Monitoring Edge-Geodetic Sets in Graphs: Extremal Graphs, Bounds, Complexity
Authors: 
Florent Foucaud, Pierre-Marie Marcille, Zin Mar Myint, R. B. Sandeep, Sagnik Sen, S. TaruniAuthors Info & Claims
Florent Foucaud, Pierre-Marie Marcille, Zin Mar Myint, R. B. Sandeep, Sagnik Sen, S. TaruniAuthors Info & ClaimsPages 29 - 43
Abstract
A monitoring edge-geodetic set, or simply an MEG-set, of a graph G is a vertex subset such that given any edge e of G, e lies on every shortest u-v path of G, for some . The monitoring edge-geodetic number of G, denoted by meg(G), is the minimum cardinality of such an MEG-set. This notion provides a graph theoretic model of the network monitoring problem.
In this article, we compare meg(G) with some other graph theoretic parameters stemming from the network monitoring problem and provide examples of graphs having prescribed values for each of these parameters. We also characterize graphs G that have V(G) as their minimum MEG-set, which settles an open problem due to Foucaud et al. (CALDAM 2023). We also provide a general upper bound for meg(G) for sparse graphs in terms of their girth, and later refine the upper bound using the chromatic number of G. We examine the change in meg(G) with respect to two fundamental graph operations: clique-sum and subdivisions. In both cases, we provide a lower and an upper bound of the possible amount of changes and provide (almost) tight examples. Finally, we prove that the decision version of the problem of finding meg(G) is NP-complete even for the family of 3-degenerate, 2 apex graphs, improving the existing result by Haslegrave (Discrete Applied Mathematics 2023).
References
[1]
Atici M On the edge geodetic number of a graph Int. J. Comput. Math. 2003 80 7 853-861
[2]
Bampas E, Bilò D, Drovandi G, Gualà L, Klasing R, and Proietti G Network verification via routing table queries J. Comput. Syst. Sci. 2015 81 1 234-248
[3]
Beerliova Z et al. Network discovery and verification IEEE J. Sel. Areas Commun. 2006 24 12 2168-2181
[4]
Bejerano, Y., Rastogi, R.: Robust monitoring of link delays and faults in IP networks. In: IEEE INFOCOM 2003. Twenty-second Annual Joint Conference of the IEEE Computer and Communications Societies (IEEE Cat. No. 03CH37428), vol. 1, pp. 134–144. IEEE (2003)
[5]
Bilò D, Erlebach T, Mihalák M, and Widmayer P Discovery of network properties with all-shortest-paths queries Theor. Comput. Sci. 2010 411 14–15 1626-1637
[6]
Chakraborty, D., Das, S., Foucaud, F., Gahlawat, H., Lajou, D., Roy, B.: Algorithms and complexity for geodetic sets on planar and chordal graphs. In: Cao, Y., Cheng, S., Li, M. (eds.), 31st International Symposium on Algorithms and Computation, ISAAC 2020, 14–18 December 2020, Hong Kong, China (Virtual Conference), volume 181 of LIPIcs, pp. 7:1–7:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)
[7]
Chakraborty D, Gahlawat H, and Roy B Algorithms and complexity for geodetic sets on partial grids Theor. Comput. Sci. 2023 979
[8]
Chartrand, G., Harary, F., Zhang, P.: On the geodetic number of a graph. Netw. Int. J. 39(1), 1–6 (2002)
[9]
Chartrand G, Palmer EM, and Zhang P The geodetic number of a graph: a survey Congressus numerantium 2002 156 37-58
[10]
Dall’Asta L, Alvarez-Hamelin I, Barrat A, Vázquez A, and Vespignani A Exploring networks with traceroute-like probes: theory and simulations Theor. Comput. Sci. 2006 355 1 6-24
[11]
Davot T, Isenmann L, and Thiebaut J Chen C-Y, Hon W-K, Hung L-J, and Lee C-W On the approximation hardness of geodetic set and its variants Computing and Combinatorics 2021 Cham Springer 76-88
[12]
Dourado MC, Protti F, Rautenbach D, and Szwarcfiter JL Some remarks on the geodetic number of a graph Discret. Math. 2010 310 4 832-837
[13]
Foucaud F, Kao S-S, Klasing R, Miller M, and Ryan J Monitoring the edges of a graph using distances Discret. Appl. Math. 2022 319 424-438
[14]
Foucaud, F., Narayanan, K., Ramasubramony Sulochana, L.: Monitoring Edge-Geodetic Sets in Graphs. In: Bagchi, A., Muthu, R. (eds.) Algorithms and Discrete Applied Mathematics. CALDAM 2023. LNCS, vol. 13947, pp. 245–256. Springer, Cham (2023).
[15]
Govindan, R., Tangmunarunkit, H.: Heuristics for internet map discovery. In: Proceedings IEEE INFOCOM 2000. Conference on Computer Communications. Nineteenth Annual Joint Conference of the IEEE Computer and Communications Societies (Cat. No. 00CH37064), vol. 3, pp. 1371–1380. IEEE (2000)
[16]
Harary F, Loukakis E, and Tsouros C The geodetic number of a graph Math. Comput. Model. 1993 17 11 89-95
[17]
Haslegrave J Monitoring edge-geodetic sets: hardness and graph products Discret. Appl. Math. 2023 340 79-84
[18]
Iršič V Strong geodetic number of complete bipartite graphs and of graphs with specified diameter Graphs Comb. 2018 34 3 443-456
[19]
Joret G and Wood DR Complete graph minors and the graph minor structure theorem J. Comb. Theory Ser. B 2013 103 1 61-74
[20]
Kellerhals L and Koana T Parameterized complexity of geodetic set J. Graph Algorithms Appl. 2022 26 4 401-419
[21]
Lovász L Graph minor theory Bull. Am. Math. Soc. 2006 43 1 75-86
[22]
Manuel P, Klavžar S, Xavier A, Arokiaraj A, and Thomas E Strong edge geodetic problem in networks Open Math. 2017 15 1 1225-1235
[23]
Robertson, N., Seymour, P.D.: Graph minors: Xvii. taming a vortex. J. Comb. Theory Ser. B 77(1), 162–210 (1999)
[24]
Robertson, N., Seymour, P.D.: Graph minors. xx. wagner’s conjecture. J. Comb. Theory Ser. B 92(2), 325–357 (2004)
[25]
Santhakumaran A and John J Edge geodetic number of a graph J. Discret. Math. Sci. Cryptogr. 2007 10 3 415-432
[26]
Yannakakis M Edge-deletion problems SIAM J. Comput. 1981 10 2 297-309
Index Terms
- Monitoring Edge-Geodetic Sets in Graphs: Extremal Graphs, Bounds, Complexity
Index terms have been assigned to the content through auto-classification.
Recommendations
A note on Reed's conjecture for triangle-free graphs
AbstractReed's Conjecture states that χ ( G ) ≤ ⌈ ( Δ ( G ) + ω ( G ) + 1 ) / 2 ⌉, where χ ( G ), Δ ( G ) and ω ( G ) are the chromatic number, maximum degree and clique number of a graph G, respectively.
In this note, we prove this conjecture for ...
Well-partitioned chordal graphs
AbstractWe introduce a new subclass of chordal graphs that generalizes the class of split graphs, which we call well-partitioned chordal graphs. A connected graph G is a well-partitioned chordal graph if there exist a partition P of the vertex ...
Extremal Edge-Girth-Regular Graphs
AbstractAn edge-girth-regular-graph is a k-regular graph of order v and girth g in which every edge is contained in distinct g-cycles. Edge-girth-regularity is shared by several interesting classes of graphs which include edge- and arc-...
Comments
Information & Contributors
Information
Published In
Feb 2024
337 pages
ISBN:978-3-031-52212-3
DOI:10.1007/978-3-031-52213-0
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.
Publisher
Springer-Verlag
Berlin, Heidelberg
Publication History
Published: 15 February 2024
Author Tags
Qualifiers
- Article
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 0Total Downloads
- Downloads (Last 12 months)0
- Downloads (Last 6 weeks)0
Reflects downloads up to 09 Nov 2024
Other Metrics
Citations
View Options
View options
Get Access
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign in