5 research outputs found

    Protecting a Graph with Mobile Guards

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    Mobile guards on the vertices of a graph are used to defend it against attacks on either its vertices or its edges. Various models for this problem have been proposed. In this survey we describe a number of these models with particular attention to the case when the attack sequence is infinitely long and the guards must induce some particular configuration before each attack, such as a dominating set or a vertex cover. Results from the literature concerning the number of guards needed to successfully defend a graph in each of these problems are surveyed.Comment: 29 pages, two figures, surve

    Solving Hard Graph Problems with Combinatorial Computing and Optimization

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    Many problems arising in graph theory are difficult by nature, and finding solutions to large or complex instances of them often require the use of computers. As some such problems are NPNP-hard or lie even higher in the polynomial hierarchy, it is unlikely that efficient, exact algorithms will solve them. Therefore, alternative computational methods are used. Combinatorial computing is a branch of mathematics and computer science concerned with these methods, where algorithms are developed to generate and search through combinatorial structures in order to determine certain properties of them. In this thesis, we explore a number of such techniques, in the hopes of solving specific problem instances of interest. Three separate problems are considered, each of which is attacked with different methods of combinatorial computing and optimization. The first, originally proposed by ErdH{o}s and Hajnal in 1967, asks to find the Folkman number Fe(3,3;4)F_e(3,3;4), defined as the smallest order of a K4K_4-free graph that is not the union of two triangle-free graphs. A notoriously difficult problem associated with Ramsey theory, the best known bounds on it prior to this work were 19leqFe(3,3;4)leq94119 leq F_e(3,3;4) leq 941. We improve the upper bound to Fe(3,3;4)leq786F_e(3,3;4) leq 786 using a combination of known methods and the Goemans-Williamson semi-definite programming relaxation of MAX-CUT. The second problem of interest is the Ramsey number R(C4,Km)R(C_4,K_m), which is the smallest nn such that any nn-vertex graph contains a cycle of length four or an independent set of order mm. With the help of combinatorial algorithms, we determine R(C4,K9)=30R(C_4,K_9)=30 and R(C4,K10)=36R(C_4,K_{10})=36 using large-scale computations on the Open Science Grid. Finally, we explore applications of the well-known Lenstra-Lenstra-Lov\u27{a}sz (LLL) algorithm, a polynomial-time algorithm that, when given a basis of a lattice, returns a basis for the same lattice with relatively short vectors. The main result of this work is an application to graph domination, where certain hard instances are solved using this algorithm as a heuristic

    Annual Report Of Research and Creative Productions, January to December, 2006

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    2006 Annual Report of Research and Creative Productions, Morehead State University, Division of Academic Affairs, Research and Creative Productions Committee
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