Volume 48, Number 4, October-December 2014
|Page(s)||497 - 507|
|Published online||10 June 2014|
1 Boğaziçi University, Department of Industrial Engineering,
34342 Bebek, Istanbul, Turkey.
2 Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia B3H 3J5, Canada.
Accepted: 17 March 2014
In this paper, we develop a divide-and-conquer approach, called block decomposition, to solve the minimum geodetic set problem. This provides us with a unified approach for all graphs admitting blocks for which the problem of finding a minimum geodetic set containing a given set of vertices (g-extension problem) can be efficiently solved. Our method allows us to derive linear time algorithms for the minimum geodetic set problem in (a proper superclass of) block-cacti and monopolar chordal graphs. Also, we show that hull sets and geodetic sets of block-cacti are the same, and the minimum geodetic set problem is NP-hard in cobipartite graphs. We conclude by pointing out several interesting research directions.
Mathematics Subject Classification: 05C12 / 05C85
Key words: Convexity / geodetic set / hull set / graph classes
© EDP Sciences, ROADEF, SMAI 2014
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