Abstract
We study the graphs G for which the hull number h(G) and the geodetic number g(G) with respect to P 3-convexity coincide. These two parameters correspond to the minimum cardinality of a set U of vertices of G such that the simple expansion process that iteratively adds to U, all vertices outside of U that have two neighbors in U, produces the whole vertex set of G either eventually or after one iteration, respectively. We establish numerous structural properties of the graphs G with h(G) = g(G), which allow the constructive characterization as well as the efficient recognition of all triangle-free such graphs. Furthermore, we characterize the graphs G that satisfy h(H) = g(H) for every induced subgraph H of G in terms of forbidden induced subgraphs.
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Centeno, C.C., Penso, L.D., Rautenbach, D., de Sá, V.G.P. (2012). Immediate versus Eventual Conversion: Comparing Geodetic and Hull Numbers in P 3-Convexity. In: Golumbic, M.C., Stern, M., Levy, A., Morgenstern, G. (eds) Graph-Theoretic Concepts in Computer Science. WG 2012. Lecture Notes in Computer Science, vol 7551. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34611-8_27
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DOI: https://doi.org/10.1007/978-3-642-34611-8_27
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-34610-1
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