Independent Covers in Plane Graphs

We initiate the study of the testability of properties in arbitrary planar graphs. We prove that bipartiteness can be tested in constant time. The previous bound for this class of graphs was O(√n), and the constant-time testability was only known for planar graphs with bounded degree. Previously used transformations of unbounded-degree sparse graphs into bounded- degree sparse graphs cannot be used to reduce the problem to the testability of bounded-degree planar graphs. Our approach extends to arbitrary minor-free graphs. Our algorithm is based on random walks. The challenge here is to analyze random walks for a class of graphs that has good separators, i.e., bad expansion. Standard techniques that use a fast convergence to a uniform distribution do not work in this case. Roughly speaking, our analysis technique self-reduces the problem of finding an odd-length cycle in a multigraph G induced by a collection of cycles to another multigraph G' induced by a set of shorter odd-length cycles, in such a way that when a random walks finds a cycle in G' with probability p >; 0, then it does so with probability λ(p) >; 0 in G. This reduction is applied until the cycles collapse to self-loops that can be easily detected.

We study the empire colouring problem (as defined by Percy Heawood in 1890) for maps containing empires formed by exactly r>1 countries each. We prove that the problem can be solved in polynomial time using s colours on maps whose underlying adjacency graph has no induced subgraph of average degree larger than s/r. However, if s≥3, the problem is NP-hard for forests of paths of arbitrary lengths (if s<r) for trees (if r≥2 and s<2r) and arbitrary planar graphs (if s<7 for r=2, and s<6r−3, for r≥3). The result for trees shows a perfect dichotomy (the problem is NP-hard if 3≤s≤2r−1 and polynomial time solvable otherwise). The one for planar graphs proves the NP-hardness of colouring with less than 7 colours graphs of thickness two and less than 6r−3 colours graphs of thickness r≥3.

A straight-line drawing $\delta$ of a planar graph $G$ need not be plane, but can be made so by \emph{untangling} it, that is, by moving some of the vertices of $G$. Let shift$(G,\delta)$ denote the minimum number of vertices that need to be moved to untangle $\delta$. We show that shift$(G,\delta)$ is NP-hard to compute and to approximate. Our hardness results extend to a version of \textsc{1BendPointSetEmbeddability}, a well-known graph-drawing problem. Further we define fix$(G,\delta)=n-shift(G,\delta)$ to be the maximum number of vertices of a planar $n$-vertex graph $G$ that can be fixed when untangling $\delta$. We give an algorithm that fixes at least $\sqrt{((\log n)-1)/\log \log n}$ vertices when untangling a drawing of an $n$-vertex graph $G$. If $G$ is outerplanar, the same algorithm fixes at least $\sqrt{n/2}$ vertices. On the other hand we construct, for arbitrarily large $n$, an $n$-vertex planar graph $G$ and a drawing $\delta_G$ of $G$ with fix$(G,\delta_G) \le \sqrt{n-2}+1$ and an $n$-vertex outerplanar graph $H$ and a drawing $\delta_H$ of $H$ with fix$(H,\delta_H) \le 2 \sqrt{n-1}+1$. Thus our algorithm is asymptotically worst-case optimal for outerplanar graphs.

An outerplanar graph is a planar graph that can be embedded in the plane in such a way that all vertices lie on the outer boundary. Outerplanar graphs express many chemical compounds. An externally extensible outerplanar graph pattern (eeo-graph pattern for short) represents a graph pattern common to a finite set of outerplanar graphs, like a dataset of chemical compounds. The eeo-graph pattern can express a substructure common to blocks that appear in outerplanar graph structured data. In this paper, we propose a polynomial time algorithm for deciding whether or not a given eeo-graph pattern matches a given connected outerplanar graph. Moreover, we show the expressiveness of the pattern class by experiments on a chemical compound database.

Edge Separators of Planar and Outerplanar Graphs With Applications

We study straight skeletons and make both theoretical and practical contributions which support new approaches to the computation of straight skeletons of arbitrary planar straight-line graphs (PSLGs). We start with an adequate extension of the concept of motorcycle graphs to PSLGs, with motorcycles starting at the reflex vertices of a PSLG, which allows us to generalize well-known results on the relation between the straight skeleton and the motorcycle graph to arbitrary PSLGs: the edges of the motorcycle graph cover a specific subset of the edges of the straight skeleton, and they form the basis of 3D slabs such that the projection of the lower envelope of those slabs to the plane forms the straight skeleton. As an immediate application we sketch how to use a graphics hardware for computing (approximate) straight skeletons of PSLGs. Further, we present and analyze a novel wavefront-type algorithm which bridges the current gap between the theory and practice of straight-skeleton computations. Our algorithm handles arbitrary PSLGs, is easy to implement, and is fast enough to handle complex data: it can be expected to run in O(n log n) time in practice for an n-vertex PSLG; its worst-case complexity is O(n2 log n). Extensive experimental results confirm an average runtime of 20 n log n µs on a standard PC for virtually all of our 13500 datasets of different characteristics. As also confirmed by our experiments, this constitutes an average gain in performance by a multiplicative factor of n, or at least one to two orders of magnitude, relative to the speed of the implementation provided by CGAL for closed polygons.

Finding regular subgraphs in both arbitrary and planar graphs

We present an algorithm to support the dynamic embedding in the plane of a dynamic graph. An edge can be inserted across a face between two vertices on the boundary (we call such a vertex pair linkable), and edges can be deleted. The planar embedding can also be changed locally by flipping components that are connected to the rest of the graph by at most two vertices. Given vertices u,v, linkable(u,v) decides whether u and v are linkable, and if so, returns a list of suggestions for the placement of (u,v) in the embedding. For non-linkable vertices u,v, we define a new query, one-flip-linkable(u,v) providing a suggestion for a flip that will make them linkable if one exists. We will support all updates and queries in O(log^2 n) time. Our time bounds match those of Italiano et al. for a static (flipless) embedding of a dynamic graph. Our new algorithm is simpler, exploiting that the complement of a spanning tree of a connected plane graph is a spanning tree of the dual graph. The primal and dual trees are interpreted as having the same Euler tour, and a main idea of the new algorithm is an elegant interaction between top trees over the two trees via their common Euler tour.

Embedding between metric spaces is a very powerful algorithmic tool and has been used for finding good approximation algorithms for several problems. In particular, embedding to an [cursive l]₁ norm has been used as the key step in an approximation algorithm for the sparsest cut problem. The sparsest cut problem, in turn, is the main ingredient of many algorithms that have a divide and conquer nature and are used in various fields. While every metric is embeddable into [cursive l]₁ with distortion O (log n) [13], and the bound is tight [39], for special classes of metrics better bounds exist. Shortest path metrics for trees and outerplanar graphs are isometrically embeddable into [cursive l]₁ [41]. Series-parallel graphs [28] and k-outerplanar graphs [19] (for constant k) are embeddable into[cursive l]₁ with constant distortion planar graphs and bounded tree-width graphs are conjectured to have constant distortion in embedding to [cursive l]₁ . Bounded tree-width graphs are one of most general graph classes on which several hard problems are tractable. We study the embedding of series-parallel graphs (or, more generally, graphs with tree-width two) into [cursive l]₁ and also the embedding between two line metrics. We then move our attention to the generalization of tree-width to digraphs and hypergraphs and study several relevant problems.

Recently, due to the rapid growth of electronic data having graph structures such as HTML and XML texts and chemical compounds, many researchers have been interested in data mining and machine learning techniques for finding useful patterns from graph-structured data (graph data). Since graph data contain a huge number of substructures and it tends to be computationally expensive to decide whether or not such data have given structural features, graph mining problems face computational difficulties. Let ${\mathcal{C}}$ be a graph class which satisfies a connected hereditary property and contains infinitely many different biconnected graphs, and for which a special kind of the graph isomorphism problem can be computed in polynomial time. In this paper, we consider learning and mining problems for ${\mathcal{C}}$ . Firstly, we define a new graph pattern, which is called a block preserving graph pattern (bp-graph pattern) for ${\mathcal{C}}$ . Secondly, we present a polynomial time algorithm for deciding whether or not a given bp-graph pattern matches a given graph in ${\mathcal{C}}$ . Thirdly, by giving refinement operators over bp-graph patterns, we present a polynomial time algorithm for finding a minimally generalized bp-graph pattern for ${\mathcal{C}}$ . Outerplanar graphs are planar graphs which can be embedded in the plane in such a way that all of vertices lie on the outer boundary. Many pharmacologic chemical compounds are known to be represented by outerplanar graphs. The class of connected outerplanar graphs ${\mathcal{O}}$ satisfies the above conditions for ${\mathcal{C}}$ . Next, we propose two incremental polynomial time algorithms for enumerating all frequent bp-graph patterns with respect to a given finite set of graphs in ${\mathcal{O}}$ . Finally, by reporting experimental results obtained by applying the two graph mining algorithms to a subset of the NCI dataset, we evaluate the performance of the two graph mining algorithms.

All finite planar graphs are subgraphs of certain planar Cayley graphs of free groups; outerplanar graphs are subgraphs of certain outerplanar Cayley graphs of free groups. It is shown for these particular planar presentations of free groups that special planar is equivalent with outerplanar.KeywordsCayley GraphCyclic PermutationSpecial PlanarOuterplanar GraphLocal FinitenessThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

On facial unique-maximum (edge-)coloring

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.

A proximity drawing of a graph is one in which pairs of adjacent vertices are drawn relatively close together according to some proximity measure while pairs of non-adjacent vertices are drawn relatively far apart. The fundamental question concerning proximity drawability is: Given a graph G and a definition of proximity, is it possible to construct a proximity drawing of G? We consider this question for outerplanar graphs with respect to an infinite family of proximity drawings called β-drawings. These drawings include as special cases the well-known Gabriel drawings (when β=1), and relative neighborhood drawings (when β=2). We first show that all biconnected outerplanar graphs are β-drawable for all values of β such that 1 ≤ β ≤ 2. As a side effect, this result settles in the affirmative a conjecture by Lubiw and Sleumer [15, 17], that any biconnected outerplanar graph admits a Gabriel drawing. We then show that there exist biconnected outerplanar graphs that do not admit any convex β-drawing for 1 ≤ β ≤ 2. We also provide upper bounds on the maximum number of biconnected components sharing the same cut-vertex in a β-drawable connected outerplanar graph. This last result is generalized to arbitrary connected planar graphs and is the first non-trivial characterization of connected β-drawable graphs. Finally, a weaker definition of proximity drawings is applied and we show that all connected outerplanar graphs are drawable under this definition.

Max point-tolerance graphs