Immediate versus Eventual Conversion: Comparing Geodetic and Hull Numbers in P3-Convexity
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.
KeywordsHull number geodetic number P3-convexity irreversible 2-threshold processes triangle-free graphs
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