@misc{Wagler2000, author = {Wagler, Annegret}, title = {The Classes of Critically and Anticritically Perfect Graphs}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-5975}, number = {00-29}, year = {2000}, abstract = {We focus on two new types of extremal graphs with respect to perfectness: critically and anticritically perfect graphs that lose their perfectness by simply deleting and adding an arbitrary edge, respectively. We present examples and study properties in order to compare critically and anticritically perfect graphs with minimally imperfect graphs, another type of extremal graphs with respect to perfectness. We discuss two attempts to characterize the classes of all critically and anticritically perfect graphs and give a brief overview on classes of perfect graphs which contain critically or anticritically perfect graphs.}, language = {en} } @misc{Wagler2000, author = {Wagler, Annegret}, title = {Critical and Anticritical Edges in Perfect Graphs}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6174}, number = {00-49}, year = {2000}, abstract = {We call an edge \$e\$ of a perfect graph \$G\$ critical if \$G-e\$ is imperfect and say further that \$e\$ is anticritical with respect to the complementary graph \$\overline G\$. We ask in which perfect graphs critical and anticritical edges occur and how to find critical and anticritical edges in perfect graphs. Finally, we study whether we can order the edges of certain perfect graphs such that deleting all the edges yields a sequence of perfect graphs ending up with a stable set.}, language = {en} } @misc{Wagler2002, author = {Wagler, Annegret}, title = {Relaxing Perfectness: Which Graphs are 'Almost' Perfect?}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6700}, number = {02-03}, year = {2002}, abstract = {For all perfect graphs, the stable set polytope STAB\$(G)\$ coincides with the fractional stable set polytope QSTAB\$(G)\$, whereas STAB\$(G) \subset\$ QSTAB\$(G)\$ holds iff \$G\$ is imperfect. Padberg asked in the early seventies for ``almost'' perfect graphs. He characterized those graphs for which the difference between STAB\$(G)\$ and QSTAB\$(G)\$ is smallest possible. We develop this idea further and define three polytopes between STAB\$(G)\$ and QSTAB\$(G)\$ by allowing certain sets of cutting planes only to cut off all the fractional vertices of QSTAB\$(G)\$. The difference between QSTAB\$(G)\$ and the largest of the three polytopes coinciding with STAB\$(G)\$ gives some information on the stage of imperfectness of the graph~\$G\$. We obtain a nested collection of three superclasses of perfect graphs and survey which graphs are known to belong to one of those three superclasses. This answers the question: which graphs are ``almost'' perfect?}, language = {en} } @misc{Wagler2002, author = {Wagler, Annegret}, title = {Antiwebs are Rank-Perfect}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6742}, number = {02-07}, year = {2002}, abstract = {\We discuss a nested collection of three superclasses of perfect graphs: near-perfect, rank-perfect, and weakly rank-perfect graphs. For that, we start with the description of the stable set polytope for perfect graphs and allow stepwise more general facets for the stable set polytopes of the graphs in each superclass. Membership in those three classes indicates how far a graph is away from being perfect. We investigate for webs and antiwebs to which of the three classes they belong. We provide a complete description of the facets of the stable set polytope for antiwebs (with help of a result due to Shepherd on near-bipartite graphs). The main result is that antiwebs are rankperfect.}, language = {en} } @misc{HougardyWagler2002, author = {Hougardy, Stefan and Wagler, Annegret}, title = {Perfectness is an Elusive Graph Property}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6787}, number = {02-11}, year = {2002}, abstract = {A graph property is called elusive (or evasive) if every algorithm for testing this property has to read in the worst case \$n\choose 2\$ entries of the adjacency matrix of the given graph. Several graph properties have been shown to be elusive, e.g. planarity (Best et al) or \$k\$-colorability (Bollobas). A famous conjecture of Karp says that every non-trivial monotone graph property is elusive. We prove that a non-monotone but hereditary graph property is elusive: perfectness.}, language = {en} } @misc{KosterWagler2006, author = {Koster, Arie M.C.A. and Wagler, Annegret}, title = {The extreme points of QSTAB(G) and its implications}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9249}, number = {06-30}, year = {2006}, abstract = {Perfect graphs constitute a well-studied graph class with a rich structure, reflected by many characterizations w.r.t different concepts. Perfect graphs are, e.g., characterized as precisely those graphs \$G\$ where the stable set polytope STAB\$(G)\$ coincides with the clique constraint stable set polytope QSTAB\$(G)\$. For all imperfect graphs STAB\$(G) \subset\$ QSTAB\$(G)\$ holds and, therefore, it is natural to measure imperfection in terms of the difference between STAB\$(G)\$ and QSTAB\$(G)\$. Several concepts have been developed in this direction, for instance the dilation ratio of STAB\$(G)\$ and QSTAB\$(G)\$ which is equivalent to the imperfection ratio imp\$(G)\$ of \$G\$. To determine imp\$(G)\$, both knowledge on the facets of STAB\$(G)\$ and the extreme points of QSTAB\$(G)\$ is required. The anti-blocking theory of polyhedra yields all {\em dominating} extreme points of QSTAB\$(G)\$, provided a complete description of the facets of STAB\$(\overline G)\$ is known. As this is typically not the case, we extend the result on anti-blocking polyhedra to a {\em complete} characterization of the extreme points of QSTAB\$(G)\$ by establishing a 1-1 correspondence to the facet-defining subgraphs of \$\overline G\$. We discuss several consequences, in particular, we give alternative proofs of several famous results.}, language = {en} } @phdthesis{Wagler2000, author = {Wagler, Annegret}, title = {Critical Edges in Perfect Graphs}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10220}, year = {2000}, language = {en} } @misc{MarencoWagler2003, author = {Marenco, Javier and Wagler, Annegret}, title = {Chromatic Scheduling Polytopes coming from the Bandwidth Allocation Problem in Point-to-Multipoint Radio AccessSystems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7614}, number = {03-39}, year = {2003}, abstract = {Point-to-Multipoint systems are one kind of radio systems supplying wireless access to voice/data communication networks. Such systems have to be run using a certain frequency spectrum, which typically causes capacity problems. Hence it is, on the one hand, necessary to reuse frequencies but, on the other hand, no interference must be caused thereby. This leads to the bandwidth allocation problem, a special case of so-called chromatic scheduling problems. Both problems are NP-hard, and there exist no polynomial time approximation algorithms with a guaranteed quality. One kind of algorithms which turned out to be successful for many other combinatorial optimization problems uses cutting plane methods. In order to apply such methods, knowledge on the associated polytopes is required. The present paper contributes to this issue, exploring basic properties of chromatic scheduling polytopes and several classes of facet-defining inequalities.}, language = {en} }