Finding regular subgraphs in both arbitrary and planar 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.

There are a number of board games such as Checkers [2], Go [5], and Gobang [8], which are known to be PSPACE-hard. This means that the problem to determine the player having a winning strategy in a given situation on an nA—n board of one of these games is as hard to solve as everything computable in polynomial space. PSPACE-completeness has been previously proven for some combinatorial games played on graphs or by logical formulas [1, 9]. In this paper we will show that the same holds for the game of Hex. The crucial point of the proof is to establish PSPACE-hardness for a generalization of Hex played on planar graphs. This will be done by showing that the problem, whether a given quantified Boolean formula in conjunctive normal form is true, is polynomial time-reducible to the decision problem for generalized Hex. In order to do this we will use methods, which were also used to prove PSPACE-completeness of planar Geography in [5]. Therefore our proof is quite different from the proof provided by Even and Tarjan [1], who showed PSPACE-completeness of generalized Hex played on arbitrary graphs. Since it is easy to see that the decision problem for Hex is in PS PACE, the decision problem for Hex is PSPACE-complete.

We describe efficient deterministic techniques for breaking symmetry in parallel. The techniques work well on rooted trees and graphs of constant degree or genus. Our primary technique allows us to 3-color a rooted tree in Ο(lg*n) time on an EREW PRAM using a linear number of processors. We apply these techniques to construct fast linear processor algorithms for several problems, including (Δ + 1)-coloring constant-degree graphs, 5-coloring planar graphs, and finding depth-first-search trees in planar graphs. We also prove lower bounds for 2-coloring directed lists and for finding maximal independent sets in arbitrary graphs.

Given an n-node plane graph G, the visibility representation of G is concerned with drawing each node of G using a horizontal line segment such that the line segments associated with any two adjacent nodes of G are vertically visible to each other. Finding most compact visibility representations of plane graphs is not only of theoretical importance but also of practical interest, and has received much attention in the community of algorithmic graph theory. In this paper, we give a linear-time algorithm to find a visibility representation of G no wider than ⌊4n/3⌋ - 2. Our result improves upon the previously known upper bound 4n/3 + 2⌈ √n⌉, providing a positive answer to a conjecture suggested in the literature about whether an upper bound 4n/3 + O(1) on the required width can be achieved for an arbitrary plane graph. In fact, our visibility representation achieves optimality in the upper bound of width because the bound differs from the previously known lower bound ⌊4n/3⌋ - 3 only by one unit.

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.

The max-bisection and min-bisection problems are to find a partition of the vertices of a graph into two equal size subsets that, respectively, maximizes or minimizes the number of edges with endpoints in both subsets. We design the first polynomial time approximation scheme for the max-bisection problem on arbitrary planar graphs solving a long-standing open problem. The method of solution involves designing exact polynomial time algorithms for computing optimal partitions of bounded treewidth graphs, in particular max- and min-bisection, which could be of independent interest. Using a similar method we design also the first polynomial time approximation scheme for max-bisection on unit disk graphs (which could also be easily extended to other geometrically defined graphs).

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.

We submit an animation of an algorithm for drawing an arbitrary planar graph with vertices located on a semi-circle and edges drawn as straight line segments with at most one bend per edge. The animation,produced in .avi format, is available at:www.cs.uleth.ca/~vpak/socg05 .

The resistance distance is widely used in random walk, electronic engineering, and complex networks. One of the main topics in the study of the resistance distance is the computation problem. The subdivision graph $S(G)$ of a graph $G$ is the graph obtained by inserting a new vertex into every edge of $G$ . Two classes of new corona graphs, the corona-vertex of the subdivision graph $G_{1}\diamondsuit G_{2}$ and the corona-edge of the subdivision graph $G_{1}\star G_{2}$ , were defined by Lu and Miao. The adjacency spectrum and the signless Laplacian spectrum of the two new graph operations were computed when $G_{1}$ is an arbitrary graph and $G_{2}$ is an $r_{2}$ -regular graph. In this paper, we give the closed-form formulas for the resistance distance and Kirchhoff index of $G_{1}\diamondsuit G_{2}$ and $G_{1}\star G_{2}$ in terms of the resistance distance and Kirchhoff index of the factor graphs.

A subset W of vertices of a plane graph G is said to be a face-independent vertex cover (FIVC) if each face-boundary contains exactly one vertex of W. A FIVC naturally corresponds to a vertex-independent face cover (VIFC) in the dual graph G * of G. In this paper we review the results on the existence of such covers in arbitrary planar graphs and in outerplanar graphs, in particular.KeywordsPlane GraphVertex CoverDual GraphTerminal VertexOuterplanar GraphThese 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.

We consider strongly regular graphs Γ e (V, E) on an even number, say 2n, of vertices which admit an automorphism group G of order n which has two orbits on V. Such graphs will be called strongly regular semi-Cayley graphs. For instance, the Petersen graph, the Hoffman–Singleton graph, and the triangular graphs T(q) with q ≡ 5 mod 8 provide examples which cannot be obtained as Cayley graphs. We give a representation of strongly regular semi-Cayley graphs in terms of suitable triples of elements in the group ring ZG. By applying characters of G, this approach allows us to obtain interesting nonexistence results if G is Abelian, in particular, if G is cyclic. For instance, if G is cyclic and n is odd, then all examples must have parameters of the form 2n e 4s2 + 4s + 2, k e 2s2 + s, λ e s2 − 1, and μ e s2s examples are known only for s e 1, 2, and 4 (together with a noncyclic example for s e 3). We also apply our results to obtain new conditions for the existence of strongly regular Cayley graphs on an even number of vertices when the underlying group H has an Abelian normal subgroup of index 2. In particular, we show the nonexistence of nontrivial strongly regular Cayley graphs over dihedral and generalized quaternion groups, as well as over two series of non-Abelian 2-groups. Up to now these have been the only general nonexistence results for strongly regular Cayley graphs over non-Abelian groupss only the first of these cases was previously known.

This paper deals with compact label-based representations for trees. Consider an n-node undirected connected graph G with a predefined numbering on the ports of each node. The all-ports tree labeling ${\mathcal L}_{all}$ gives each node v of G a label containing the port numbers of all the tree edges incident to v. The upward tree labeling ${\mathcal L}_{up}$ labels each node v by the number of the port leading from v to its parent in the tree. Our measure of interest is the worst case and total length of the labels used by the scheme, denoted Mup(T) and Sup(T) for ${\mathcal L}_{up}$ and Mall(T) and Sall(T) for ${\mathcal L}_{all}$. The problem studied in this paper is the following: Given a graph G and a predefined port labeling for it, with the ports of each node v numbered by 0,...,deg(v) – 1, select a rooted spanning tree for G minimizing (one of) these measures. We show that the problem is polynomial for Mup(T), Sup(T) and Sall(T) but NP-hard for Mall(T) (even for 3-regular planar graphs). We show that for every graph G and port numbering there exists a spanning tree T for which Sup(T) = O(n log log n). We give a tight bound of O(n) in the cases of complete graphs with arbitrary labeling and arbitrary graphs with symmetric port assignments. We conclude by discussing some applications for our tree representation schemes.

On the hull number on cycle convexity of graphs

The Star Arboricity of Graphs

Often in problems with a parameter k like k-CNFSat and k-colorability, larger values of k make the problem harder. This is not always the case. Consider the problem of determining whether a planar graph has a k-coloring. The problem is trivial for k = 1, easy for k = 2 (check by DFS or BFS whether the graph is bipartite, i.e. has no odd cycles), and trivial for k = 4 or greater by the Four Color Theorem, which says that every planar graph is 4-colorable. This leaves k = 3. We show below that 3-colorability of planar graphs is no easier than 3-colorability of arbitrary graphs. This result is due to Garey, Johnson, and Stockmeyer [40]; see also Lichtenstein [72] for some other NP-completeness results involving planar graphs.