Abstract: This paper proposes a novel message passing neural (MPN) architecture Conv-MPN, which reconstructs an outdoor building as a planar graph from a single RGB image. Conv-MPN is specifically designed for cases where nodes of a graph have explicit spatial embedding. In our problem, nodes correspond to building edges in an image. Conv-MPN is different from MPN in that 1) the feature associated with a node is represented as a feature volume instead of a 1D vector; and 2) convolutions encode messages instead of fully connected layers. Conv-MPN learns to select a true subset of nodes (i.e., building edges) to reconstruct a building planar graph. Our qualitative and quantitative evaluations over 2,000 buildings show that Conv-MPN makes significant improvements over the existing fully neural solutions. We believe that the paper has a potential to open a new line of graph neural network research for structured geometry reconstruction.

Abstract: We introduce a general technique for proving estimates for certain random planar maps which belong to the $$\gamma $$ -Liouville quantum gravity (LQG) universality class for $$\gamma \in (0,2)$$ . The family of random planar maps we consider are those which can be encoded by a two-dimensional random walk with i.i.d. increments via a mating-of-trees bijection, and includes the uniform infinite planar triangulation (UIPT; $$\gamma =\sqrt{8/3}$$ ); and planar maps weighted by the number of different spanning trees ( $$\gamma =\sqrt{2}$$ ), bipolar orientations ( $$\gamma =\sqrt{4/3}$$ ), or Schnyder woods ( $$\gamma =1$$ ) that can be put on the map. Using our technique, we prove estimates for graph distances in the above family of random planar maps. In particular, we obtain non-trivial upper and lower bounds for the cardinality of a graph distance ball consistent with the Watabiki (Prog Theor Phys Suppl 114:1–17, 1993) prediction for the Hausdorff dimension of $$\gamma $$ -LQG and we establish the existence of an exponent for certain distances in the map. The basic idea of our approach is to compare a given random planar map M to a mated-CRT map—a random planar map constructed from a correlated two-dimensional Brownian motion—using a strong coupling (Zaitsev in ESAIM Probab Stat 2:41–108, 1998) of the encoding walk for M and the Brownian motion used to construct the mated-CRT map. This allows us to deduce estimates for graph distances in M from the estimates for graph distances in the mated-CRT map which we proved (using continuum theory) in a previous work. In the special case when $$\gamma =\sqrt{8/3}$$ , we instead deduce estimates for the $$\sqrt{8/3}$$ -mated-CRT map from known results for the UIPT. The arguments of this paper do not directly use SLE/LQG, and can be read without any knowledge of these objects.

Abstract: This paper tackles a 2D architecture vectorization problem, whose task is to infer an outdoor building architecture as a 2D planar graph from a single RGB image. We provide a new benchmark with ground-truth annotations for 2,001 complex buildings across the cities of Atlanta, Paris, and Las Vegas. We also propose a novel algorithm utilizing 1) convolutional neural networks (CNNs) that detects geometric primitives and infers their relationships and 2) an integer programming (IP) that assembles the information into a 2D planar graph. While being a trivial task for human vision, the inference of a graph structure with an arbitrary topology is still an open problem for computer vision. Qualitative and quantitative evaluations demonstrate that our algorithm makes significant improvements over the current state-of-the-art, towards an intelligent system at the level of human perception. We will share code and data.

Abstract: We prove that the simple random walk on the uniform infinite planar triangulation (UIPT) typically travels graph distance at most $$n^{1/4 + o_n(1)}$$ in n units of time. Together with the complementary lower bound proven by Gwynne and Miller (2017) this shows that the typical graph distance displacement of the walk after n steps is $$n^{1/4 + o_n(1)}$$ , as conjectured by Benjamini and Curien (Geom Funct Anal 2(2):501–531, 2013. arXiv:1202.5454 ). More generally, we show that the simple random walks on a certain family of random planar maps in the $$\gamma $$ -Liouville quantum gravity (LQG) universality class for $$\gamma \in (0,2)$$ —including spanning tree-weighted maps, bipolar-oriented maps, and mated-CRT maps—typically travels graph distance $$n^{1/d_\gamma + o_n(1)}$$ in n units of time, where $$d_\gamma $$ is the growth exponent for the volume of a metric ball on the map, which was shown to exist and depend only on $$\gamma $$ by Ding and Gwynne (Commun Math Phys 374:1877–1934, 2018. arXiv:1807.01072 ). Since $$d_\gamma > 2$$ , this shows that the simple random walk on each of these maps is subdiffusive. Our proofs are based on an embedding of the random planar maps under consideration into $${\mathbb {C}}$$ wherein graph distance balls can be compared to Euclidean balls modulo subpolynomial errors. This embedding arises from a coupling of the given random planar map with a mated-CRT map together with the relationship of the latter map to SLE-decorated LQG.

Abstract: For a fixed connected graph H, the {H}-M-Deletion problem asks, given a graph G, for the minimum number of vertices that intersect all minor models of H in G. It is known that this problem can be solved in time f(tw) · nO(1), where tw is the treewidth of G. We determine the asymptotically optimal function f(tw), for each possible choice of H. Namely, we prove that, under the ETH, f(tw) = 2Θ(tw) if H is a contraction of the chair or the banner, and f(tw) = 2Θ(tw·log tw) otherwise. Prior to this work, such a complete characterization was only known when H is a planar graph with at most five vertices. For the upper bounds, we present an algorithm in time 2Θ(tw·log tw) ·nO(1) for the more general problem where all minor models of connected graphs in a finite family F need to be hit. We combine several ingredients such as the machinery of boundaried graphs in dynamic programming via representatives, the Flat Wall Theorem, Bidimensionality, the irrelevant vertex technique, treewidth modulators, and protrusion replacement. In particular, this algorithm vastly generalizes a result of Jansen et al. [SODA 2014] for the particular case F = {K5, K3,3}. For the lower bounds, our reductions are based on a generic construction building on the one given by the authors in [IPEC 2018], which uses the framework introduced by Lokshtanov et al. [SODA 2011] to obtain superexponential lower bounds.

Abstract: We study the following version of cut sparsification. Given a large edge-weighted network $G$ with $k$ terminal vertices, compress it into a smaller network $H$ with the same terminals, such that e...

Abstract: We introduce the framework of discrete holomorphic functions on t-embeddings of weighted bipartite planar graphs; t-embeddings also appeared under the name Coulomb gauges in a recent work arXiv:1810.05616. We argue that this framework is particularly relevant for the analysis of scaling limits of the height fluctuations in the corresponding dimer models. In particular, it unifies both Kenyon's interpretation of dimer observables as derivatives of harmonic functions on T-graphs and the notion of s-holomorphic functions originated in Smirnov's work on the critical Ising model. We develop an a priori regularity theory for such functions and provide a meta-theorem on convergence of the height fluctuations to the Gaussian Free Field.

Abstract: An embedding of a graph in a book consists of a linear order of its vertices along the spine of the book and of an assignment of its edges to the pages of the book, so that no two edges on the same page cross. The book thickness of a graph is the minimum number of pages over all its book embeddings. Accordingly, the book thickness of a class of graphs is the maximum book thickness over all its members. In this paper, we address a long-standing open problem regarding the exact book thickness of the class of planar graphs, which previously was known to be either three or four. We settle this problem by constructing planar graphs that require four pages in all of their book embeddings, thus establishing that the book thickness of the class of planar graphs is four.

Abstract: Given query access to an undirected graph G, we consider the problem of computing a (1 ± e)-approximation of the number of k-cliques in G. The standard query model for general graphs allows for degree queries, neighbor queries, and pair queries. Let n be the number of vertices, m be the number of edges, and nk be the number of k-cliques. Previous work by Eden, Ron and Seshadhri (STOC 2018) gives an [MATH HERE]-time algorithm for this problem (we use O*(·) to suppress poly(log n, 1/e,kk) dependencies). Moreover, this bound is nearly optimal when the expression is sublinear in the size of the graph. Our motivation is to circumvent this lower bound, by parameterizing the complexity in terms of graph arboricity. The arboricity of G is a measure for the graph density "everywhere". There is a very rich family of graphs with bounded arboricity, including all minor-closed graph classes (such as planar graphs and graphs with bounded treewidth), bounded degree graphs, preferential attachment graphs and more. We design an algorithm for the class of graphs with arboricity at most α, whose running time is [MATH HERE]. We also prove a nearly matching lower bound. For all graphs, the arboricity is [MATH HERE], so this bound subsumes all previous results on sublinear clique approximation. As a special case of interest, consider minor-closed families of graphs, which have constant arboricity. Our result implies that for any minor-closed family of graphs, there is a (1 ± e)-approximation algorithm for nk that has running time [MATH HERE]. Such a bound was not known even for the special (classic) case of triangle counting in planar graphs.

Abstract: In this paper, the picture fuzzy planar graph is defined along with some basic properties of picture fuzzy graph. The notion of picture fuzzy planar graphs is introduced and the terms such as strong (weak) edges, strong (weak) picture fuzzy planar graphs, strength of an edge, degree of planarity, picture fuzzy faces, strong (weak) picture fuzzy faces, picture fuzzy dual graphs and picture fuzzy combinatorial dual graphs are defined. Some theorems on degree of planarity and picture fuzzy combinatorial dual graphs have been established. An application of picture fuzzy planar graph in the design of road maps is given.

Abstract: For an integer $t$, a graph $G$ is called {\em{$C_{>t}$-free}} if $G$ does not contain any induced cycle on more than~$t$ vertices. We prove the following statement: for every pair of integers $d$ and $t$ and a CMSO$_2$ statement~$\phi$, there exists an algorithm that, given an $n$-vertex $C_{>t}$-free graph $G$ with weights on vertices, finds in time $n^{O(\log^4 n)}$ a maximum-weight vertex subset $S$ such that $G[S]$ has degeneracy at most $d$ and satisfies $\phi$. The running time can be improved to $n^{O(\log^2 n)}$ assuming $G$ is $P_t$-free, that is, $G$ does not contain an induced path on $t$ vertices. This expands the recent results of the authors [to appear at FOCS 2020 and SOSA 2021] on the {\sc{Maximum Weight Independent Set}} problem on $P_t$-free graphs in two directions: by encompassing the more general setting of $C_{>t}$-free graphs, and by being applicable to a much wider variety of problems, such as {\sc{Maximum Weight Induced Forest}} or {\sc{Maximum Weight Induced Planar Graph}}.

Abstract: In the Disjoint Paths problem, the input is an undirected graph G on n vertices and a set of k vertex pairs, {s i ,t i } i=1 k , and the task is to find k pairwise vertex-disjoint paths such that the i’th path connects s i to t i . In this paper, we give a parameterized algorithm with running time 2 O(k 2) n O(1) for Planar Disjoint Paths, the variant of the problem where the input graph is required to be planar. Our algorithm is based on the unique linkage/treewidth reduction theorem for planar graphs by Adler et al. [JCTB 2017], the algebraic co-homology based technique developed by Schrijver [SICOMP 1994] for Disjoint Paths on directed planar graphs, and one of the key combinatorial insights developed by Cygan et al. [FOCS 2013] in their algorithm for Disjoint Paths on directed planar graphs. To the best of our knowledge our algorithm is the first parameterized algorithm to exploit that the treewidth of the input graph is small in a way completely different from the use of dynamic programming.

Abstract: We show that for every integer $n\geq 1$ there exists a graph $G_n$ with $n^{1 + o(1)}$ edges such that every $n$-vertex planar graph is isomorphic to a subgraph of $G_n$. The best previous bound on the number of edges was $O(n^{3/2})$, proved by Babai, Erd\H{o}s, Chung, Graham, and Spencer in 1982.

Abstract: There are many representations of planar graphs but few are as elegant as Turan’s (1984): it is simple and practical, uses only four bits per edge, can handle multi-edges and can store any specified embedding. Its main disadvantage has been that “it does not allow efficient searching” (Jacobson, 1989). In this paper we show how to add a sublinear number of bits to Turan’s representation such that it supports fast navigation, thus overcoming this disadvantage. Other data structures for planar embeddings may be asymptotically faster or smaller but ours is simpler, and that can be a theoretical as well as a practical advantage: e.g., we show how our structure can be built efficiently in parallel.

Abstract: It was recently proved that every planar graph is a subgraph of the strong product of a path and a graph with bounded treewidth. This paper surveys generalisations of this result for graphs on surfaces, minor-closed classes, various non-minor-closed classes, and graph classes with polynomial growth. We then explore how graph product structure might be applicable to more broadly defined graph classes. In particular, we characterise when a graph class defined by a cartesian or strong product has bounded or polynomial expansion. We then explore graph product structure theorems for various geometrically defined graph classes, and present several open problems.

Abstract: Naor, Parter, and Yogev [SODA 2020] recently designed a compiler for automatically translating standard centralized interactive protocols to distributed interactive protocols, as introduced by Kol, Oshman, and Saxena [PODC 2018]. In particular, by using this compiler, every linear-time algorithm for deciding the membership to some fixed graph class can be translated into a $\mathsf{dMAM}(O(\log n))$ protocol for this class, that is, a distributed interactive protocol with $O(\log n)$-bit proof size in $n$-node graphs, and three interactions between the (centralizer) computationally-unbounded but non-trustable prover Merlin, and the (decentralized) randomized computationally-limited verifier Arthur. As a corollary, there is a $\mathsf{dMAM}(O(\log n))$ protocol for the class of planar graphs, as well as for the class of graphs with bounded genus. We show that there exists a distributed interactive protocol for the class of graphs with bounded genus performing just a single interaction, from the prover to the verifier, yet preserving proof size of $O(\log n)$ bits. This result also holds for the class of graphs with bounded demi-genus, that is, graphs that can be embedded on a non-orientable surface of bounded genus. The interactive protocols described in this paper are actually proof-labeling schemes, i.e., a subclass of interactive protocols, previously introduced by Korman, Kutten, and Peleg [PODC 2005]. In particular, these schemes do not require any randomization from the verifier, and the proofs may often be computed a priori, at low cost, by the nodes themselves. Our results thus extend the recent proof-labeling scheme for planar graphs by Feuilloley et al. [PODC 2020], to graphs of bounded genus, and to graphs of bounded demigenus.

Abstract: This paper addresses a long standing, widely studied, open question: Given a planar 3-graph G (i.e., a planar graph with vertex degree at most three), what is the best computational upper bound to compute a bend-minimum planar orthogonal drawing of G in the variable embedding setting? In this setting the algorithm can choose among the exponentially many planar embeddings of G the one that leads to an orthogonal drawing with the minimum number of bends. We answer the question by describing a linear-time algorithm that computes a bend-minimum planar orthogonal drawing of G. Also, if G is not K4, the drawing has at most one bend per edge. The existence of an orthogonal drawing Γ of a planar 3-graph such that Γ has the minimum number of bends and at most one bend per edge was previously unknown.

Abstract: We survey results at the intersection of topological graph theory and the game of Cops and Robbers, focusing on results, conjectures, and open problems for the cop number of a graph embedded on a surface. After a discussion on results for planar graphs, we consider graphs of higher genus. In 2001, Schroeder conjectured that if a graph has genus $g,$ then its cop number is at most $g + 3.$ While Schroeder's bound is known to hold for planar and toroidal graphs, the case for graphs with higher genus remains open. We consider the capture time of graphs on surfaces and examine results for embeddings of graphs on non-orientable surfaces. We present a conjecture by the second author, and in addition, we survey results for the lazy cop number, directed graphs, and Zombies and Survivors.

Abstract: In this research work, the notion of complex Pythagorean fuzzy planar graph (CPFPG), an extension of Pythagorean fuzzy planar graph, is presented to study the planarity. The planarity of these graphs is based on the extended range of degree from real to complex plane with unit circle. Here, the ideas of complex Pythagorean fuzzy multigraphs (CPFMGs), complex Pythagorean fuzzy planar graphs (CPFPGs) and some distinguished aspects of these graphs are presented by inspecting the complex Pythagorean fuzzy planarity value using weak and strong edges. A close relation is established between CPFPGs and dual graphs. Discussion about the non-planarity of graphs and the concepts of co-weak isomorphism, isomorphism and weak isomorphism for CPFPGs are also added. Furthermore, an application based on the proposed idea is presented to illustrate the efficiency of given model. The comparison study is also given to validate the consistency and superiority of our model.

Abstract: Is perfect matching in NC? That is, is there a deterministic fast parallel algorithm for it? This has been an outstanding open question in theoretical computer science for over three decades, ever since the discovery of RNC perfect matching algorithms. Within this question, the case of planar graphs has remained an enigma: On the one hand, counting the number of perfect matchings is far harder than finding one (the former is #P-complete and the latter is in P), and on the other, for planar graphs, counting has long been known to be in NC whereas finding one has resisted a solution. In this article, we give an NC algorithm for finding a perfect matching in a planar graph. Our algorithm uses the above-stated fact about counting perfect matchings in a crucial way. Our main new idea is an NC algorithm for finding a face of the perfect matching polytope at which a set (which could be polynomially large) of conditions, involving constraints of the polytope, are simultaneously satisfied. Several other ideas are also needed, such as finding, in NC, a point in the interior of the minimum-weight face of this polytope and finding a balanced tight odd set.

Abstract: More than half a century after its first formulation, the Weisfeiler-Leman (WL) algorithm is still an important combinatorial technique whenever graphs or other relational structures are to be classified. However, despite its simple algebraic description and its variety of applications, we still lack a precise understanding of the expressive power of the algorithm.This column introduces the reader to the basic concepts of the WL algorithm and discusses its dimension as a parameter to capture the structural complexity of an input graph. Specifically, I present a survey of work regarding the WL dimension conducted with my co-authors. First, I outline the proof that the 3-dimensional WL algorithm (3-WL) is able to identify every planar graph. The proof version presented here relies on strong insights about the ability of 2-WL to decompose graphs. Afterwards, I highlight the most important ingredients of the generalisation of our bound to graphs that are parameterised by their Euler genus.Further details as well as a study of other aspects of the WL algorithm can be found in my dissertation [Kiefer 2020].

Abstract: A proper vertex coloring of a graph G is acyclic if there is no bicolored cycle in G. In other words, each cycle of G must be colored with at least three colors. Given a list assignment L = {L(v): v ∈ V}, if there exists an acyclic coloring π of G such that π(v) ∈ L(v) for all v ∈ V, then we say that G is acyclically L-colorable. If G is acyclically L-colorable for any list assignment L with ∣L(v)∣ ⩾ k for all v ∈ V, then G is acyclically k-choosable. In 2006, Montassier, Raspaud and Wang conjectured that every planar graph without 4-cycles is acyclically 4-choosable. However, this has been as yet verified only for some restricted classes of planar graphs. In this paper, we prove that every planar graph with neither 4-cycles nor intersecting i-cycles for each i ∈ {3, 5} is acyclically 4-choosable.

Abstract: We revisit the well-studied problem of triangle count estimation in graph streams. Given a graph represented as a stream of $m$ edges, our aim is to compute a $(1\pm\varepsilon)$-approximation to the triangle count $T$, using a small space algorithm. For arbitrary order and a constant number of passes, the space complexity is known to be essentially $\Theta(\min(m^{3/2}/T, m/\sqrt{T}))$ (McGregor et al., PODS 2016, Bera et al., STACS 2017). We give a (constant pass, arbitrary order) streaming algorithm that can circumvent this lower bound for \emph{low degeneracy graphs}. The degeneracy, $\kappa$, is a nuanced measure of density, and the class of constant degeneracy graphs is immensely rich (containing planar graphs, minor-closed families, and preferential attachment graphs). We design a streaming algorithm with space complexity $\widetilde{O}(m\kappa/T)$. For constant degeneracy graphs, this bound is $\widetilde{O}(m/T)$, which is significantly smaller than both $m^{3/2}/T$ and $m/\sqrt{T}$. We complement our algorithmic result with a nearly matching lower bound of $\Omega(m\kappa/T)$.

Abstract: Let $f(n,H)$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph. The order of magnitude of $f(n,P_k)$, where $P_k$ is a path of length $k$, is $n^{{\lfloor{\frac{k}{2}}\rfloor}+1}$. In this paper we determine the asymptotic value of $f(n,P_4)$ and give conjectures for longer paths.

Abstract: We consider the problem of counting the number of copies of a fixed graph $H$ within an input graph $G$. This is one of the most well-studied algorithmic graph problems, with many theoretical and practical applications. We focus on solving this problem when the input $G$ has bounded degeneracy. This is a rich family of graphs, containing all graphs without a fixed minor (e.g. planar graphs), as well as graphs generated by various random processes (e.g. preferential attachment graphs). We say that $H$ is easy if there is a linear-time algorithm for counting the number of copies of $H$ in an input $G$ of bounded degeneracy. A seminal result of Chiba and Nishizeki from '85 states that every $H$ on at most 4 vertices is easy. Bera, Pashanasangi, and Seshadhri recently extended this to all $H$ on 5 vertices, and further proved that for every $k > 5$ there is a $k$-vertex $H$ which is not easy. They left open the natural problem of characterizing all easy graphs $H$. Bressan has recently introduced a framework for counting subgraphs in degenerate graphs, from which one can extract a sufficient condition for a graph $H$ to be easy. Here we show that this sufficient condition is also necessary, thus fully answering the Bera--Pashanasangi--Seshadhri problem. We further resolve two closely related problems; namely characterizing the graphs that are easy with respect to counting induced copies, and with respect to counting homomorphisms. Our proofs rely on several novel approaches for proving hardness results in the context of subgraph-counting.