.. Domination-romaine-faible-sur-les-graphes-d-'intervalles, 99 4.3.1 Description de l'algorithme, p.106

@. Pour-tout-d, ?. {0, and 1. , 2n}, MinR[1][d] et MinR[2][d] sont les indices des deux intervalles x et x tels que x, x ? D >d , x x et r(x), r(x ) sont minimums

@. Pour-tout-d, ?. {0, and 1. , 2n}, MinL[1][d ] et minL[2][d ] sont les indices des deux intervalles ? et ? tels que ?, ? ? D >d , ? ? et l(?), l(? ) sont minimum

. Finalement, est-à-dire qu'il n'existe pas de ? tel que ? y et d < l(?) ? d) peut être fait en

G. Nous-considérons-ici-un-exemple-de-graphe-d-'intervalles, dont le modèle d'intervalles est défini comme suit : ? I = { v 1 , v 2 , v 3 , v 4 , v 5 , v 6 } ; ? l(v 1 ) = 4 ; l(v 2 ) = 10 ; l(v 3 ) = 3 ; l(v 4 ) = 1 ; l(v 5 ) = 8 ; l(v 6 ) = 2 ; ? r(v 1 ) = 7, pp.11-14

. Construction-de-maxr and . Pour-tout-i-?-{1, 6}, MaxR[i] correspond à l'indice de l'intervalle v tel que v ? N [v i ] et r(v) est maximum Connected Tropical Subgraphs in Vertex-Colored Graphs, 9th International Colloquium on Graph Theory and Combinatorics, p.14, 2014.

M. [. Abu-khzam, A. E. Liedloff, and . Mouawad, An exact algorithm for connected red???blue dominating set, Journal of Discrete Algorithms, vol.9, issue.3, pp.252-262, 2011.
DOI : 10.1016/j.jda.2011.03.006
URL : https://hal.archives-ouvertes.fr/hal-00461066

]. C. Ber76 and . Berge, Graphs and Hypergraphs. Graphs and Hypergraphs, 1976.

V. [. Brandstädt, J. P. Le, and . Spinrad, Graph Classes : A Survey, Society for Industrial and Applied Mathematics, 1999.
DOI : 10.1137/1.9780898719796

N. Betzler, R. Van-bevern, M. R. Fellows, C. Komusiewicz, and R. Niedermeier, Parameterized Algorithmics for Finding Connected Motifs in Biological Networks, IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol.8, issue.5, pp.1296-1308, 2011.
DOI : 10.1109/TCBB.2011.19

]. H. Bvr08, J. M. Bodlaender, and . Van-rooij, Design by Measure and Conquer, A Faster Exact Algorithm for Dominating Set, STACS 2008, pp.657-668, 2008.

]. H. Bvr11, J. M. Bodlaender, and . Van-rooij, Exact algorithms for dominating set, Discrete Applied Mathematics, vol.159, issue.17, pp.2147-2164, 2011.

. Ccc-+-13-]-m, M. Chapelle, J. Cochefert, D. Couturier, M. Kratsch et al., Exact Algorithms for Weak Roman Domination, 24th International Workshop On Combinatorial Algorithms, IWOCA '13, pp.81-93, 2013.

M. Chapelle, M. Cochefert, D. Kratsch, R. Letourneur, and M. Liedloff, Exact Exponential Algorithms to Find a Tropical Connected Set of Minimum Size, 9th International Symposium on Parameterized and Exact Computation, pp.147-158, 2014.
DOI : 10.1007/978-3-319-13524-3_13
URL : https://hal.archives-ouvertes.fr/hal-01105083

. J. Bibliographie-[-cdjhh04-]-e, P. A. Cockayne, S. M. Dreyer-jr, S. T. Hedetniemi, and . Hedetniemi, Roman Domination in Graphs, Discrete Mathematics, vol.278, pp.1-311, 2004.

O. [. Cockayne and . Favaron, Secure domination, weak roman domination and forbidden subgraphs, Bulletin of the Institute of Combinatorics and its Applications, 2003.

. J. Cgg-+-05-]-e, P. J. Cockayne, W. R. Grobler, J. Gründlingh, J. H. Munganga et al., Protection of a Graph, Utilitas Mathematica, vol.67, pp.19-32, 2005.

J. Couturier, P. Heggernes, P. Van-'t-hof, and D. Kratsch, Minimal Dominating Sets in Graph Classes: Combinatorial Bounds and Enumeration, 38th International Conference on Current Trends in Theory and Practice of Computer Science, pp.202-213, 2012.
DOI : 10.1137/0210022

J. Couturier, P. Heggernes, P. Van-'t-hof, and D. Kratsch, Minimal Dominating Sets in Graph Classes: Combinatorial Bounds and Enumeration, Theoretical Computer Science, vol.10, issue.3, pp.82-94, 2013.
DOI : 10.1137/0210022

E. W. Chambers, W. B. Kinnersley, N. Prince, and D. B. West, Extremal Problems for Roman Domination, SIAM Journal on Discrete Mathematics, vol.23, issue.3, pp.1575-1586, 2009.
DOI : 10.1137/070699688

[. Couturier, R. Letourneur, and M. Liedloff, On the number of minimal dominating sets on some graph classes, Theoretical Computer Science, vol.562, pp.634-642, 2015.
DOI : 10.1016/j.tcs.2014.11.006
URL : https://hal.archives-ouvertes.fr/hal-01105097

]. S. Coo71 and . Cook, The complexity of theorem-proving procedures, Proceedings of the Third Annual ACM Symposium on Theory of Computing, STOC '71, pp.151-158, 1971.

]. B. Cou90 and . Courcelle, The monadic second-order logic of graphs. recognizable sets of finite graphs, Information and Computation, vol.85, issue.1, pp.12-75, 1990.

G. [. Dondi, S. Fertin, and . Vialette, Complexity issues in vertex-colored graph pattern matching, Journal of Discrete Algorithms, vol.9, issue.1, pp.82-99, 2011.
DOI : 10.1016/j.jda.2010.09.002
URL : https://hal.archives-ouvertes.fr/hal-00606154

M. [. Downey, A. Fellows, G. Vardy, and . Whittle, The Parametrized Complexity of Some Fundamental Problems in Coding Theory, SIAM Journal on Computing, vol.29, issue.2, pp.545-570, 1999.
DOI : 10.1137/S0097539797323571

M. [. Demaine and . Hajiaghayi, The Bidimensionality Theory and Its Algorithmic Applications, The Computer Journal, vol.51, issue.3, pp.292-302, 2008.
DOI : 10.1093/comjnl/bxm033

]. R. Die12 and . Diestel, Graph Theory of Graduate texts in mathematics, 2012.

]. H. Fer08 and . Fernau, Roman domination : a parameterized perspective, International Journal of Computer Mathematics, vol.85, issue.1, pp.25-38, 2008.

. R. Bibliographie-[-ffhv07-]-m, G. Fellows, D. Fertin, S. Hermelin, and . Vialette, Sharp Tractability Borderlines for Finding Connected Motifs in Vertex-Colored Graphs, Automata, Languages and Programming, pp.340-351, 2007.

M. R. Fellows, G. Fertin, D. Hermelin, and S. Vialette, Upper and lower bounds for finding connected motifs in vertex-colored graphs, Journal of Computer and System Sciences, vol.77, issue.4, pp.799-811, 2011.
DOI : 10.1016/j.jcss.2010.07.003
URL : https://hal.archives-ouvertes.fr/hal-00606148

F. [. Fomin, D. Grandoni, and . Kratsch, Measure and Conquer: Domination ??? A Case Study, Automata, Languages and Programming, pp.191-203, 2005.
DOI : 10.1007/11523468_16

F. [. Fomin, D. Grandoni, and . Kratsch, Measure and Conquer : A Simple O(2.0288?n) Independent Set Algorithm, Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm, SODA '06, pp.18-25, 2006.

F. [. Fomin, D. Grandoni, and . Kratsch, A measure & conquer approach for the analysis of exact algorithms, Journal of the ACM, vol.56, issue.5, 2009.
DOI : 10.1145/1552285.1552286

. V. Fgk-+-13-]-f, F. Fomin, D. Grandoni, D. Kratsch, S. Lokshtanov et al., Computing optimal steiner trees in polynomial space, Algorithmica, vol.65, issue.3, pp.584-604, 2013.

S. [. Fomin, A. V. Gaspers, I. Pyatkin, and . Razgon, On the Minimum Feedback Vertex Set Problem: Exact and Enumeration Algorithms, Algorithmica, vol.117, issue.2, pp.293-307, 2008.
DOI : 10.1007/s00453-007-9152-0

F. V. Fomin, F. Grandoni, A. V. Pyatkin, and A. A. Stepanov, Combinatorial bounds via measure and conquer, ACM Transactions on Algorithms, vol.5, issue.1, 2008.
DOI : 10.1145/1435375.1435384

. V. Fhk-+-11-]-f, P. Fomin, D. Heggernes, C. Kratsch, Y. Papadopoulos et al., Enumerating Minimal Subset Feedback Vertex Sets, WADS, pp.399-410, 2011.

D. [. Fomin and . Kratsch, Exact Exponential Algorithms, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00085561

H. Fernau, J. Kneis, D. Kratsch, A. Langer, M. Liedloff et al., An Exact Algorithm for the Maximum Leaf Spanning Tree Problem, Parameterized and Exact Computation, pp.161-172, 2009.
DOI : 10.1007/978-3-642-11269-0_13
URL : https://hal.archives-ouvertes.fr/hal-00460824

. V. Bibliographie-[-fkw04-]-f, D. Fomin, G. J. Kratsch, and . Woeginger, Exact (Exponential) Algorithms for the Dominating Set Problem, Graph-Theoretic Concepts in Computer Science, number 3353 in Lecture Notes in Computer Science, pp.245-256, 2004.

V. [. Fernandes, M. Lacroix, and . Sagot, Motif search in graphs : application to metabolic networks, Computational Biology and Bioinformatics IEEE/ACM Transactions on, vol.3, issue.4, pp.360-368, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00427984

D. [. Fomin and . Thilikos, A Simple and Fast Approach for Solving Problems on Planar Graphs, STACS 2004, number 2996 in Lecture Notes in Computer Science, pp.56-67, 2004.
DOI : 10.1007/978-3-540-24749-4_6

Y. [. Fomin and . Villanger, Finding Induced Subgraphs via Minimal Triangulations, STACS, pp.383-394, 2010.
URL : https://hal.archives-ouvertes.fr/inria-00455360

W. Goddard, S. M. Hedetniemi, and S. T. Hedetniemi, Eternal security in graphs, J. Combin. Math. Combin. Comput, vol.52, pp.169-180, 2005.

D. [. Garey and . Johnson, Computers and Intractability : A Guide to the Theory of NP-Completeness, 1979.

W. [. Goldwasser and . Klostermeyer, Tight bounds for eternal dominating sets in graphs, Discrete Mathematics, vol.308, issue.12, pp.2589-2593, 2008.
DOI : 10.1016/j.disc.2007.06.005

C. [. Grobler and . Mynhardt, Secure domination critical graphs, Discrete Mathematics, vol.309, issue.19, pp.5820-5827, 2009.
DOI : 10.1016/j.disc.2008.05.050

]. M. Gol04 and . Golumbic, Algorithmic Graph Theory and Perfect Graphs : Second Edition, Annals of Discrete Mathematics. Elsevier Science, 2004.

]. F. Gra04 and . Grandoni, Exact Algorithms for Hard Graph Problems, 2004.

]. F. Gra06 and . Grandoni, A note on the complexity of minimum dominating set, Journal of Discrete Algorithms, vol.4, issue.2, pp.209-214, 2006.

F. [. Guillemot and . Sikora, Finding and counting vertex-colored subtrees, In Mathematical Foundations of Computer Science, pp.405-416, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00455134

M. [. Hedetniemi and . Henning, Defending the Roman Empire?A new strategy, Discrete Mathematics, vol.266, issue.1-3, pp.239-251, 2003.

T. W. Haynes, S. T. Hedetniemi, and P. J. Slater, Domination in Graphs, 1998.
DOI : 10.1201/b16132-65

T. W. Haynes, S. T. Hedetniemi, and P. J. Slater, Fundamentals of domination in graphs, 1998.

. [. Bibliographie, R. M. Ideker, J. Karp, R. Scott, and . Sharan, Efficient algorithms for detecting signaling pathways in protein interaction networks, Journal of Computational Biology, vol.13, issue.2, pp.133-144, 2006.

R. [. Impagliazzo and . Paturi, Complexity of k-SAT, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317), pp.367-375, 2001.
DOI : 10.1109/CCC.1999.766282

R. [. Impagliazzo, F. Paturi, and . Zane, Which problems have strongly exponential complexity?, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280), pp.512-530, 2001.
DOI : 10.1109/SFCS.1998.743516

]. Y. Iwa11 and . Iwata, A Faster Algorithm for Dominating Set Analyzed by the Potential Method

]. R. Kar72 and . Karp, Reducibility among Combinatorial Problems The IBM Research Symposia Series, Complexity of Computer Computations, pp.85-103, 1972.

T. Kloks, M. Liedloff, J. Liu, and S. Peng, Efficient algorithms for Roman domination on some classes of graphs, Discrete Applied Mathematics, vol.156, issue.18, pp.3400-3415, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00460842

. M. Klm-+-13-]-m, V. Kanté, A. Limouzy, L. Mary, T. Nourine et al., On the Enumeration and Counting of Minimal Dominating sets in Interval and Permutation Graphs, ISAAC, pp.339-349, 2013.

M. M. Kanté, V. Limouzy, A. Mary, and L. Nourine, Enumeration of Minimal Dominating Sets and Variants, FCT, pp.298-309, 2011.
DOI : 10.1023/A:1016783217662

M. M. Kanté, V. Limouzy, A. Mary, and L. Nourine, On the Neighbourhood Helly of Some Graph Classes and Applications to the Enumeration of Minimal Dominating Sets, ISAAC, pp.289-298, 2012.
DOI : 10.1007/978-3-642-35261-4_32

]. M. Krz13 and . Krzywkowski, Trees having many minimal dominating sets, Information Processing Letters, vol.113, issue.8, pp.276-279, 2013.

]. M. Lie08 and . Liedloff, Finding a dominating set on bipartite graphs, Inf. Process. Lett, vol.107, issue.5, pp.154-157, 2008.

R. [. Lipton and . Tarjan, Applications of a Planar Separator Theorem, SIAM Journal on Computing, vol.9, issue.3, pp.615-627, 1980.
DOI : 10.1137/0209046

M. [. Lonc and . Truszczynski, On the number of minimal transversals in 3-uniform hypergraphs, Discrete Mathematics, vol.308, issue.16, pp.3668-3687, 2008.
DOI : 10.1016/j.disc.2007.07.024

M. J. Bibliographie, L. Moon, and . Moser, On cliques in graphs, Israel Journal of Mathematics, vol.3, issue.1, pp.23-28, 1965.

P. [. Mai and . Pushpam, Weak Roman domination in graphs, Discussiones Mathematicae Graph Theory, vol.31, issue.1, pp.161-170, 2011.

T. [. Mcmorris, T. Warnow, and . Wimer, Triangulating Vertex-Colored Graphs, SIAM Journal on Discrete Mathematics, vol.7, issue.2, pp.296-306, 1994.
DOI : 10.1137/S0895480192229273

]. J. Ned12 and . Nederlof, Fast Polynomial-Space Algorithms Using Inclusion-Exclusion, Algorithmica, vol.65, issue.4, pp.868-884, 2012.

J. Nederlof, T. C. Van-dijk, and J. M. Van-rooij, Inclusion/Exclusion Meets Measure and Conquer, Algorithmica, vol.142, issue.1???3, pp.554-565, 2009.
DOI : 10.1007/s00453-013-9759-2

]. C. Pap07 and . Papadimitriou, Computational complexity, 2007.

K. [. Revelle and . Rosing, Defendens Imperium Romanum: A Classical Problem in Military Strategy, The American Mathematical Monthly, vol.107, issue.7, 2000.
DOI : 10.2307/2589113

]. I. Ste99 and . Stewart, Exact exponential-time algorithms for domination problems in graphs, Defend the Roman Empire ! Scientific American J. M. M. van Rooij, vol.6, issue.281, pp.136-139, 1999.

]. D. Wes00 and . West, Introduction to Graph Theory. Pearson, Upper Saddle River, N.J, 2000.

]. P. Wol91 and . Wolper, Introduction à la calculabilité. Collection iia. InterEditions, 1991.