Planar Graphs Research Articles

Small Planar Hypohamiltonian Graphs

ABSTRACTA graph is hypohamiltonian if it is non‐hamiltonian, but the deletion of every single vertex gives a Hamiltonian graph. Until now, the smallest known planar hypohamiltonian graph had 40 vertices, a result due to Jooyandeh, McKay, Östergård, Pettersson, and Zamfirescu. That result is here improved upon by two planar hypohamiltonian graphs on 34 vertices. We exploited a special subgraph contained in two graphs of Jooyandeh et al., and modified it to construct the two 34‐vertex graphs and six planar hypohamiltonian graphs on 37 vertices. Each of the 34‐vertex graphs has 26 cubic vertices, improving upon the result of Jooyandeh et al. that planar hypohamiltonian graphs have 30 cubic vertices. We use the 34‐vertex graphs to construct hypohamiltonian graphs of order 34 with crossing number 1, improving the best‐known bound of 36 due to Wiener. Whether there exists a planar hypohamiltonian graph on 41 vertices was an open question. We settled this question by applying an operation introduced by Thomassen to the 37‐vertex graphs to obtain several planar hypohamiltonian graphs on 41 vertices. The 25 planar hypohamiltonian graphs on 40 vertices of Jooyandeh et al. have no nontrivial automorphisms. The result is here improved upon by six planar hypohamiltonian graphs on 40 vertices with nontrivial automorphisms.

Creating Entangled Logical Qubits in the Heavy-Hex Lattice with Topological Codes

Designs for quantum error correction depend strongly on the connectivity of the qubits. For solid-state qubits, the most straightforward approach is to have connectivity constrained to a planar graph. Practical considerations may also further restrict the connectivity, resulting in a relatively sparse graph such as the heavy-hexagonal (“heavy-hex”) architecture of current IBM Quantum devices. In such cases, it is hard to use all qubits to their full potential. Instead, in order to emulate the denser connectivity required to implement well-known quantum error-correcting codes, many qubits remain effectively unused. In this work, we show how this bug can be turned into a feature. By using the unused qubits of one code to execute another, two codes can be implemented on top of each other, allowing easy application of fault-tolerant entangling gates and measurements. We demonstrate this by realizing a surface code and a Bacon-Shor code on a 133-qubit IBM Quantum device. Using transversal controlled-X () gates and lattice surgery, we demonstrate entanglement between these logical qubits with code distance up to d=4 and five rounds of stabilizer-measurement cycles. The nonplanar coupling between the qubits allows us to simultaneously measure the logical XX, YY, and ZZ observables. With this, we verify the violation of Bell’s inequality for both the d=2 case with postselection featuring a fidelity of 94% and the d=3 instance using only quantum error correction. Published by the American Physical Society 2024

On Layered Area-Proportional Rectangle Contact Representations

Semantic word clouds visualize the semantic relatedness between the words of a text by placing pairs of related words close to each other. Formally, the problem of drawing semantic word clouds corresponds to drawing a rectangle contact representation of a graph whose vertices correlate to the words to be displayed and whose edges indicate that two words are semantically related. The goal is to maximize the number of realized contacts while avoiding any false adjacencies. We consider a variant of this problem that restricts input graphs to be layered and all rectangles to be of equal height, called Maximum Layered Contact Representation Of Word Networks or Max-LayeredCrown, as well as the variant Max-IntLayeredCrown, which restricts the problem to only rectangles of integer width and the placement of those rectangles to integer coordinates.We classify the corresponding decision problem k-IntLayeredCrown as NP-complete even for internally triangulated planar graphs and k-LayeredCrown as NP-complete for planar graphs. We introduce three algorithms: a 1/2-approximation for Max-LayeredCrown of internally triangulated planar graphs, and a PTAS and an XP algorithm for Max-IntLayeredCrown with rectangle width polynomial in n.

Complexity issues concerning the quadruple Roman domination problem in graphs

Given a graph G with vertex set V(G), a mapping h:V(G)→{0,1,2,3,4,5} is called a quadruple Roman dominating function (4RDF) for G if it holds the following. Every vertex x such that h(x)∈{0,1,2,3} satisfies that h(N[x])=∑v∈N[x]h(v)≥|{y:y∈N(x)andh(y)≠0}|+4, where N(x) and N[x] stands for the open and closed neighborhood of x, respectively. The smallest possible weight ∑x∈V(G)h(x) among all possible 4RDFs h for G is the quadruple Roman domination number of G, denoted by γ[4R](G).This work is focused on complexity aspects for the problem of computing the value of this parameter for several graph classes. Specifically, it is shown that the decision problem concerning γ[4R](G) is NP-complete when restricted to star convex bipartite, comb convex bipartite, split and planar graphs. In contrast, it is also proved that such problem can be efficiently solved for threshold graphs where an exact solution is demonstrated, while for graphs having an efficient dominating set, tight upper and lower bounds in terms of the classical domination number are given. In addition, some approximation results to the problem are given. That is, we show that the problem cannot be approximated within (1−ϵ)ln⁡|V| for any ϵ>0 unless P=NP. An approximation algorithm for it is proposed, and its APX-completeness proved, whether graphs of maximum degree four are considered. Finally, an integer linear programming formulation for our problem is presented.

Powers of planar graphs, product structure, and blocking partitions

Powers of planar graphs, product structure, and blocking partitions

Statistical Physics and Random Surfaces

This conference featured a diverse group of participants, from various career stages, to study problems in several hot topics that have grown increasingly prominent and interrelated in recent years. These included the following: (1) random surface models (Liouville quantum gravity, random planar maps, etc.) (2) random curve models (Schramm–Loewner evolution, conformal loop ensembles, etc.) (3) gauge theory models (various forms of lattice Yang–Mills, other approximations) (4) spin models and height functions (including delocalization problems) (5) dimer models (two and higher dimensions versions) (6) conformal field theory (Liouville theory, other theories related to SLE) The conference enabled participants to communicate about the most recent break-throughs and lay the groundwork for further collaborative progress.

Surface Reconstruction Using Rotation Systems

Inspired by the seminal result that a graph and an associated rotation system uniquely determine the topology of a closed manifold, we propose a combinatorial method for reconstruction of surfaces from points. Our method constructs a spanning tree and a rotation system. Since the tree is trivially a planar graph, its rotation system determines a genus zero surface with a single face which we proceed to incrementally refine by inserting edges to split faces. In order to raise the genus, special handles are added in a later stage by inserting edges between different faces and thus merging them. We apply our method to a wide range of input point clouds in order to investigate its effectiveness, and we compare our method to several other surface reconstruction methods. It turns out that our approach has two specific benefits over these other methods. First, the output mesh preserves the most information from the input point cloud. Second, our method provides control over the topology of the reconstructed surface. Code is available on https://github.com/cuirq3/RsR.

On MAX–SAT with cardinality constraint

We consider the weighted MAX–SAT problem with an additional constraint that at mostk variables can be set to true. We call this problem k–WMAX–SAT. This problem admits a (1−1e)-factor approximation algorithm in polynomial time [Sviridenko, Algorithmica (2001)] and it is proved that there is no (1−1e+ϵ)-factor approximation algorithm in f(k)⋅no(k) time for Maximum Coverage, the unweighted monotone version of k–WMAX–SAT [Manurangsi, SODA 2020]. Therefore, we study two restricted versions of the problem in the realm of parameterized complexity.1.When the input is an unweighted 2–CNF formula (the problem is called k–MAX–2SAT), we design an efficient polynomial-size approximate kernelization scheme. That is, we design a polynomial-time algorithm that given a 2–CNF formula ψ and ϵ>0, compresses the input instance to a 2–CNF formula ψ⋆ such that any c-approximate solution of ψ⋆ can be converted to a c(1−ϵ)-approximate solution of ψ in polynomial time.2.When the input is a planar CNF formula, i.e., the variable-clause incidence graph is a planar graph, we show the following results:•There is an FPT algorithm for k–WMAX–SAT on planar CNF formulas that runs in 2O(k)⋅(C+V) time.•There is a polynomial-time approximation scheme for k–WMAX–SAT on planar CNF formulas that runs in time 2O(1ϵ)⋅k⋅(C+V). The above-mentioned C and V are the number of clauses and variables of the input formula respectively.

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.

Product structure of graphs with an excluded minor

This paper shows that K t K_t -minor-free (and K s , t K_{s, t} -minor-free) graphs G G are subgraphs of products of a tree-like graph H H (of bounded treewidth) and a complete graph K m K_m . Our results include optimal bounds on the treewidth of H H and optimal bounds (to within a constant factor) on m m in terms of the number of vertices of G G and the treewidth of G G . These results follow from a more general theorem whose corollaries include a strengthening of the celebrated separator theorem of Alon, Seymour, and Thomas [J. Amer. Math. Soc. 3 (1990), 801–808] and the Planar Graph Product Structure Theorem of Dujmović et al. [J. ACM 67 (2020)].

Variable Degeneracy of Planar Graphs without Chorded 6-Cycles

Variable Degeneracy of Planar Graphs without Chorded 6-Cycles

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.

2-Distance (Δ + 1)-coloring of sparse graphs using the potential method

A 2-distance k-coloring of a graph is a proper k-coloring of the vertices where vertices at distance at most 2 cannot share the same color. We prove the existence of a 2-distance (Δ+1)-coloring for graphs with maximum average degree less than 187 and maximum degree Δ≥7. As a corollary, every planar graph with girth at least 9 and Δ≥7 admits a 2-distance (Δ+1)-coloring. The proof uses the potential method to reduce new configurations compared to classic approaches on 2-distance coloring.

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.