Discussiones Mathematicae Graph Theory 27(2) (2007)
313-321
DOI: https://doi.org/10.7151/dmgt.1363
ON Θ-GRAPHS OF PARTIAL CUBES
Sandi Klavžar Department of Mathematics and Computer Science |
Matjaz Kovse Institute of Mathematics, Physics and Mechanics |
Abstract
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.Keywords: intersection graph, partial cube, median graph, expansion theorem, Cartesian product of graphs.
2000 Mathematics Subject Classification: 05C75, 05C12.
References
[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 |
[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. |
[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. |
[4] | B. Bresar and T. Kraner Sumenjak, Θ-graphs of partial cubes and strong edge colorings, Ars Combin., to appear. |
[5] | V.D. Chepoi, d-Convexity and isometric subgraphs of Hamming graphs, Cybernetics 1 (1988) 6-9, doi: 10.1007/BF01069520. |
[6] | M.M. Deza and M. Laurent, Geometry of cuts and metrics, Algorithms and Combinatorics, 15 (Springer-Verlag, Berlin, 1997). |
[7] | D. Djoković, Distance preserving subgraphs of hypercubes, J. Combin. Theory (B) 14 (1973) 263-267, doi: 10.1016/0095-8956(73)90010-5. |
[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. |
[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. |
[10] | W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition (Wiley, New York, 2000). |
[11] | S. Klavžar and M. Kovse, Partial cubes and their τ-graphs, European J. Combin. 28 (2007) 1037-1042. |
[12] | S. Klavžar and H.M. Mulder, Partial cubes and crossing graphs, SIAM J. Discrete Math. 15 (2002) 235-251, doi: 10.1137/S0895480101383202. |
[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. |
[14] | H.M. Mulder, The structure of median graphs, Discrete Math. 24 (1978) 197-204, doi: 10.1016/0012-365X(78)90199-1. |
[15] | H.M. Mulder, The Interval Function of a Graph (Math. Centre Tracts 132, Mathematisch Centrum, Amsterdam, 1980). |
[16] | M. van de Vel, Theory of Convex Structures (North-Holland, Amsterdam, 1993). |
[17] | A. Vesel, Characterization of resonance graphs of catacondensed hexagonal graphs, MATCH Commun. Math. Comput. Chem. 53 (2005) 195-208. |
[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. |
Received 25 April 2006
Revised 28 December 2006
Accepted 28 December 2006
Close