Geodesic
Helly and strong Helly numbers of $B_k$-EPG and $B_k$-VPG graphs
Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 4, pp. 1237-1252 Cet article a éte moissonné depuis la source Library of Science

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EPG graphs were introduced by Golumbic, Lypshteyn, and Stern in 2009 and consist of the intersection graphs of sets of paths on the orthogonal grid, whose intersections are taken considering the edges of the paths. If the intersections of the paths consider the vertices and not the edges, the resulting graph class is called VPG graphs. A path P is a B_k-path if it contains at most k bends. B_k-EPG and B_k-VPG graphs are the intersection graphs of B_k-paths on the orthogonal grid, considering the intersection of edges and vertices, respectively. A family ℱ is h-Helly when every h-intersecting subfamily ℱ' of it satisfies core(ℱ')≠∅. If for every subfamily ℱ' of ℱ, there are h subsets whose core equals the core of ℱ', then ℱ is said to be strong h-Helly. The Helly number of the family ℱ is the least integer h, such that ℱ is h-Helly. Similarly, the strong Helly number of ℱ is the least h, for which ℱ is strong h-Helly. In this paper, we solve the problem of determining both the Helly and strong Helly numbers, for B_k-EPG, and B_k-VPG graphs, for each value k.
Keywords: EPG, VPG, path, grid, bend, Helly number, strong Helly 
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Bornstein, Claudson; Morgenstern, Gila; Santos, Tanilson D.; Souza, Ueverton S.; Szwarcfiter, Jayme L. Helly and strong Helly numbers of $B_k$-EPG and $B_k$-VPG graphs. Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 4, pp. 1237-1252. http://geodesic.mathdoc.fr/item/DMGT_2023_43_4_a19/

[1] A. Asinowski, E. Cohen, M.C. Golumbic, V. Limouzy, M. Lipshteyn and M. Stern, String graphs of k-bend paths on a grid, Electron. Notes Discrete Math. 37 (2011) 141–146. https://doi.org/10.1016/j.endm.2011.05.025

[2] A. Asinowski, E. Cohen, M.C. Golumbic, V. Limouzy, M. Lipshteyn and M. Stern, Vertex intersection graphs of paths on a grid, J. Graph Algorithms Appl. 16 (2012) 129–150. https://doi.org/10.7155/jgaa.00253

[3] C. Berge and P. Duchet, A generalization of Gilmore's theorem, in: Recent Advances in Graph Theory, M. Fiedler (Ed(s)), (Acad. Praha, Prague 1975) 49–55.

[4] C.F. Bornstein, M.C. Golumbic, T.D. Santos, U.S. Souza and J.L. Szwarcfiter, The complexity of Helly-B1 EPG graph recognition, Discrete Math. Theor. Comput. Sci. 22 (2020) #19. https://doi.org/10.23638/DMTCS-22-1-19

[5] M.C. Dourado, M.C. Lin, F. Protti and J.L. Szwarcfiter, Improved algorithms for recognizing p-Helly and hereditary p-Helly hypergraphs, Inform. Process. Lett. 108 (2008) 247–250. https://doi.org/10.1016/j.ipl.2008.05.013

[6] M.C. Dourado, F. Protti and J.L. Szwarcfiter, On the strong p-Helly property, Discrete Appl. Math. 156 (2008) 1053–1057. https://doi.org/10.1016/j.dam.2007.05.047

[7] M.C. Dourado, F. Protti and J.L. Szwarcfiter, Complexity aspects of the Helly property: graphs and hypergraphs, Electron. J. Combin. (2009) #DS17. https://doi.org/10.37236/38

[8] P. Duchet, Proprieté de Helly et problèmes de représentations, Problémes Combinatoires et Théorie de Graphs, Colloquium International CNRS 260 (1978) 117–118.

[9] M.C. Golumbic and R.E. Jamison, The edge intersection graphs of paths in a tree, J. Combin. Theory Ser. B 38 (1985) 8–22. https://doi.org/10.1016/0095-8956(85)90088-7

[10] M.C. Golumbic, M. Lipshteyn and M. Stern, Edge intersection graphs of single bend paths on a grid, Networks 54 (2009) 130–138. https://doi.org/10.1002/net.20305

[11] M.C. Golumbic, M. Lipshteyn and M. Stern, Single bend paths on a grid have strong Helly number 4: errata atque emendationes ad ``edge intersection graphs of single bend paths on a grid'', Networks 62 (2013) 161–163. https://doi.org/10.1002/net.21509

[12] M.C. Golumbic and G. Morgenstern, Edge intersection graphs of paths on a grid, in: 50 Years of Combinatorics, Graph Theory, and Computing, F. Chung, R. Graham, F. Hoffman, L. Hogben, R. Mullin and D. West (Ed(s)), (Chapman and Hall/CRC 2019) 193–209.

[13] J. Kratochv{'\i}l, String graphs. II.} Recognizing string graphs is NP-hard, J. Combin. Theory Ser. B 52 (1991) 67–78. https://doi.org/10.1016/0095-8956(91)90091-W

[14] M. Schaefer, E. Sedgwick and D. Štefankovič, Recognizing string graphs in NP, J. Comput. System Sci. 67 (2003) 365–380. https://doi.org/10.1016/S0022-0000(03)00045-X