Abstract: We propose a new geometric spanner, for wireless ad hoc networks, which can be constructed efficiently in a distributed manner. It combines the connected dominating set and the local Delaunay graph to form the backbone of a wireless network. This new spanner has the following attractive properties: (1) the backbone is a planar graph; (2) the node degree of the backbone is bounded from above by a positive constant; (3) it is a spanner both for hops and length; moreover, we show that, given any two nodes u and /spl upsi/, there is a path connecting them in the backbone such that its length is no more than 6 times that of the shortest path and the number of links is no more than 3 times that of the shortest path; (4) it can be constructed locally and is easy to maintain when the nodes move around; and (5) we show that the computation cost of each node is at most O(d log d), where d is its l-hop neighbors in the original unit disk graph, and the communication cost of each node is bounded by a constant. Simulation results are also presented for studying its practical performance.

Abstract: On a periodic planar graph whose edge weights satisfy a certain simple geometric condition, the discrete Laplacian and Dirac operators have the property that their determinants and inverses only depend on the local geometry of the graph. We obtain explicit expressions for the logarithms of the (normalized) determinants, as well as the inverses of these operators. We relate the logarithm of the determinants to the volume plus mean curvature of an associated hyperbolic ideal polyhedron. In the associated dimer and spanning tree models, for which the determinants of the Dirac operator and the Laplacian respectively play the role of the partition function, this allows us to compute the entropy and correlations in terms of the local geometry. In addition, we define a continuous family of special discrete holomorphic functions which, via convolutions, gives a general process for constructing discrete holomorphic functions and discrete harmonic functions on critical planar graphs.

Abstract: On a periodic planar graph whose edge weights satisfy a certain simple geometric condition, the discrete Laplacian and d-bar operators have the property that their determinants and inverses only depend on the local geometry of the graph. We obtain explicit expressions for the logarithms of the (normalized) determinants, as well as the inverses of these operators. We relate the logarithm of the determinants to the volume plus mean curvature of an associated hyperbolic ideal polyhedron. In the associated dimer and spanning tree models, for which the determinants of d-bar and the Laplacian respectively play the role of the partition function, this allows us to compute the entropy and correlations in terms of the local geometry. In addition, we define a continuous family of special discrete analytic functions, which, via convolutions gives a general process for constructing discrete analytic functions and discrete harmonic functions on critical planar graphs.

Abstract: Given an n-vertex outer-planar graph G and a set P of n points in the plane, we present an O(n log3 n) time and O(n) space algorithm to compute a straight-line embedding of G in P, improving upon the algorithm in [8,12] that requires O(n2) time. Our algorithm is near-optimal as there is an ω (n log n) lower bound for the problem [4]. We present a simpler O(nd) time and O(n) space algorithm to compute a straight-line embedding of G in P where log n ≥ d ≥ 2n is the length of the longest vertex disjoint path in the dual of G. Therefore, the time complexity of the simpler algorithm varies between O(n log n) and O(n2) depending on the value of d. More efficient algorithms are presented for certain restricted cases. If the dual of G is a path, then an optimal Θ (n log n) time algorithm is presented. If the given point set is in convex position then we show that O(n) time suffices.

Abstract: We present an algorithm that determines in polytime whether a graph contains an even hole. The algorithm is based on a decomposition theorem for even-hole-free graphs obtained in Part I of this work. We also give a polytime algorithm to find an even hole in a graph when one exists. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 238–266, 2002

Abstract: Earlier researchers have studied the set of orientations of a connected finite graph $G$, and have shown that any two such orientations having the same flow-difference around all closed loops can be obtained from one another by a succession of local moves of a simple type. Here I show that the set of orientations of $G$ having the same flow-differences around all closed loops can be given the structure of a distributive lattice. The construction generalizes partial orderings that arise in the study of alternating sign matrices. It also gives rise to lattices for the set of degree-constrained factors of a bipartite planar graph; as special cases, one obtains lattices that arise in the study of plane partitions and domino tilings. Lastly, the theory gives a lattice structure to the set of spanning trees of a planar graph.

Abstract: A bisection of a graph with n vertices is a partition of its vertices into two sets, each of size n/2. The bisection cost is the number of edges connecting the two sets. It is known that finding a bisection of minimum cost is NP-hard. We present an algorithm that finds a bisection whose cost is within ratio of O(log2 n) from the minimum. For graphs excluding any fixed graph as a minor (e.g., planar graphs) we obtain an improved approximation ratio of O(log n). The previously known approximation ratio for bisection was roughly $\sqrt{n}$.

Abstract: In this study, we provide methods for drawing a tree with n vertices on a convex polygon, without crossings and using the minimum number of edges of the polygon. We apply the results to obtain planar packings of two trees in some specific cases. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 172–181, 2002

Abstract: In ℍ2 × ℝ” one has catenoids, helicoids and Scherk-type surfaces. A Jenkins-Serrin type theorem holds here. Moreover there exist complete minimal graphs in ℍ2 with arbitrary continuous asymptotic values. Finally, a graph on a domain of ℍ2 cannot have an isolated singularity.

Abstract: It is proven that if G is a 3-connected claw-free graph which is also H1-free (where H1 consists of two disjoint triangles connected by an edge), then G is hamiltonian-connected. Also, examples will be described that determine a finite family of graphs ${\cal L}$ such that if a 3-connected graph being claw-free and L-free implies G is hamiltonian-connected, then L $\in \cal L$. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 104–119, 2002 The first four authors dedicate this paper to Henk Jan Veldman, a valued colleague and beloved friend who died October 12, 1998.

Abstract: We present a novel approach to the aesthetic drawing of undirected graphs. The method has two phases: first embed the graph in a very high dimension and then project it into the 2-D plane using principal components analysis. Running time is linear in the graph size, and experiments we have carried out show the ability of the method to draw graphs of 105 nodes in few seconds. The new method appears to have several advantages over classical methods, including a significantly better running time, a useful inherent capability to exhibit the graph in various dimensions, and an effective means for interactive exploration of large graphs.

Abstract: A robot has to visit all nodes and traverse all edges of an unknown undirected connected graph, using as few edge traversals as possible. The quality of an exploration algorithm A is measured by comparing its cost (number of edge traversals) to that of the optimal algorithm having full knowledge of the graph. The ratio between these costs, maximized over all starting nodes in the graph and over all graphs in a given class U, is called the overhead of algorithm A for the class U of graphs. We construct natural exploration algorithms, for various classes of graphs, that have smallest, or - in one case - close to smallest, overhead. An important contribution of this paper is establishing lower bounds that prove optimality of these exploration algorithms.

Abstract: Given a plane graph, a $k$-star at $u$ is a set of $k$ vertices with a common neighbour $u$; and a bunch is a maximal collection of paths of length at most two in the graph, such that all paths have the same end vertices and the edges of the paths form consecutive edges (\,in the natural order in the plane graph\,) around the two end vertices. We first prove a theorem on the structure of plane graphs in terms of stars and bunches. The result states that a plane graph contains a $(d-1)$-star centred at a vertex of degree $d\leq5$ and the sum of the degrees of the vertices in the star is bounded, or there exists a large bunch.

Abstract: In 1983, Conway and Gordon [J Graph Theory 7 (1983), 445–453] showed that every (tame) spatial embedding of K7, the complete graph on 7 vertices, contains a knotted cycle. In this paper, we adapt the methods of Conway and Gordon to show that K3,3,1,1 contains a knotted cycle in every spatial embedding. In the process, we establish that if a graph satisfies a certain linking condition for every spatial embedding, then the graph must have a knotted cycle in every spatial embedding. © 2002 Wiley Periodicals, Inc. J Graph Theory 39: 178–187, 2002; DOI 10.1002-jgt.10017

Abstract: The acyclic list chromatic number of every planar graph is proved to be at most 7. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 83–90, 2002

Abstract: Let G be a planar graph and let g(G) and Δ(G) be its girth and maximum degree, respectively. We show that G has an edge-partition into a forest and a subgraph H so that (i) Δ(H) ≤ 4 if g(G) ≥ 5; (ii) Δ(H) ≤ 2 if g(G) ≥ 7; (iii) Δ(H)≤ 1 if g(G) ≥ 11; (iv) Δ(H) ≤ 7 if G does not contain 4-cycles (though it may contain 3-cycles). These results are applied to find the following upper bounds for the game coloring number colg(G) of a planar graph G: (i) colg(G) ≤ 8 if g(G) ≥ 5; (ii) colg(G)≤ 6 if g(G) ≥ 7; (iii) colg(G) ≤ 5 if g(G) ≥ 11; (iv) colg(G) ≤ 11 if G does not contain 4-cycles (though it may contain 3-cycles). © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 307–317, 2002

Abstract: Recently, there has been a lot of interest and progress in lowering the worst-case time complexity for the PLANAR DOMINATING SET problem. In this paper, we present improved parameterized algorithms for the PLANAR DOMINATING SET problem. In particular, given a planar graph G and a positive integer k, we can compute a dominating set of size bounded by k or report that no such set exists in time O(2 27 n), where n is the number of vertices in G. Our algorithms induce a significant improvement over the previous best algorithm for the problem.

Abstract: Let G be a planar graph without two triangles sharing a common vertex. We prove that (1) G is 4-choosable and (2) G is edge-$(\Delta(G)+1)$-choosable when its maximum degree $\Delta(G) e 5$.

Abstract: Recently, there has been a lot of interest and progress in lowering the worst-case time complexity for the PLANAR DOMINATING SET problem. In this paper, we present improved parameterized algorithms for the PLANAR DOMINATING SET problem. In particular, given a planar graph G and a positive integer k, we can compute a dominating set of size bounded by k or report that no such set exists in time O(227?kn), where n is the number of vertices in G. Our algorithms induce a significant improvement over the previous best algorithm for the problem.

Abstract: We show that graph-theoretic thickness and geometric thickness are not asymptotically equivalent: for every t, there exists a graph with thickness three and geometric thickness ? t.

Abstract: Given a set S of n points in the plane, we give an O(n log n)- time algorithm that constructs a plane t-spanner for S, with t ? 10.02, such that the degree of each point of S is bounded from above by 27, and the total edge length is proportional to the weight of a minimum spanning tree of S. These constants are all worst case constants that are artifacts of our proofs. In practice, we believe them to be much smaller. Previously, no algorithms were known for constructing plane t-spanners of bounded degree.

Abstract: In a level directed acyclic graph G = (V;E) the vertex set V is partitioned into k ≤ |V | levels V1; V2... Vk such that for each edge (u, v) ∈ E with u ∈ Vi and v ∈; Vj we have i < j. The level planarity testing problem is to decide if G can be drawn in the plane such that for each level Vi, all v ∈ Vi are drawn on the line li = {(x, k - i) | x ∈ ℝ}, the edges are drawn monotonically with respect to the vertical direction, and no edges intersect except at their end vertices. In order to draw a level planar graph without edge crossings, a level planar embedding of the level graph has to be computed. Level planar embeddings are characterized by linear orderings of the vertices in each Vi (1 ≤ i ≤ k). We present an O(|V |) time algorithm for embedding level planar graphs. This approach is based on a level planarity test by Junger, Leipert, and Mutzel [6].

Abstract: A realizer of a maximal plane graph is a set of three particular spanning trees. It has been used in several graph algorithms and particularly in graph drawing algorithms. We propose colored flips on realizers to generalize Wagner's theorem on maximal planar graphs to realizers. From this result, it is proved that ?0 + ?1 + ?2 - ? = ni is the number of inner nodes in the tree Ti, ? is the number of three colored faces in the realizer and n is the number of vertices. As an application of this formula, we show that orderly spanning trees with at most ? 2n+1-?/3 ? leaves can be computed in linear time.

Abstract: We present a nondeterministic model of computation based on reversing edge directions in weighted directed graphs with minimum in-flow constraints on vertices. Deciding whether this simple graph model can be manipulated in order to reverse the direction of a particular edge is shown to be PSPACE-complete by a reduction from Quantified Boolean Formulas. We prove this result in a variety of special cases including planar graphs and highly restrictedv ertex configurations, some of which correspond to a kind of passive constraint logic. Our framework is inspired by (and indeed a generalization of) the "Generalized Rush Hour Logic" developed by Flake and Baum [2].We illustrate the importance of our model of computation by giving simple reductions to show that multiple motion-planning problems are PSPACE-hard. Our main result along these lines is that classic unrestricted sliding-block puzzles are PSPACE-hard, even if the pieces are restrictedto be all dominoes (1×2 blocks) andthe goal is simply to move a particular piece. No prior complexity results were known about these puzzles. This result can be seen as a strengthening of the existing result that the restricted Rush Hour? puzzles are PSPACE-complete [2], of which we also give a simpler proof. Finally, we strengthen the existing result that the pushing-blocks puzzle Sokoban is PSPACE-complete [1], by showing that it is PSPACE-complete even if no barriers are allowed.

Abstract: A three-dimensional (straight-line grid) drawing of a graph represents the vertices by points in Z3 and the edges by noncrossing line segments. This research is motivated by the following open problem due to Felsner, Liotta, and Wismath [Graph Drawing '01, Lecture Notes in Comput. Sci., 2002]: does every n-vertex planar graph have a three-dimensional drawing with O(n) volume? We prove that this question is almost equivalent to an existing one-dimensional graph layout problem. A queue layout consists of a linear order ? of the vertices of a graph, and a partition of the edges into queues, such that no two edges in the same queue are nested with respect to ?. The minimum number of queues in a queue layout of a graph is its queue-number. Let G be an n-vertex member of a proper minor-closed family of graphs (such as a planar graph). We prove that G has a O(1) × O(1) × O(n) drawing if and only if G has O(1) queue-number. Thus the above question is almost equivalent to an open problem of Heath, Leighton, and Rosenberg [SIAM J. Discrete Math., 1992], who ask whether every planar graph has O(1) queue-number? We also present partial solutions to an open problem of Ganley and Heath [Discrete Appl. Math., 2001], who ask whether graphs of bounded tree-width have bounded queue-number? We prove that graphs with bounded path-width, or both bounded tree-width and bounded maximum degree, have bounded queue-number. As a corollary we obtain three-dimensional drawings with optimal O(n) volume, for series-parallel graphs, and graphs with both bounded tree-width and bounded maximum degree.