The Θ-graph Θ(G) of a partial cube G is the intersection graph of the equivalence classes of the Djoković-Winkler relation. Θ-graphs that are 2-connected, trees, or complete graphs are characterized. In particular, Θ(G) is complete if and only if G can be obtained from K₁ by a sequence of (newly introduced) dense expansions. Θ-graphs are also compared with familiar concepts of crossing graphs and τ-graphs.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1363,
author = {Sandi Klav\v zar and Matjaz Kovse},
title = {On $\theta$-graphs of partial cubes},
journal = {Discussiones Mathematicae Graph Theory},
volume = {27},
year = {2007},
pages = {313-321},
zbl = {1135.05063},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1363}
}
Sandi Klavžar; Matjaz Kovse. On θ-graphs of partial cubes. Discussiones Mathematicae Graph Theory, Tome 27 (2007) pp. 313-321. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1363/
[000] [1] B. Bresar, Coloring of the Θ-graph of a median graph, Problem 2005.3, Maribor Graph Theory Problems. http://www-mat.pfmb.uni-mb.si/personal/klavzar/MGTP/index.html.
[001] [2] B. Bresar, W. Imrich and S. Klavžar, Tree-like isometric subgraphs of hypercubes, Discuss. Math. Graph Theory 23 (2003) 227-240, doi: 10.7151/dmgt.1199. | Zbl 1055.05129
[002] [3] B. Bresar and S. Klavžar, Crossing graphs as joins of graphs and Cartesian products of median graphs, SIAM J. Discrete Math. 21 (2007) 26-32, doi: 10.1137/050622997. | Zbl 1141.05071
[003] [4] B. Bresar and T. Kraner Sumenjak, Θ-graphs of partial cubes and strong edge colorings, Ars Combin., to appear. | Zbl 1224.05160
[004] [5] V.D. Chepoi, d-Convexity and isometric subgraphs of Hamming graphs, Cybernetics 1 (1988) 6-9, doi: 10.1007/BF01069520.
[005] [6] M.M. Deza and M. Laurent, Geometry of cuts and metrics, Algorithms and Combinatorics, 15 (Springer-Verlag, Berlin, 1997). | Zbl 0885.52001
[006] [7] D. Djoković, Distance preserving subgraphs of hypercubes, J. Combin. Theory (B) 14 (1973) 263-267, doi: 10.1016/0095-8956(73)90010-5. | Zbl 0245.05113
[007] [8] R.J. Faudree, A. Gyarfas, R.H. Schelp and Z. Tuza, Induced matchings in bipartite graphs, Discrete Math. 78 (1989) 83-87, doi: 10.1016/0012-365X(89)90163-5. | Zbl 0709.05026
[008] [9] W. Imrich and S. Klavžar, A convexity lemma and expansion procedures for bipartite graphs, European J. Combin. 19 (1998) 677-685, doi: 10.1006/eujc.1998.0229. | Zbl 0918.05085
[009] [10] W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition (Wiley, New York, 2000).
[010] [11] S. Klavžar and M. Kovse, Partial cubes and their τ-graphs, European J. Combin. 28 (2007) 1037-1042. | Zbl 1120.05027
[011] [12] S. Klavžar and H.M. Mulder, Partial cubes and crossing graphs, SIAM J. Discrete Math. 15 (2002) 235-251, doi: 10.1137/S0895480101383202. | Zbl 1003.05089
[012] [13] F.R. McMorris, H.M. Mulder and F.R. Roberts, The median procedure on median graphs, Discrete Appl. Math. 84 (1998) 165-181, doi: 10.1016/S0166-218X(98)00003-1. | Zbl 0906.05023
[013] [14] H.M. Mulder, The structure of median graphs, Discrete Math. 24 (1978) 197-204, doi: 10.1016/0012-365X(78)90199-1. | Zbl 0394.05038
[014] [15] H.M. Mulder, The Interval Function of a Graph (Math. Centre Tracts 132, Mathematisch Centrum, Amsterdam, 1980). | Zbl 0446.05039
[015] [16] M. van de Vel, Theory of Convex Structures (North-Holland, Amsterdam, 1993).
[016] [17] A. Vesel, Characterization of resonance graphs of catacondensed hexagonal graphs, MATCH Commun. Math. Comput. Chem. 53 (2005) 195-208. | Zbl 1077.05088
[017] [18] P. Winkler, Isometric embeddings in products of complete graphs, Discrete Appl. Math. 7 (1984) 221-225, doi: 10.1016/0166-218X(84)90069-6. | Zbl 0529.05055