05C69 Dominating sets, independent sets, cliques
Refine
Document Type
- ZIB-Report (3)
Language
- English (3)
Has Fulltext
- yes (3)
Is part of the Bibliography
- no (3)
Keywords
- (non-)rank facet (2)
- rank-perfect graph (2)
- stable set polytope (2)
- web (2)
- circulants (1)
- normal graphs (1)
- perfect graphs (1)
Institute
- ZIB Allgemein (3)
Normal graphs are defined in terms of cross-intersecting set families: a graph is normal if it admits a clique cover $\cal Q$ and a stable set cover $\cal S$ s.t.~every clique in $\cal Q$ intersects every stable set in $\cal S$. Normal graphs can be considered as closure of perfect graphs by means of co-normal products (Körner 1973) and graph entropy (Czisz\'ar et al. 1990). Perfect graphs have been recently characterized as those graphs without odd holes and odd antiholes as induced subgraphs (Strong Perfect Graph Theorem, Chudnovsky et al. 2002). Körner and de Simone observed that $C_5$, $C_7$, and $\overline C_7$ are minimal not normal and conjectured, as generalization of the Strong Perfect Graph Theorem, that every $C_5$, $C_7$, $\overline C_7$- free graph is normal (Normal Graph Conjecture, Körner and de Simone 1999). We prove this conjecture for a first class of graphs that generalize both odd holes and odd antiholes, the circulants, by characterizing all the normal circulants.
Graphs with circular symmetry, called webs, are relevant for describing the stable set polytopes of two larger graph classes, quasi-line graphs [{\sl Giles and Trotter 1981, Oriolo 2001}] and claw-free graphs [{\sl Galluccio and Sassano 1997, Giles and Trotter 1981}]. Providing a decent linear description of the stable set polytopes of claw-free graphs is a long-standing problem [{\sl Grötschel, Lov\'asz, and Schrijver 1988}]. However, even the problem of finding all facets of stable set polytopes of webs is open. So far, it is only known that stable set polytopes of webs with clique number $\leq 3$ have rank facets only [{\sl Dahl 1999, Trotter 1975}] while there are examples with clique number $\geq 4$ having non-rank facets [{\sl e.g. Liebling et al. 2003, Oriolo 2001, P\^echer and Wagler 2003}]. In this paper, we provide a construction for non-rank facets of stable set polytopes of webs. We use this construction to prove, for several fixed values of $\omega$ including all odd values at least 5, that there are only finitely many webs with clique number $\omega$ whose stable set polytopes admit rank facets only.
Graphs with circular symmetry, called webs, are relevant w.r.t. describing the stable set polytopes of two larger graph classes, quasi-line graphs and claw-free graphs. Providing a decent linear description of the stable set polytopes of claw-free graphs is a long-standing problem. However, even the problem of finding all facets of stable set polytopes of webs is open. So far, it is only known that stable set polytopes of webs with clique number $\leq 3$ have rank facets only while there are examples with clique number $>4$ having non-rank facets.