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In this paper we discuss the bounds of and relations among various kinds of intersection numbers of graphs. Especially, we address extremal graphs with respect to the established bounds. The uniqueness of the minimum-size intersection representations for some graphs is also studied. In the course of this work, we introduce a superclass of chordal graphs, defined in terms of a generalization of simplicial vertex and perfect elimination ordering. Keywords: chordal graph, diamond-free elimination graph, edge clique graph, fractional intersection number, Helly property, intersection graph, intersection number, purple graph, maximal clique, maximal clique irreducible graph, ST weighting algorithm, unique representability. [ABSTRACT FROM AUTHOR]

In this paper, we consider the recognition problem on three classes of perfect graphs, namely, the HH-free, the HHD-free, and the Welsh-Powell opposition graphs (or WPO-graphs). In particular, we prove properties of the chordal completion of a graph and show that a modified version of the classic linear-time algorithm for testing for a perfect elimination ordering can be efficiently used to determine in O(n min{m α(n, n), m + n log n}) time whether a given graph G on n vertices and m edges contains a house or a hole; this implies an O(n min{m α (n, n), m + n log n})-time and O(n + m)-space algorithm for recognizing HH-free graphs, and in turn leads to an HHD-free graph recognition algorithm exhibiting the same time and space complexity. We also show that determining whether the complement G of the graph G is HH-free can be efficiently resolved in O(nm) time using O(n²) space, which leads to an O(nm)-time and O(n²) space algorithm for recognizing WPO-graphs. The previously best algorithms for recognizing HH-free, HHD-free, and WPO-graphs required O(n³) time and O(n²) space. [ABSTRACT FROM AUTHOR]