Abstract
A graph is called t-perfect, if its stable set polytope is defined by non-negativity, edge and odd-cycle inequalities. We characterise the class of all claw-free t-perfect graphs by forbidden t-minors, and show that they are 3-colourable. Moreover, we determine the chromatic number of claw-free h-perfect graphs and give a polynomial-time algorithm to compute an optimal colouring.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barahona F., Mahjoub A.R.: Compositions of graphs and polyhedra II: stable sets. SIAM J. Discrete Math. 7, 359–371 (1994)
Boulala M., Uhry J.P.: Polytope des indépendants d’un graphe série-parallèle. Discrete. Math. 27, 225–243 (1979)
Cao D., Nemhauser G.L.: Polyhedral characterizations and perfection of line graphs. Discrete Appl. Math. 81, 141–154 (1998)
Chudnovsky M., Cornuéjols G., Liu X., Seymour P., Vuskovic K.: Recognizing berge graphs. Combinatorica 25, 143–187 (2005)
Chudnovsky M., Ovetsky A.: Coloring quasi-line graphs. J. Comb. Theory Ser. B 54, 41–50 (2007)
Chudnovsky, M., Seymour, P.: The structure of claw-free graphs, Surveys in Combinatorics 2005, vol. 327, London Mathematical Society Lecture Note, pp. 153–171 (2005)
Chudnovsky M., Seymour R., Robertson N., Thomas R.: The strong perfect graph theorem. Ann. Math. 164, 51–229 (2006)
Chvátal V.: On certain polytopes associated with graphs. J. Comb. Theory Ser. B 18, 138–154 (1975)
Dahl G.: Stable set polytopes for a class of circulant graphs. SIAM J. Optim. 9, 493–503 (1999)
Diestel R.: Graph Theory. 3rd edn. Springer-Verlag, Berlin (2005)
Edmonds J.: Maximum matching and a polyhedron with 0, 1-vertices. J. Res. Natl. Bur. Stand. Sect. B 69, 125–130 (1965)
Eisenbrand F., Oriolo G., Stauffer G., Ventura P.: The stable set polytope of quasi-line graphs. Combinatorica 28, 45–67 (2008)
Fonlupt J., Uhry J.P.: Transformations which preserve perfectness and h-perfectness of graphs. Ann. Discrete Math. 16, 83–95 (1982)
Fulkerson D.R.: Blocking and anti-blocking pairs of polyhedra. Math. Program. 1, 168–194 (1971)
Fulkerson D.R.: Anti-blocking polyhedra. J. Comb. Theory Ser. B 12, 50–71 (1972)
Galluccio A., Sassano A.: The rank facets of the stable set polytope for claw-free graphs. J. Comb. Theory Ser. B 69, 1–38 (1997)
Gerards, A.M.H.: Graphs and polyhedra—binary spaces and cutting planes. CWI Tract 73, CWI, Amsterdam (1990)
Gerards A.M.H., Shepherd F.B.: The graphs with all subgraphs t-perfect. SIAM J. Discrete Math. 11, 524–545 (1998)
Grötschel M., Lovász L., Schrijver A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1, 169–197 (1981)
Grötschel M., Lovász L., Schrijver A.: Relaxations of vertex packing. J. Comb. Theory Ser. B 40, 330–343 (1986)
Grötschel M., Lovász L., Schrijver A.: Geometric Algorithms and Combinatorial Optimization. Springer-Verlag, Berlin (1988)
Harary F.: Graph Theory. Addison Wesley, Reading, MA (1972)
Lovász L.: A note on factor-critical graphs. Stud. Sci. Mathematicum Hung. 7, 279–280 (1972)
Mahjoub A.R.: On the stable set polytope of a series-parallel graph. Math. Program. 40, 53–57 (1988)
Pulleyblank, W., Edmonds, J.: Facets of 1-matching polyhedra, Hypergraph Seminar. In: Berge, C., Ray-Chaudhuri, D. (eds.) Springer, pp. 214–242 (1974)
Roussopoulos N.D.: A max{m, n} algorithm for determining the graph H from its line graph G. Inf. Process. Lett. 2, 108–112 (1973)
Sbihi N., Uhry J.P.: A class of h-perfect graphs. Discrete Math. 51, 191–205 (1984)
Sebö, A.: Personal communication
Schrijver A.: Combinatorial Optimization. Polyhedra and Efficiency. Springer-Verlag, Berlin (2003)
Shepherd F.B.: Applying Lehman’s theorems to packing problems. Math. Prog. 71, 353–367 (1995)
Author information
Authors and Affiliations
Corresponding author
Additional information
M. Stein was supported by Fondecyt grant 11090141.