ANNEGRET K . WAGLER Critical and Anticritical Edges with respect to Perfectness
نویسنده
چکیده
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 Hougardy & Wagler [32] 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 Hayward [15] and Spinrad & Sritharan [27] 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 [33]. 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 [31, 32]. An application of the concept of critically and anticritically perfect graphs is a result due to Hougardy & Wagler [23] showing that perfectness is an elusive graph property.
منابع مشابه
On Critically Perfect Graphs on Critically Perfect Graphs
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 relate critically perfect graphs to well-known classes of perfect graphs, investigate the structure of the class of critically perfect graphs, and study operations preserving critical perfectness.
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