Geodetic Number versus Hull Number in $P_3$-Convexity

• Published in: SIAM journal on discrete mathematics Vol. 27; no. 2; pp. 717 - 731
• Main Authors: Centeno, C. C., Penso, L. D., Rautenbach, D., Pereira de Sá, V. G.
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2013
• Subjects: Character recognition; Construction; Equality; Graphs; Hulls; Hulls (structures); Iterative methods; Mathematical analysis; Network topologies; Recognition; Terminology
• ISSN: 0895-4801; 1095-7146
• DOI: 10.1137/110859014

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 which iteratively adds to $U$ all vertices outside of $U$ having 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 $h(G)=g(G)$, allowing for the constructive characterization as well as the efficient recognition of all such graphs that are triangle-free. Furthermore, we characterize---in terms of forbidden induced subgraphs---the graphs $G$ that satisfy $h(G')=g(G')$ for every induced subgraph $G'$ of $G$. [PUBLICATION ABSTRACT]