑘 be the family of Theta graphs on 𝑘 ≥ 4 vertices, that is, graphs obtained by joining a pair of non-consecutive of a 𝑘-cycle with an edge. Lan, Shi and Song determined an upper bound exp (𝑛, Θ6) ≤ 18𝑛/7−36𝑛/7, but for large 𝑛, they did not verify that the bound is sharp. In this paper, we improve their bound by proving exp (𝑛, Θ6) ≤ 18𝑛/−48𝑛/7 and then we demonstrate the existence of infinitely many positive integer 𝑛 and an 𝑛-vertex Θ6-free planar graph attaining the bound.

How to reconstruct a planar map from its branching curve

We present an algorithm for constructing a map P 2 → P 2 \mathbb {P}^2\to \mathbb {P}^2 with a given branching curve. The stepping stone is the ramification curve, which is obtained as the linear normalisation of the branching curve.

A Whitney Type Theorem for Surfaces: Characterising Graphs with Locally Planar Embeddings

AbstractGiven a graph G and a parameter r, we define the r-local matroid of G to be the matroid generated by its cycles of length at most r. Extending Whitney’s abstract planar duality theorem from 1932, we prove that for every r the r-local matroid of G is co-graphic if and only if G admits a certain type of embedding in a surface, which we call r-planar embedding. The maximum value of r such that a graph G admits an r-planar embedding is closely related to face-width, and such embeddings for this maximum value of r are quite often embeddings of minimum genus. Unlike minimum genus embeddings, these r-planar embeddings can be computed in polynomial time. This provides the first systematic and polynomially computable method to construct for every graph G a surface so that G embeds in that surface in an optimal way (phrased in our notion of r-planarity).

Cycles in 3‐connected claw‐free planar graphs and 4‐connected planar graphs without 4‐cycles

AbstractThe cycle spectrum of a graph is the set of the cycle lengths in . Let be a graph class. For any integer , define to be the least integer such that for any with circumference at least . Denote by , and the classes of 3‐connected planar graphs, 3‐connected cubic planar graphs, 3‐connected claw‐free planar graphs, and 3‐connected claw‐free cubic planar graphs, respectively. The values of and were known for all . In the first part of this article, we prove the claw‐free version of these results by giving the values of and for all . In the second part we study the cycle spectra of 4‐connected planar graphs without 4‐cycles. Bondy conjectured that every 4‐connected planar graph has all cycle lengths from 3 to except possibly one even length. The truth of this conjecture would imply that every 4‐connected planar graph without 4‐cycles has all cycle lengths other than 4. It was already known that . We prove that can be embedded in the plane such that for any has a ‐cycle and all vertices not in lie in the exterior of .

Defective acyclic colorings of planar graphs

AbstractThis paper studies two variants of defective acyclic coloring of planar graphs. For a graph and a coloring of , a 2‐colored cycle (2CC) transversal is a subset of that intersects every 2‐colored cycle. Let be a positive integer. We denote by the minimum integer such that has a proper ‐coloring which has a 2CC transversal of size , and by the minimum size of a subset of such that is acyclic ‐colorable. We prove that for any ‐vertex 3‐colorable planar graph and for any planar graph provided that . We show that these upper bounds are sharp: there are infinitely many planar graphs attaining these upper bounds. Moreover, the minimum 2CC transversal can be chosen in such a way that induces a forest. We also prove that for any planar graph and .

Local certification of graph decompositions and applications to minor-free classes

Local certification consists in assigning labels to the vertices of a network to certify that some given property is satisfied, in such a way that the labels can be checked locally. In the last few years, certification of graph classes received considerable attention. The goal is to certify that a graph G belongs to a given graph class G. Such certifications with labels of size O(log⁡n) (where n is the size of the network) exist for trees, planar graphs and graphs embedded on surfaces. Feuilloley et al. ask if this can be extended to any class of graphs defined by a finite set of forbidden minors.In this work, we develop new decomposition tools for graph certification, and apply them to show that for every small enough minor H, H-minor-free graphs can indeed be certified with labels of size O(log⁡n). We also show matching lower bounds using a new proof technique.

Approximation algorithms for maximum weighted internal spanning trees in regular graphs and subdivisions of graphs

Abstract Let $G$ be a vertex-weighted connected graph of $n$ vertices and let $T$ be a spanning tree of $G$. We call $T$ a maximum weighted internal spanning tree of $G$ if the sum of the weights of the internal vertices of $T$ is the maximum over all spanning trees of $G$. The maximum weighted internal spanning tree (MaxwIST) problem asks to find such a spanning tree $T$ of $G$. The problem is NP-hard. We give an $O(dn)$ time approximation algorithm for $d$-regular graphs of $n=|V|$ vertices that computes a spanning tree with total weight of the internal vertices is at least $\frac{\beta _{d}}{\beta _{d} +d-2} - \epsilon $ of the total weight of all the vertices of the graph for any $\epsilon>0$, where $\beta _{d} = (d-1)H_{d-1}$, and $H_{d-1} = \sum _{i=1}^{d-1} i^{-1}$ is the $(d-1)$th harmonic number. For every $d \geq 3$ and $n_{0} \geq 1$, we show the construction of a $d$-regular graph of at least $n_{0}$ vertices, such that for any of its spanning trees, $\frac{w(I)}{w(V)}\le \frac{d}{d+1}$ holds. We give an $O(dn)$ time approximation algorithm for subdivisions of $d$-regular graphs, where the ratio of the internal weight of the spanning tree with the total vertex weight of the graph is at least $\frac{d-1}{2d-3} - \epsilon $ for $\epsilon>0$. We extend our study to $x$-subdivisions of Hamiltonian and hypoHamiltonian graphs, where each edge of the original Hamiltonian or hypoHamiltonian graph has been subdivided at least $x$ times. For those two graph classes, we show that there exists a spanning tree with internal vertex weight at least $1-\frac{2}{x-1}$ of the total vertex weight of the graph. Furthermore, we give $O(n)$ time algorithm for $x$-subdivisions of biconnected outerplanar graphs and $4$-connected planar graphs to achieve the above bound.

Face irregular evaluations of family of grids

Let G = ( V , E , F ) be a connected plane graph. Let ρ : α ( V ) ∪ β ( E ) ∪ γ ( F ) → { 1 , 2 , … , k } be a k-labeling where α , β , γ ∈ { 0 , 1 } and ( α , β , γ ) ≠ ( 0 , 0 , 0 ) . The weight of a face f under ρ is defined as Wt ( f ) = α ∑ v ∼ f ρ ( v ) + β ∑ e ∼ f ρ ( e ) + γ ρ ( f ) . Then ρ is called a face irregular labeling of type ( α , β , γ ) if Wt ( f ) ≠ Wt ( g ) for any two faces f ≠ g . The face irregular strength of G is the smallest integer k such that G admits a face irreg ular k-labeling. In this paper we determine the face irregular strength of the grid graph P n + 1 □ P m + 1 under a labeling of type ( α , β , γ ) .

A graph theoretic approach for generating T-shaped floor plans

A non-rectangular floor plan (NRFP) is one with a rectilinear exterior boundary containing rectangular modules. An NRFP is identified as a T-shaped floor plan if the rectilinear exterior boundary forms a T-shape with two concave corners. A T-shaped floor plan is further classified as aligned or non-aligned based on the alignment of its two concave corners. In this work, we aim to investigate graph-theoretic characteristics of properly triangulated plane graphs (PTPGs) for the existence of corresponding aligned and non-aligned T-shaped floor plans. Also, we provide an O(n3) algorithm for generating T-shaped floor plans (both aligned and non-aligned) for any PTPG with six or fewer corner-implying paths (CIPs), if they exist. Moreover, we claim that the resultant T-shaped floor plans are non-trivial. A T-shaped floor plan is considered non-trivial if the count of concave corners (two) at its exterior boundary can only be lowered by disrupting the modules' horizontal and vertical adjacencies.

True Contraction Decomposition and Almost ETH-Tight Bipartization for Unit-Disk Graphs

We prove a structural theorem for unit-disk graphs, which (roughly) states that given a set \(\mathcal{D}\) of \(n\) unit disks inducing a unit-disk graph \(G_{\mathcal{D}}\) and a number \(p\in[n]\) , one can partition \(\mathcal{D}\) into \(p\) subsets \(\mathcal{D}_{1},\dots,\mathcal{D}_{p}\) such that for every \(i\in[p]\) and every \(\mathcal{D}^{\prime}\subseteq\mathcal{D}_{i}\) , the graph obtained from \(G_{\mathcal{D}}\) by contracting all edges between the vertices in \(\mathcal{D}_{i}\backslash\mathcal{D}^{\prime}\) admits a tree decomposition in which each bag consists of \(O(p+|\mathcal{D}^{\prime}|)\) cliques. Our theorem can be viewed as an analog for unit-disk graphs of the structural theorems for planar graphs and almost-embeddable graphs proved recently by Marx et al. [SODA ’22] and Bandyapadhyay et al. [SODA ’22]. By applying our structural theorem, we give several new combinatorial and algorithmic results for unit-disk graphs. On the combinatorial side, we obtain the first Contraction Decomposition Theorem for unit-disk graphs, resolving an open question in the work by Panolan et al. [SODA ’19]. On the algorithmic side, we obtain a new algorithm for bipartization (also known as odd cycle transversal) on unit-disk graphs, which runs in \(2^{O(\sqrt{k}\log k)}\cdot n^{O(1)}\) time, where \(k\) denotes the solution size. Our algorithm significantly improves the previous slightly subexponential-time algorithm given by Lokshtanov et al. [SODA ’22] which runs in \(2^{O(k^{27/28})}\cdot n^{O(1)}\) time. We also show that the problem cannot be solved in \(2^{o(\sqrt{k})}\cdot n^{O(1)}\) time assuming the Exponential Time Hypothesis, which implies that our algorithm is almost optimal.

Twin-width of planar graphs; a short proof

The fascinating question of the maximum value of twin-width on planar graphs is nowadays not far from the final resolution; there is a lower bound of 7 coming from a construction by Král’ and Lamaison (2022), and an upper bound of 8 by Hliněný and Jedelský (2022). The upper bound (currently best) of 8, however, is rather complicated and involved. In the paper we give a short and simple self-contained proof that the twin-width of planar graphs is at most 11. We believe that this short proof can also shed more light on the topic of upper bound(s) on the twin-width of planar and beyond-planar graphs in general.

Embedding of Neutrosophic Graphs on Topological Surfaces

Background: A planar graph (PG) is a graph with no intersecting edges. Particular to both crisp and neutrosophic graphs (NG) is the planar graph, in contrast to crisp planar graphs. NPGs allow for the intersection of neutrosophic edge NEs, since the value of planarity in these graphs is the degree of planarity of the intersected NEs. The NPGs are often represented on a flat surface. Materials and Methods: This study discusses how to embed NGs on surfaces such as spheres and m-toruses by defining the degree of intersection of the neutrosophic edges of NGs with finding the faces on the given graph structures using Euler's theorems. Here, the proofs of Euler's theorems help us find, given the total NFV of G, the interval containing that value. Result: As result of this work obtained that for any two isomorphic planer graphs, they have the same planarity value. For any neutrosophic planer graph with f = (1,1,1) can be embedded in the plane if it can be embedded in the sphere and according to NPGs, for planar and spherical surfaces, equivalent theorems to Euler's formula are proved and shown. Conclusion: It concludes that by using neutrosophic sets and crisp graphs to construct neutrosophic graphs with the benefit of Euler’s theorem, it can provide the concept of embedding neutrosophic graphs in different topological surfaces such as a plane, sphere, and m-torus.

OpenChemIE: An Information Extraction Toolkit for Chemistry Literature.

Information extraction from chemistry literature is vital for constructing up-to-date reaction databases for data-driven chemistry. Complete extraction requires combining information across text, tables, and figures, whereas prior work has mainly investigated extracting reactions from single modalities. In this paper, we present OpenChemIE to address this complex challenge and enable the extraction of reaction data at the document level. OpenChemIE approaches the problem in two steps: extracting relevant information from individual modalities and then integrating the results to obtain a final list of reactions. For the first step, we employ specialized neural models that each address a specific task for chemistry information extraction, such as parsing molecules or reactions from text or figures. We then integrate the information from these modules using chemistry-informed algorithms, allowing for the extraction of fine-grained reaction data from reaction condition and substrate scope investigations. Our machine learning models attain state-of-the-art performance when evaluated individually, and we meticulously annotate a challenging dataset of reaction schemes with R-groups to evaluate our pipeline as a whole, achieving an F1 score of 69.5%. Additionally, the reaction extraction results of OpenChemIE attain an accuracy score of 64.3% when directly compared against the Reaxys chemical database. OpenChemIE is most suited for information extraction on organic chemistry literature, where molecules are generally depicted as planar graphs or written in text and can be consolidated into a SMILES format. We provide OpenChemIE freely to the public as an open-source package, as well as through a web interface.

On quasi-polynomials counting planar tight maps

On quasi-polynomials counting planar tight maps

Some Convex Polytopes Joining Paths and Their Metric Dimension

In recent years, there is a lot of interest in the topic of conveying the groups of planar graphs with an unvarying metric dimension. A few types of planar graphs have recently had their locating number (or metric dimension) determined, and an outstanding problem concerning these graphs was brought up that: Illustrate the types of planar graphs Υ that can be generated from a graph Φ through the addition of more edges to Φ , such that d i m ( Φ ) = d i m ( Υ ) and V ( Φ ) = V ( Υ ) . While proceeding in a similar directives, we identify two families of radially identical planar graphs with unaltered metric dimension in this study: ϝ n , m and ℷ n , m . We do this by establishing that d i m ( ϝ n , m ) = d i m ( ℷ n , m ) and V ( ϝ n , m ) = V ( ℷ n , m ) , respectively. We acquire another family of a radially symmetrical plane graph (i.e., ℸ n , m ) with a constant metric dimension. We show that all the vertices of these classes of the plane graphs can potentially be identified with just three well-chosen nodes.

Biequivalent Planar Graphs

We define biequivalent planar graphs, which are a generalisation of the uniform polyhedron graphs, as planar graphs made out of two families of equivalent nodes. Such graphs are required to identify polyhedral cages with geometries suitable for artificial protein cages. We use an algebraic method, which is followed by an algorithmic method, to determine all such graphs with up to 300 nodes each with valencies ranging between three and six. We also present a graphic representation of every graph found.

Inverse Fuzzy Mixed Planar Graphs with Application

Inverse Fuzzy Mixed Planar Graphs with Application

Good r-divisions Imply Optimal Amortized Decremental Biconnectivity

We present a data structure that, given a graph G of n vertices and m edges, and a suitable pair of nested r-divisions of G, preprocesses G in O(m+n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(m+n)$$\\end{document} time and handles any series of edge-deletions in O(m) total time while answering queries to pairwise biconnectivity in worst-case O(1) time. In case the vertices are not biconnected, the data structure can return a cutvertex separating them in worst-case O(1) time. As an immediate consequence, this gives optimal amortized decremental biconnectivity, 2-edge connectivity, and connectivity for large classes of graphs, including planar graphs and other minor free graphs.