19 research outputs found

    On Two-Path Convexity in Multipartite Tournaments

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    Abstract In the context of two-path convexity, we study the rank, Helly number, Radon number, Caratheodory number, and hull number for multipartite tournaments. We show the maximum Caratheodory number of a multipartite tournament is 3. We then derive tight upper bounds for rank in both general multipartite tournaments and clone-free multipartite tournaments. We show that these same tight upper bounds hold for the Helly number, Radon number, and hull number. We classify all clone-free multipartite tournaments of maximum Helly number, Radon number, hull number, and rank. Finally we determine all convexly independent sets of clone-free multipartite tournaments of maximum rank

    2006 (Fall)

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    Abstracts of the talks givn at the 2006 Fall Colloquium

    On the hull and interval numbers of oriented graphs

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    In this work, for a given oriented graph DD, we study its interval and hull numbers, denoted by in(D){in}(D) and hn(D){hn}(D), respectively, in the geodetic, P3{P_3} and P3∗{P_3^*} convexities. This last one, we believe to be formally defined and first studied in this paper, although its undirected version is well-known in the literature. Concerning bounds, for a strongly oriented graph DD, we prove that hng(D)≤m(D)−n(D)+2{hn_g}(D)\leq m(D)-n(D)+2 and that there is a strongly oriented graph such that hng(D)=m(D)−n(D){hn_g}(D) = m(D)-n(D). We also determine exact values for the hull numbers in these three convexities for tournaments, which imply polynomial-time algorithms to compute them. These results allows us to deduce polynomial-time algorithms to compute hnP3(D){hn_{P_3}}(D) when the underlying graph of DD is split or cobipartite. Moreover, we provide a meta-theorem by proving that if deciding whether ing(D)≤k{in_g}(D)\leq k or hng(D)≤k{hn_g}(D)\leq k is NP-hard or W[i]-hard parameterized by kk, for some i∈Z+∗i\in\mathbb{Z_+^*}, then the same holds even if the underlying graph of DD is bipartite. Next, we prove that deciding whether hnP3(D)≤k{hn_{P_3}}(D)\leq k or hnP3∗(D)≤k{hn_{P_3^*}}(D)\leq k is W[2]-hard parameterized by kk, even if the underlying graph of DD is bipartite; that deciding whether inP3(D)≤k{in_{P_3}}(D)\leq k or inP3∗(D)≤k{in_{P_3^*}}(D)\leq k is NP-complete, even if DD has no directed cycles and the underlying graph of DD is a chordal bipartite graph; and that deciding whether inP3(D)≤k{in_{P_3}}(D)\leq k or inP3∗(D)≤k{in_{P_3^*}}(D)\leq k is W[2]-hard parameterized by kk, even if the underlying graph of DD is split. We also argue that the interval and hull numbers in the oriented P3P_3 and P3∗P_3^* convexities can be computed in polynomial time for graphs of bounded tree-width by using Courcelle's theorem

    Master index to volumes 251-260

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    NÚMERO ENVOLTÓRIO NA CONVEXIDADE P3: RESULTADOS E APLICAÇÕES

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    Este artigo apresenta uma revisão sistemática da literatura sobre os resultados e aplicações do número envoltório na convexidade P3 em grafos. A determinação deste parâmetro é equivalente ao problema de se encontrar o menor número de vértices de um grafo que permitam disseminar uma informação, influência, ou contaminação, para todos os vértices do grafo. Em particular, esta revisão descreve um panorama sobre estudos teóricos e aplicados acerca do número envoltório P3 considerando a modelagem de fenômenos sociais. Os resultados mostram que o parâmetro é pouco explorado em sociologia computacional para a modelagem de fenômenos sociais. Por outro lado, com o surgimento das redes sociais, pesquisas teóricas têm sido impulsionadas nas últimas décadas. Pesquisadores têm direcionado esforços com o objetivo de contribuir para a solução de problemas relacionados à influência social e disseminação de informação. Entretanto, ainda há espaço para estudos envolvendo o número envoltório na convexidade P3

    Math Department Newsletter, 2007

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    Subject Index Volumes 1–200

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    On sufficient conditions for Hamiltonicity in dense graphs

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    We study structural conditions in dense graphs that guarantee the existence of vertex-spanning substructures such as Hamilton cycles. It is easy to see that every Hamiltonian graph is connected, has a perfect fractional matching and, excluding the bipartite case, contains an odd cycle. Our main result in turn states that any large enough graph that robustly satisfies these properties must already be Hamiltonian. Moreover, the same holds for embedding powers of cycles and graphs of sublinear bandwidth subject to natural generalisations of connectivity, matchings and odd cycles. This solves the embedding problem that underlies multiple lines of research on sufficient conditions for Hamiltonicity in dense graphs. As applications, we recover and establish Bandwidth Theorems in a variety of settings including Ore-type degree conditions, P\'osa-type degree conditions, deficiency-type conditions, locally dense and inseparable graphs, multipartite graphs as well as robust expanders
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