240 research outputs found

    Coloring polygon visibility graphs and their generalizations

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    Curve pseudo-visibility graphs generalize polygon and pseudo- polygon visibility graphs and form a hereditary class of graphs. We prove that every curve pseudo-visibility graph with clique number ω has chromatic number at most 3 · 4ω−1. The proof is carried through in the setting of ordered graphs; we identify two conditions satisfied by every curve pseudo- visibility graph (considered as an ordered graph) and prove that they are sufficient for the claimed bound. The proof is algorithmic: both the clique number and a coloring with the claimed number of colors can be computed in polynomial time

    Proof of Koml\'os's conjecture on Hamiltonian subsets

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    Koml\'os conjectured in 1981 that among all graphs with minimum degree at least dd, the complete graph Kd+1K_{d+1} minimises the number of Hamiltonian subsets, where a subset of vertices is Hamiltonian if it contains a spanning cycle. We prove this conjecture when dd is sufficiently large. In fact we prove a stronger result: for large dd, any graph GG with average degree at least dd contains almost twice as many Hamiltonian subsets as Kd+1K_{d+1}, unless GG is isomorphic to Kd+1K_{d+1} or a certain other graph which we specify.Comment: 33 pages, to appear in Proceedings of the London Mathematical Societ

    Surfaces, Tree-Width, Clique-Minors, and Partitions

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    In 1971, Chartrand, Geller, and Hedetniemi conjectured that the edge set of a planar graph may be partitioned into two subsets, each of which induces an outerplanar graph. Some partial results towards this conjecture are presented. One such result, in which a planar graph may be thus edge partitioned into two series-parallel graphs, has nice generalizations for graphs embedded onto an arbitrary surface and graphs with no large clique-minor. Several open questions are raised. © 2000 Academic Press

    Mobile vs. point guards

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    We study the problem of guarding orthogonal art galleries with horizontal mobile guards (alternatively, vertical) and point guards, using "rectangular vision". We prove a sharp bound on the minimum number of point guards required to cover the gallery in terms of the minimum number of vertical mobile guards and the minimum number of horizontal mobile guards required to cover the gallery. Furthermore, we show that the latter two numbers can be calculated in linear time.Comment: This version covers a previously missing case in both Phase 2 &

    Algorithms and complexity analyses for some combinational optimization problems

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    The main focus of this dissertation is on classical combinatorial optimization problems in two important areas: scheduling and network design. In the area of scheduling, the main interest is in problems in the master-slave model. In this model, each machine is either a master machine or a slave machine. Each job is associated with a preprocessing task, a slave task and a postprocessing task that must be executed in this order. Each slave task has a dedicated slave machine. All the preprocessing and postprocessing tasks share a single master machine or the same set of master machines. A job may also have an arbitrary release time before which the preprocessing task is not available to be processed. The main objective in this dissertation is to minimize the total completion time or the makespan. Both the complexity and algorithmic issues of these problems are considered. It is shown that the problem of minimizing the total completion time is strongly NP-hard even under severe constraints. Various efficient algorithms are designed to minimize the total completion time under various scenarios. In the area of network design, the survivable network design problems are studied first. The input for this problem is an undirected graph G = (V, E), a non-negative cost for each edge, and a nonnegative connectivity requirement ruv for every (unordered) pair of vertices &ruv. The goal is to find a minimum-cost subgraph in which each pair of vertices u,v is joined by at least ruv edge (vertex)-disjoint paths. A Polynomial Time Approximation Scheme (PTAS) is designed for the problem when the graph is Euclidean and the connectivity requirement of any point is at most 2. PTASs or Quasi-PTASs are also designed for 2-edge-connectivity problem and biconnectivity problem and their variations in unweighted or weighted planar graphs. Next, the problem of constructing geometric fault-tolerant spanners with low cost and bounded maximum degree is considered. The first result shows that there is a greedy algorithm which constructs fault-tolerant spanners having asymptotically optimal bounds for both the maximum degree and the total cost at the same time. Then an efficient algorithm is developed which finds fault-tolerant spanners with asymptotically optimal bound for the maximum degree and almost optimal bound for the total cost

    Some results on cubic graphs

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    Pursuing a question of Oxley, we investigate whether the edge set of a graph admits a bipartition so that the contraction of either partite set produces a series-parallel graph. While Oxley\u27s question in general remains unanswered, our investigations led to two graph operations (Chapters 2 and 4) which are of independent interest. We present some partial results toward Oxley\u27s question in Chapter 3. The central results of the dissertation involve an operation on cubic graphs called the switch; in the literature, a similar operation is known as the edge slide. In Chapter 2, the author proves that we can transform, with switches, any connected, cubic graph on n vertices into any other connected, cubic graph on n vertices. Furthermore, connectivity, up to internal 4-connectedness, can be preserved during the operations. In 2007, Demaine, Hajiaghayi, and Mohar proved the following: for a fixed genus g and any integer k greater than or equal to 2, and for every graph G of Euler genus at most g, the edges of G can be partitioned into k sets such that contracting any one of the sets produces a graph of tree-width at most O(g^2 k). In Chapter 3 we sharpen this result, when k=2, for the projective plane (g=1) and the torus (g=2). During early simultaneous investigations of Jaeger\u27s Dual-Hamiltonian conjecture and Oxley\u27s question, we obtained a simple structure theorem on cubic, internally 4-connected graphs. That result is found in Chapter 4
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