Abstract: We give the first polynomial-time approximation schemes (PTASs) for the following problems: (1) uniform facility location in edge-weighted planar graphs; (2) $k$-median and $k$-means in edge-weight...

Abstract: Over 50 years ago, Lovasz proved that two graphs are isomorphic if and only if they admit the same number of homomorphisms from any graph [Acta Math. Hungar. 18 (1967), pp. 321--328]. In this work we prove that two graphs are quantum isomorphic (in the commuting operator framework) if and only if they admit the same number of homomorphisms from any planar graph. As there exist pairs of non-isomorphic graphs that are quantum isomorphic, this implies that homomorphism counts from planar graphs do not determine a graph up to isomorphism. Another immediate consequence is that determining whether there exists some planar graph that has a different number of homomorphisms to two given graphs is an undecidable problem, since quantum isomorphism is known to be undecidable. Our characterization of quantum isomorphism is proven via a combinatorial characterization of the intertwiner spaces of the quantum automorphism group of a graph based on counting homomorphisms from planar graphs. This result inspires the definition of "graph categories" which are analogous to, and a generalization of, partition categories that are the basis of the definition of easy quantum groups. Thus we introduce a new class of "graph-theoretic quantum groups" whose intertwiner spaces are spanned by maps associated to (bi-labeled) graphs. Finally, we use our result on quantum isomorphism to prove an interesting reformulation of the Four Color Theorem: that any planar graph is 4-colorable if and only if it has a homomorphism to a specific Cayley graph on the symmetric group $S_4$ which contains a complete subgraph on four vertices but is not 4-colorable.

Abstract: Computing the Strongly-Connected Components (SCCs) in a graph G=(V,E) is known to take only O(m+n) time using an algorithm by Tarjan from 1972[SICOMP 72] where m = |E|, n=|V|. For fully-dynamic graphs, conditional lower bounds provide evidence that the update time cannot be improved by polynomial factors over recomputing the SCCs from scratch after every update. Nevertheless, substantial progress has been made to find algorithms with fast update time for decremental graphs, i.e. graphs that undergo edge deletions. In this paper, we present the first algorithm for general decremental graphs that maintains the SCCs in total update time O(m), thus only a polylogarithmic factor from the optimal running time. Previously such a result was only known for the special case of planar graphs [Italiano et al, STOC 17]. Our result should be compared to the formerly best algorithm for general graphs achieving O(m√n) total update time by Chechik et.al. [FOCS 16] which improved upon a breakthrough result leading to O(mn0.9 + o(1)) total update time by Henzinger, Krinninger and Nanongkai [STOC 14, ICALP 15]; these results in turn improved upon the longstanding bound of O(mn) by Roditty and Zwick [STOC 04]. All of the above results also apply to the decremental Single-Source Reachability (SSR) problem, which can be reduced to decrementally maintaining SCCs. A bound of O(mn) total update time for decremental SSR was established already in 1981 by Even and Shiloach [JACM 81].

Abstract: We present new tradeoffs between space and query-time for exact distance oracles in directed weighted planar graphs. These tradeoffs are almost optimal in the sense that they are within polylogarithmic, subpolynomial or arbitrarily small polynomial factors from the naive linear space, constant query-time lower bound. These tradeoffs include: (i) an oracle with space O(n1+є) and query-time O(1) for any constant є>0, (ii) an oracle with space O(n) and query-time O(nє) for any constant є>0, and (iii) an oracle with space n1+o(1) and query-time no(1).

Abstract: DP-coloring (also known as correspondence coloring) of a simple graph is a generalization of list coloring. It is known that planar graphs without 4-cycles adjacent to triangles are 4-choosable, and planar graphs without 4-cycles are DP-4-colorable. In this paper, we show that planar graphs without 4-cycles adjacent to triangles are DP-4-colorable, which implies the two results above.

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: The modular graph $C_{a,b,c,d}$ on the torus is a three loop planar graph in which two of the vertices have coordination number four, while the others have coordination number two. We obtain an eigenvalue equation satisfied by $C_{a,b,c,d}$ for generic values of $a,b,c$ and $d$, where the source terms involve various modular graphs. This is obtained by varying the graph with respect to the Beltrami differential on the toroidal worldsheet. Use of several auxiliary graphs at various intermediate stages of the analysis is crucial in obtaining the equation. In fact, the eigenfunction is not simply $C_{a,b,c,d}$ but involves subtracting from it specific sums of squares of non--holomorphic Eisenstein series characterized by $a,b,c$ and $d$.

Abstract: We study soliton solutions of matrix Kadomtsev–Petviashvili (KP) equations in a tropical limit, in which their support at fixed time is a planar graph and polarizations are attached to its constituting lines. There is a subclass of “pure line soliton solutions” for which we find that, in this limit, the distribution of polarizations is fully determined by a Yang–Baxter map. For a vector KP equation, this map is given by an R-matrix, whereas it is a nonlinear map in the case of a more general matrix KP equation. We also consider the corresponding Korteweg–deVries reduction. Furthermore, exploiting the fine structure of soliton interactions in the tropical limit, we obtain an apparently new solution of the tetrahedron (or Zamolodchikov) equation. Moreover, a solution of the functional tetrahedron equation arises from the parameter dependence of the vector KP R-matrix.

Abstract: The Minimum Dominating Set (MDS) problem is a fundamental and challenging problem in distributed computing. While it is well known that minimum dominating sets cannot be well approximated locally on general graphs, in recent years there has been much progress on computing good local approximations on sparse graphs and in particular on planar graphs. In this article, we study distributed and deterministic MDS approximation algorithms for graph classes beyond planar graphs. In particular, we show that existing approximation bounds for planar graphs can be lifted to bounded genus graphs and more general graphs, which we call locally embeddable graphs, and present(1) a local constant-time, constant-factor MDS approximation algorithm on locally embeddable graphs, and(2) a local O(logan)-time (1+e)-approximation scheme for any e > 0 on graphs of bounded genus.Our main technical contribution is a new analysis of a slightly modified variant of an existing algorithm by Lenzen et al. [21]. Interestingly, unlike existing proofs for planar graphs, our analysis does not rely on direct topological arguments but on combinatorial density arguments only.

Abstract: A colouring of a graph is "nonrepetitive" if for every path of even order, the sequence of colours on the first half of the path is different from the sequence of colours on the second half. We show that planar graphs have nonrepetitive colourings with a bounded number of colours, thus proving a conjecture of Alon, Grytczuk, Haluszczak and Riordan (2002). We also generalise this result for graphs of bounded Euler genus, graphs excluding a fixed minor, and graphs excluding a fixed topological minor.

Abstract: The modular graph Ca,b,c,d on the torus is a three loop planar graph in which two of the vertices have coordination number four, while the others have coordination number two. We obtain an eigenvalue equation satisfied by Ca,b,c,d for generic values of a, b, c and d, where the source terms involve various modular graphs. This is obtained by varying the graph with respect to the Beltrami differential on the toroidal worldsheet. Use of several auxiliary graphs at various intermediate stages of the analysis is crucial in obtaining the equation. In fact, the eigenfunction is not simply Ca,b,c,d but involves subtracting from it specific sums of squares of non-holomorphic Eisenstein series characterized by a, b, c and d.

Abstract: A vertex coloring $\phi$ of a graph $G$ is $p$-centered if for every connected subgraph $H$ of $G$ either $\phi$ uses more than $p$ colors on $H$ or there is a color that appears exactly once on $H$. Centered colorings form one of the families of parameters that allow to capture notions of sparsity of graphs: A class of graphs has bounded expansion if and only if there is a function $f$ such that for every $p\geq1$, every graph in the class admits a $p$-centered coloring using at most $f(p)$ colors. In this paper, we give upper bounds for the maximum number of colors needed in a $p$-centered coloring of graphs from several widely studied graph classes. We show that: (1) planar graphs admit $p$-centered colorings with $\mathcal{O}(p^3\log p)$ colors where the previous bound was $\mathcal{O}(p^{19})$; (2) bounded degree graphs admit $p$-centered colorings with $\mathcal{O}(p)$ colors while it was conjectured that they may require exponential number of colors in $p$; (3) graphs avoiding a fixed graph as a topological minor admit $p$-centered colorings with a polynomial in $p$ number of colors. All these upper bounds imply polynomial algorithms for computing the colorings. Prior to this work there were no non-trivial lower bounds known. We show that: (4) there are graphs of treewidth $t$ that require $\binom{p+t}{t}$ colors in any $p$-centered coloring and this bound matches the upper bound; (5) there are planar graphs that require $\Omega(p^2\log p)$ colors in any $p$-centered coloring.

Abstract: A graph $G$ is $(d_1,\ldots,d_k)$-colorable if its vertex set can be partitioned into $k$ sets $V_1,\ldots,V_k$, such that for each $i\in\{1, \ldots, k\}$, the subgraph of $G$ induced by $V_i$ has maximum degree at most $d_i$. The Four Color Theorem states that every planar graph is $(0,0,0,0)$-colorable, and a classical result of Cowen, Cowen, and Woodall shows that every planar graph is $(2,2,2)$-colorable. In this paper, we extend both of these results to graphs on surfaces. Namely, we show that every graph embeddable on a surface of Euler genus $g>0$ is $(0,0,0,9g-4)$-colorable and $(2,2,9g-4)$-colorable. Moreover, these graphs are also $(0,0,O(\sqrt{g}),O(\sqrt{g}))$-colorable and $(2,O(\sqrt{g}),O(\sqrt{g}))$-colorable. We also prove that every triangle-free graph that is embeddable on a surface of Euler genus $g$ is $(0, 0, O(g))$-colorable. This is an extension of Gr\"{o}tzsch's Theorem, which states that triangle-free planar graphs are $(0, 0, 0)$-colorable. Finally, we prove that every graph of girth at least 7 that is embeddable on a surface of Euler genus $g$ is $(0,O(\sqrt{g}))$-colorable. All these results are best possible in several ways as the girth condition is sharp, the constant maximum degrees cannot be improved, and the bounds on the maximum degrees depending on $g$ are tight up to a constant multiplicative factor.

Abstract: In this paper, we study distributed graph algorithms in networks in which the nodes have a limited communication capacity. Many distributed systems are built on top of an underlying networking infrastructure, for example by using a virtual communication topology known as an overlay network. Although this underlying network might allow each node to directly communicate with a large number of other nodes, the amount of communication that a node can perform in a fixed amount of time is typically much more limited. We introduce the Node-Capacitated Clique model as an abstract communication model, which allows us to study the effect of nodes having limited communication capacity on the complexity of distributed graph computations. In this model, the n nodes of a network are connected as a clique and communicate in synchronous rounds. In each round, every node can exchange messages of $O(log n)$ bits with at most $O(log n)$ other nodes. When solving a graph problem, the input graph G is defined on the same set of n nodes, where each node knows which other nodes are its neighbors in G. To initiate research on the Node-Capacitated Clique model, we present distributed algorithms for the Minimum Spanning Tree (MST), BFS Tree, Maximal Independent Set, Maximal Matching, and Vertex Coloring problems. We show that even with only $O(log n)$ concurrent interactions per node, the MST problem can still be solved in polylogarithmic time. In all other cases, the runtime of our algorithms depends linearly on the arboricity of G, which is a constant for many important graph families such as planar graphs.

Abstract: We study an alternating sign matrix analogue of the Chan–Robbins–Yuen polytope, which we call the ASM-CRY polytope. We show that this polytope has Catalan many vertices and its volume is equal to the number of standard Young tableaus of staircase shape; we also determine its Ehrhart polynomial. We achieve the previous by proving that the members of a family of faces of the alternating sign matrix polytope which includes ASM-CRY are both order and flow polytopes. Inspired by the above results, we relate three established triangulations of order and flow polytopes, namely Stanley’s triangulation of order polytopes, the Postnikov–Stanley triangulation of flow polytopes and the Danilov–Karzanov–Koshevoy triangulation of flow polytopes. We show that when a graph G is a planar graph, in which case the flow polytope $${{\mathcal {F}}}_G$$ is also an order polytope, Stanley’s triangulation of this order polytope is one of the Danilov–Karzanov–Koshevoy triangulations of $${{\mathcal {F}}}_G$$ . Moreover, for a general graph G we show that the set of Danilov–Karzanov–Koshevoy triangulations of $${{\mathcal {F}}}_G$$ equals the set of framed Postnikov–Stanley triangulations of $${{\mathcal {F}}}_G$$ . We also describe explicit bijections between the combinatorial objects labeling the simplices in the above triangulations.

Abstract: The Capacitated Vehicle Routing problem is to find a minimum-cost set of tours that collectively cover clients in a graph, such that each tour starts and ends at a specified depot and is subject to a capacity bound on the number of clients it can serve. In this paper, we present a polynomial-time approximation scheme (PTAS) for instances in which the input graph is planar and the capacity is bounded. Previously, only a quasipolynomial-time approximation scheme was known for these instances. To obtain this result, we show how to embed planar graphs into bounded-treewidth graphs while preserving, in expectation, the client-to-client distances up to a small additive error proportional to client distances to the depot.

Abstract: Given a graph $H$, a graph is $H$-free if it does not contain $H$ as a subgraph. We continue to study the topic of "extremal" planar graphs, that is, how many edges can an $H$-free planar graph on $n$ vertices have? We define $ex_{_\mathcal{P}}(n,H)$ to be the maximum number of edges in an $H$-free planar graph on $n $ vertices. We first obtain several sufficient conditions on $H$ which yield $ex_{_\mathcal{P}}(n,H)=3n-6$ for all $n\ge |V(H)|$. We discover that the chromatic number of $H$ does not play a role, as in the celebrated Erdős-Stone Theorem. We then completely determine $ex_{_\mathcal{P}}(n,H)$ when $H$ is a wheel or a star. Finally, we examine the case when $H$ is a $(t, r)$-fan, that is, $H$ is isomorphic to $K_1+tK_{r-1}$, where $t\ge2$ and $r\ge 3$ are integers. However, determining $ex_{_\mathcal{P}}(n,H)$, when $H$ is a planar subcubic graph, remains wide open.

Abstract: A generalization of list-coloring, now known as DP-coloring, was recently introduced by Dvořak and Postle (Comb Theory Ser B 129:38–54, 2018). Essentially, DP-coloring assigns an arbitrary matching between lists of colors at adjacent vertices, as opposed to only matching identical colors as is done for list-coloring. Several results on list-coloring of planar graphs have since been extended to the setting of DP-coloring (Liu and Li, Discrete Math 342:623–627, 2019; Liu et al., Discrete Math 342(1):178–189, 2019; Kim and Ozeki, A note on a Brooks type theorem for DP-coloring, arXiv:1709.09807 , 2019; Kim and Yu, Planar graphs without 4-cycles adjacent to triangles are DP-4-colorable, arXiv:1712.08999 , 2019; Sittitrai and Nakprasit, Every planar graph without i-cycles adjacent simultaneously to j-cycles and k-cycles is DP-4-colorable when $$\{i,j,k\}=\{3,4,5\}$$ , arXiv:1801.06760 , 2019; Yin and Yu, Planar graphs without cycles of lengths 4 and 5 and close triangles are DP-3-colorable, arXiv:1809.00925 , 2019). We note that list-coloring results do not always extend to DP-coloring results, as shown in Bernshteyn and Kostochka (On differences between DP-coloring and list coloring, arXiv:1705.04883 , 2019). Our main result in this paper is to prove that every planar graph without cycles of length $$\{4, a, b, 9\}$$ for $$a, b \in \{6, 7, 8\}$$ is DP-3-colorable, extending three existing results (Shen and Wang, Inf Process Lett 104:146–151, 2007; Wang and Shen, Discrete Appl Math 159:232–239, 2011; Whang et al., Inf Process Lett 105:206–211, 2008) on 3-choosability of planar graphs.

Abstract: Dujmovic et al. (FOCS 2019) recently proved that every planar graph is a subgraph of the strong product of a graph of bounded treewidth and a path. This tool has been used to solve longstanding problems on queue layouts, non-repetitive colouring, $p$-centered colouring, and implicit graph encoding. We generalise this result for $k$-planar graphs, where a graph is $k$-planar if it has a drawing in the plane in which each edge is involved in at most $k$ crossings. In particular, we prove that every $k$-planar graph is a subgraph of the strong product of a graph of treewidth $O(k^5)$ and a path. This is the first result of this type for a non-minor-closed class of graphs. It implies, amongst other results, that $k$-planar graphs have non-repetitive chromatic number upper-bounded by a function of $k$. All these results generalise for drawings of graphs on arbitrary surfaces. In fact, we work in a much more general setting based on so-called shortcut systems that are of independent interest.

Abstract: There are several ways to generalize graph coloring to signed graphs. Macajova, Raspaud and Skoviera introduced one of them and conjectured that in this setting, for signed planar graphs four colors are always enough, generalising thereby The Four Color Theorem. We disprove the conjecture.

Abstract: Hakimi and Schmeichel considered the problem of maximizing the number of cycles of a given length in an $n$-vertex planar graph. They determined this number exactly for triangles and 4-cycles and conjectured the solution to the problem for 5-cycles. We confirm their conjecture.