A graph G is a general partition graph if there is some set S and an assignment of non-empty subsets Sx ⊆ S to the vertices of G such that two vertices x and y are adjacent if and only if Sx ∩ Sy ≠ φ and for every maximal independent set M the set {Sm | m ∈ M} is a partition of S. For every minor closed family of graphs there exists a polynomial time algorithm that checks if an element of the family is a general partition graph. The triangle condition says that for every maximal independent set M and for every edge (x, y) with x, y ∉ M there is a vertex m ∈ M such that {x, y, m} induces a triangle in G. It is known that the triangle condition is necessary for a graph to be a general partition graph (but in general not sufficient). We show that for AT-free graphs this condition is also sufficient and this leads to an efficient algorithm that demonstrates whether or not an AT-free graph is a general partition graph. We show that the triangle condition can be checked in polynomial time for planar graphs and circle graphs. It is unknown if the triangle condition is also a sufficient condition for planar graphs to be a general partition graph. For circle graphs we show that the triangle condition is not sufficient. © Springer-Verlag Berlin Heidelberg 2003.
CITATION STYLE
Kloks, T., Lee, C. M., Liu, J., & Müller, H. (2003). On the recognition of general partition graphs. Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2880, 273–283. https://doi.org/10.1007/978-3-540-39890-5_24
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