Seidel Minor, Permutation Graphs and Combinatorial Properties

A permutation graph is an intersection graph of segments lying between two parallel lines. A Seidel complementation of a finite graph at a vertex v consists in complementing the edges between the neighborhood and the non-neighborhood of v. Two graphs are Seidel complement equivalent if one can be obtained from the other by a sequence of Seidel complementations. In this paper we introduce the new concept of Seidel complementation and Seidel minor. We show that this operation preserves cographs and the structure of modular decomposition. The main contribution of this paper is to provide a new and succinct characterization of permutation graphs namely, a graph is a permutation graph if and only if it does not contain any of the following graphs: C 5, C 7, $XF_{6}^{2}$ , $XF_{5}^{2n+3}$ , $C_{2n}, n\geqslant6$ and their complements as a Seidel minor. This characterization is in a sense similar to Kuratowski’s characterization [Kur30] of planar graphs by forbidden topological minors.