Abstract
Perfect graphs constitute a well-studied graph class with a rich structure, which is reflected by many characterizations with respect to different concepts. Perfect graphs are, for instance, precisely those graphs G where the stable set polytope STAB(G) equals the fractional stable set polytope QSTAB(G). The dilation ratio \({\rm min}\{t : {\rm QSTAB}(G) \subseteq t\,{\rm STAB}(G)\}\) of the two polytopes yields the imperfection ratio of G. It is NP-hard to compute and, for most graph classes, it is even unknown whether it is bounded. For graphs G such that all facets of STAB(G) are rank constraints associated with antiwebs, we characterize the imperfection ratio and bound it by 3/2. Outgoing from this result, we characterize and bound the imperfection ratio for several graph classes, including near-bipartite graphs and their complements, namely quasi-line graphs, by means of induced antiwebs and webs, respectively.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berge C. (1961). Färbungen von Graphen, deren sämtliche bzw. deren ungerade Kreise starr sind. Wiss. Zeitschrift der Martin-Luther-Universität Halle-Wittenberg 10: 114–115
Cheng E. and de Vries S. (2001). Antiweb inequalities: strength and intractability. Congr. Numeration 152: 5–19
Chudnovsky M. and Ovetsky A. (2007). Coloring quasi-line graphs. J. Graph. Theory 54: 41–50
Chudnovsky M., Robertson N., Seymour P. and Thomas R. (2006). The strong perfect graph theorem. Ann Math 164: 51–229
Chudnovsky, M., Seymour, P.: The structure of claw-free graphs. Surv. Combinat. 153–171 (2005)
Chvátal V. (1975). On certain polytopes associated with graphs. J. Comb. Theory (B) 18: 138–154
Edmonds J.R. (1965). Maximum matching and a polyhedron with (0,1) vertices. J. Res. Nat. Bur. Stand. 69: 125–130
Eisenbrand F., Oriolo G., Stauffer G. and Ventura P. (2005). Circular ones matrices and the stable set polytope of quasi-line graphs. Lect. Notes Comput. Sci. 3509: 291–305
Gerke S. and McDiarmid C. (2001). Graph imperfection. J. Comb. Theory (B) 83: 58–78
Gerke S. and McDiarmid C. (2001). Graph imperfection II. J. Comb. Theory (B) 83: 79–101
Grötschel M., Lovász L. and Schrijver A. (1988). Geometric Algorithms and Combinatorial Optimization. Springer, Heidelberg
Mycielski J. (1955). Sur le coloriage des graphs. Colloq. Math. 3: 161–162
Reed B. and Ramirez-Alfonsin J. (2001). Perfect Graphs. Wiley, New York
Shepherd F.B. (1995). Applying Lehman’s theorem to packing problems. Math. Program. 71: 353–367
Wagler A. (2004). Antiwebs are rank-perfect. 4OR 2: 149–152
Wagler A. (2005). On rank-perfect subclasses of near-bipartite graphs. 4OR 3: 329–336
Zhu X. (2001). Circular chromatic number: a survey. Discrete Math. 229: 371–410
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Coulonges, S., Pêcher, A. & Wagler, A.K. Characterizing and bounding the imperfection ratio for some classes of graphs. Math. Program. 118, 37–46 (2009). https://doi.org/10.1007/s10107-007-0182-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-007-0182-9