# A New Characterization of Unichord-Free Graphs

Discussiones Mathematicae Graph Theory (2015)

- Volume: 35, Issue: 4, page 765-771
- ISSN: 2083-5892

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topTerry A. McKee. "A New Characterization of Unichord-Free Graphs." Discussiones Mathematicae Graph Theory 35.4 (2015): 765-771. <http://eudml.org/doc/276023>.

@article{TerryA2015,

abstract = {Unichord-free graphs are defined as having no cycle with a unique chord. They have appeared in several papers recently and are also characterized by minimal separators always inducing edgeless subgraphs (in contrast to characterizing chordal graphs by minimal separators always inducing complete subgraphs). A new characterization of unichord-free graphs corresponds to a suitable reformulation of the standard simplicial vertex characterization of chordal graphs.},

author = {Terry A. McKee},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {chordal graph; unichord-free graph; minimal separator},

language = {eng},

number = {4},

pages = {765-771},

title = {A New Characterization of Unichord-Free Graphs},

url = {http://eudml.org/doc/276023},

volume = {35},

year = {2015},

}

TY - JOUR

AU - Terry A. McKee

TI - A New Characterization of Unichord-Free Graphs

JO - Discussiones Mathematicae Graph Theory

PY - 2015

VL - 35

IS - 4

SP - 765

EP - 771

AB - Unichord-free graphs are defined as having no cycle with a unique chord. They have appeared in several papers recently and are also characterized by minimal separators always inducing edgeless subgraphs (in contrast to characterizing chordal graphs by minimal separators always inducing complete subgraphs). A new characterization of unichord-free graphs corresponds to a suitable reformulation of the standard simplicial vertex characterization of chordal graphs.

LA - eng

KW - chordal graph; unichord-free graph; minimal separator

UR - http://eudml.org/doc/276023

ER -

## References

top- [1] A. Berry, J.-P. Bordat and O. Cogis, Generating all the minimal separators of a graph, Internat. J. Found. Comput. Sci. 11 (2000) 397-403. doi:10.1142/S0129054100000211[Crossref] Zbl1320.05120
- [2] A. Brandstädt, F.F. Dragan, V.B. Le and T. Szymczak, On stable cutsets in graphs, Discrete Appl. Math. 105 (2000) 39-50. doi:10.1016/S0166-218X(00)00197-9[Crossref] Zbl0962.68138
- [3] A. Brandstädt, V.B. Le, and J.P. Spinrad, Graph Classes: A Survey (Society for Industrial and Applied Mathematics, Philadelphia, 1999). doi:10.1137/1.9780898719796[Crossref]
- [4] S. Klein and C.M.H. de Figueiredo, The NP-completeness of multi-partite cutset testing, Congr. Numer. 119 (1996) 217-222. Zbl0897.68070
- [5] T. Kloks and D. Kratsch, Listing all minimal separators of a graph, SIAM J. Comput. 27 (1998) 605-613. doi:10.1137/S009753979427087X[Crossref] Zbl0907.68136
- [6] R.C.S. Machado and C.M.H. de Figueiredo, Total chromatic number of unichord- free graphs, Discrete Appl. Math. 159 (2011) 1851-1864. doi:10.1016/j.dam.2011.03.024[Crossref][WoS] Zbl1228.05157
- [7] R.C.S. Machado, C.M.H. de Figueiredo and N. Trotignon, Complexity of colouring problems restricted to unichord-free and {square,unichord}-free graphs, Discrete Appl. Math. 164 (2014) 191-199. doi:10.1016/j.dam.2012.02.016[WoS][Crossref] Zbl1321.05089
- [8] R.C.S. Machado, C.M.H. de Figueiredo and K. Vušković, Chromatic index of graphs with no cycle with unique chord, Theoret. Comput. Sci. 411 (2010) 1221-1234. doi:10.1016/j.tcs.2009.12.018[WoS][Crossref] Zbl1213.05150
- [9] T.A. McKee, Independent separator graphs, Util. Math. 73 (2007) 217-224. Zbl1140.05037
- [10] T.A. McKee, When all minimal vertex separators induce complete or edgeless sub- graphs, Discrete Math. Algorithms Appl. 5 (2013) #1350015. doi:10.1142/S1793830913500158[Crossref] Zbl1276.05063
- [11] T.A. McKee and F.R. McMorris, Topics in Intersection Graph Theory (Society for Industrial and Applied Mathematics, Philadelphia, 1999). doi:10.1137/1.9780898719802[Crossref] Zbl0945.05003
- [12] Y. Shibata, On the tree representation of chordal graphs, J. Graph Theory 12 (1988) 421-428. doi:10.1002/jgt.3190120313[Crossref] Zbl0654.05022
- [13] R.E. Tarjan, Decomposition by clique separators, Discrete Math. 55 (1985) 221-232. doi:10.1016/0012-365X(85)90051-2[Crossref]
- [14] N. Trotignon and K. Vušković, A structure theorem for graphs with no cycle with a unique chord and its consequences, J. Graph Theory 63 (2010) 31-67. doi:10.1002/jgt.20405[Crossref] Zbl1186.05104
- [15] S.H. Whitesides, An algorithm for finding clique cut-sets, Inform. Process. Lett. 12 (1981) 31-32. doi:10.1016/0020-0190(81)90072-7 [Crossref] Zbl0454.68078

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