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.
Mots clés : convexity, geodetic set, hull set, graph classes
@article{RO_2014__48_4_497_0,
author = {Ekim, T{\i}naz and Erey, Aysel},
title = {Block decomposition approach to compute a minimum geodetic set},
journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
pages = {497--507},
publisher = {EDP-Sciences},
volume = {48},
number = {4},
year = {2014},
doi = {10.1051/ro/2014019},
mrnumber = {3264390},
zbl = {1301.05100},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ro/2014019/}
}
TY - JOUR AU - Ekim, Tınaz AU - Erey, Aysel TI - Block decomposition approach to compute a minimum geodetic set JO - RAIRO - Operations Research - Recherche Opérationnelle PY - 2014 SP - 497 EP - 507 VL - 48 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ro/2014019/ DO - 10.1051/ro/2014019 LA - en ID - RO_2014__48_4_497_0 ER -
%0 Journal Article %A Ekim, Tınaz %A Erey, Aysel %T Block decomposition approach to compute a minimum geodetic set %J RAIRO - Operations Research - Recherche Opérationnelle %D 2014 %P 497-507 %V 48 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ro/2014019/ %R 10.1051/ro/2014019 %G en %F RO_2014__48_4_497_0
Ekim, Tınaz; Erey, Aysel. Block decomposition approach to compute a minimum geodetic set. RAIRO - Operations Research - Recherche Opérationnelle, Tome 48 (2014) no. 4, pp. 497-507. doi : 10.1051/ro/2014019. http://www.numdam.org/articles/10.1051/ro/2014019/
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