A directed path graph is the intersection graph of a family of directed subpaths of a directed tree. A rooted path graph is the intersection graph of a family of directed subpaths of a rooted tree. Rooted path graphs are directed path graphs. Several characterizations are known for directed path graphs: one by forbidden induced subgraphs and one by forbidden asteroids. It is an open problem to find such characterizations for rooted path graphs. For this purpose, we are studying in this paper directed path graphs that are non rooted path graphs. We prove that such graphs always contain an asteroidal quadruple.
@article{bwmeta1.element.doi-10_7151_dmgt_1821,
author = {Marisa Gutierrez and Benjamin L\'ev\^eque and Silvia B. Tondato},
title = {Asteroidal Quadruples in non Rooted Path Graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {35},
year = {2015},
pages = {603-614},
zbl = {1327.05289},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1821}
}
Marisa Gutierrez; Benjamin Lévêque; Silvia B. Tondato. Asteroidal Quadruples in non Rooted Path Graphs. Discussiones Mathematicae Graph Theory, Tome 35 (2015) pp. 603-614. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1821/
[1] T. Kloks, D. Kratsch, and H. Muller, Asteroidal sets in graphs, Lecture Notes in Comput. Sci. 1335 (1997) 229-241. doi:10.1007/BFb0024501[Crossref] | Zbl 0897.05076
[2] K. Cameron, C.T. Hóang and B. Lévêque, Asteroids in rooted and directed path graphs, Electron. Notes Discrete Math. 32 (2009) 67-74. doi:10.1016/j.endm.2009.02.010[Crossref] | Zbl 1267.05174
[3] K. Cameron, C.T. Hóang and B. Lévêque, Characterizing directed path graphs by forbidden asteroids, J. Graph Theory 68 (2011) 103-112. doi:10.1002/jgt.20543[Crossref][WoS] | Zbl 1230.05139
[4] S. Chaplick, M. Gutierrez, B.Lévêque and S.B.Tondato, From path graphs to di- rected path graphs, Lecture Notes in Comput. Sci. 6410 (2010) 256-265. doi:10.1007/978-3-642-16926-7 24[Crossref] | Zbl 1310.05196
[5] F. Gavril, The intersection graphs of subtrees in trees are exactly the chordal graphs, J. Combin. Theory Ser. B 16 (1974) 47-56. doi:10.1016/0095-8956(74)90094-X[Crossref]
[6] C.G. Lekkerkerker and J.Ch. Boland, Representation of finite graph by a set of intervals on the real line, Fund. Math. (1962) 45-64. | Zbl 0105.17501
[7] B.Lévêque, F. Maffray and M. Preissmann, Characterizing path graphs by forbidden induced subgraphs, J. Graph Theory 62 (2009) 369-384. doi:10.1002/jgt.20407[Crossref] | Zbl 1221.05220
[8] I. Lin, T. McKee and D.B. West, The leafage of a chordal graphs, Discuss. Math. Graph Theory 18 (1998) 23-48. doi:10.7151/dmgt.1061[Crossref] | Zbl 0912.05053
[9] C. Monma and V. Wei, Intersection graphs of paths in a tree, J. Combin. Theory Ser. B 41 (1986) 141-181. doi:10.1016/0095-8956(86)90042-0[Crossref]
[10] B.S. Panda, The forbidden subgraph characterization of directed vertex graphs, Discrete Math. 196 (1999) 239-256. doi:10.1016/S0012-365X(98)00127-7[Crossref]