@misc{KosterWagler2005, author = {Koster, Arie M.C.A. and Wagler, Annegret}, title = {Comparing Imperfection Ratio and Imperfection Index for Graph Classes}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8836}, number = {05-50}, year = {2005}, abstract = {Perfect graphs constitute a well-studied graph class with a rich structure, 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)\$ coincides with the fractional stable set polytope \$QSTAB(G)\$. For all imperfect graphs \$G\$ it holds that \$STAB(G) \subset QSTAB(G)\$. It is, therefore, natural to use the difference between the two polytopes in order to decide how far an imperfect graph is away from being perfect; we discuss three different concepts, involving the facet set of \$STAB( G)\$, the disjunctive index of \$QSTAB(G)\$, and the dilation ratio of the two polytopes. Including only certain types of facets for \$STAB(G)\$, we obtain graphs that are in some sense close to perfect graphs, for example minimally immperfect graphs, and certain other classes of so-called rank-perfect graphs. The imperfection ratio has been introduced by (Gerke and McDiarmid, 2001) as the dilation ratio of \$STAB(G)\$ and \$QSTAB(G)\$, whereas (Aguilera et al., 2003) suggest to take the disjunctive index of \$Q STAB(G)\$ as the imperfection index of \$G\$. For both invariants there exist no general upper bounds, but there are bounds known for the imperfection ratio of several graph classes (Coulonges et al. 2005, Gerke and McDiarmid, 2001). Outgoing from a graph-theoretical interpretation of the imperfection index, we conclude that the imperfection index is NP-hard to compute and we prove that there exists no upper bound on the imperfect ion index for those graph classes with a known bounded imperfection ratio. Comparing the two invariants on those classes, it seems that the imperfection index measures imperfection much more roughly than the imperfection ratio; therefoe, discuss possible directions for refinements.}, language = {en} } @misc{TuzaWagler1997, author = {Tuza, Zsolt and Wagler, Annegret}, title = {Minimally non-preperfect graphs of small maximum degree}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2971}, number = {SC-97-28}, year = {1997}, abstract = {A graph \$G\$ is called preperfect if each induced subgraph \$G' \subseteq G\$ of order at least 2 has two vertices \$x,y\$ such that either all maximum cliques of \$G'\$ containing \$x\$ contain \$y\$, or all maximum indepentent sets of \$G'\$ containing \$y\$ contain \$x\$, too. Giving a partial answer to a problem of Hammer and Maffray [Combinatorica 13 (1993), 199-208], we describe new classes of minimally non-preperfect graphs, and prove the following characterizations: \begin{itemize} \item[(i)] A graph of maximum degree 4 is minimally non-preperfect if and only if it is an odd cycle of length at least 5, or the complement of a cycle of length 7, or the line graph of a 3-regular 3-connected bipartite graph. \item[(ii)] If a graph \$G\$ is not an odd cycle and has no isolated vertices, then its line graph is minimally non-preperfect if and only if \$G\$ is bipartite, 3-edge-connected, regular of degree \$d\$ for some \$d \ge 3\$, and contains no 3-edge-connected \$d'\$-regular subgraph for any \$3 \le d' \le d\$. \end{itemize}}, language = {en} } @misc{Wagler1996, author = {Wagler, Annegret}, title = {On Critically Perfect Graphs}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-2609}, number = {SC-96-50}, year = {1996}, abstract = {A perfect graph is critical if the deletion of any edge results in an imperfect graph. We give examples of such graphs and prove some basic properties. We investigate the relationship of critically perfect graphs to well-known classes of perfect graphs and study operations preserving critical perfectness.}, language = {en} } @misc{Wagler1997, author = {Wagler, Annegret}, title = {Critical Edges in Perfect Graphs and Some Polyhedral Consequences}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3153}, number = {SC-97-46}, year = {1997}, abstract = {An edge \$e\$ of a perfect graph \$G\$ is called critical if \$G-e\$ is imperfect. For certain graphs \$G-e\$ of this type, we determine all minimally imperfect subgraphs. We use this knowledge to describe inequalities inducing facets of the stable set polytope associated with \$G-e\$.\\}, language = {en} } @misc{PecherWagler2003, author = {Pecher, Arnaud and Wagler, Annegret}, title = {On Non-Rank Facets of Stable Set Polytopes of Webs with Clique Number Four}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7238}, number = {03-01}, year = {2003}, abstract = {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.}, language = {en} } @misc{PecherWagler2003, author = {P{\^e}cher, Arnaud and Wagler, Annegret}, title = {A construction for non-rank facets of stable set polytopes of webs}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7437}, number = {03-21}, year = {2003}, abstract = {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 Gr{\"o}tschel, 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.}, language = {en} } @misc{Wagler2003, author = {Wagler, Annegret}, title = {Critical and Anticritical Edges with respect to Perfectness}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7446}, number = {03-22}, year = {2003}, abstract = {We call an edge \$e\$ of a perfect graph \$G\$ critical if \$G-e\$ is imperfect and call \$e\$ anticritical if \$G+e\$ is imperfect. The present paper surveys several questions in this context. We ask in which perfect graphs critical and anticritical edges occur and how to detect such edges. The main result by [{\sl Wagler, PhD thesis 2000}] shows that a graph does not admit any critical edge if and only if it is Meyniel. The goal is to order the edges resp.~non-edges of certain perfect graphs s.t. deleting resp.~adding all edges in this order yields a sequence of perfect graphs only. Results of [{\sl Hayward 1985}] and [{\sl Spinrad \& Sritharan 1995}] show the existence of such edge orders for weakly triangulated graphs; the line-perfect graphs are precisely these graphs where all edge orders are perfect [{\sl Wagler 2001}]. Such edge orders cannot exist for every subclass of perfect graphs that contains critically resp.~anticritically perfect graphs where deleting resp.~adding an arbitrary edge yields an imperfect graph. We present several examples and properties of such graphs, discuss constructions and characterizations from [{\sl Wagler 1999, Wagler PhD thesis 2000}]. An application of the concept of critically and anticritically perfect graphs is a result due to [{\sl Hougardy \& Wagler 2002}] showing that perfectness is an elusive graph property.}, language = {en} } @misc{Wagler2004, author = {Wagler, Annegret}, title = {The Normal Graph Conjecture is true for Circulants}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7813}, number = {04-06}, year = {2004}, abstract = {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 (K{\"o}rner 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). K{\"o}rner 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, K{\"o}rner 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.}, language = {en} } @misc{Wagler2001, author = {Wagler, Annegret}, title = {Rank-Perfect and Weakly Rank-Perfect Graphs}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6448}, number = {01-18}, year = {2001}, abstract = {An edge of a perfect graph \$G\$ is critical if \$G-e\$ is imperfect. We would like to decide whether \$G - e\$ is still {\sl almost perfect} or already {\sl very imperfect}. Via relaxations of the stable set polytope of a graph, we define two superclasses of perfect graphs: rank-perfect and weakly rank-perfect graphs. Membership in those two classes indicates how far an imperfect graph is away from being perfect. We study the cases, when a critical edge is removed from the line graph of a bipartite graph or from the complement of such a graph.}, language = {en} } @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} }