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The structure of permutation graphs
Abstract
The class of permutation graphs has been studied extensively for more than two decades. The most popular representational tool employed is the permutation or matching diagram. However, the matching-diagram does not capture all the structural information of a permutation graph. The study structures involving vertex ordering in two dimensions that determine adjacency and more generally distance between vertices in a permutation graph. In this context, we explore the Euclidean representation for its power to display these structures of a permutation graph in a two dimensional space. We demonstrate that problems involving adjacency and distance can be easily handled with the Euclidean representation. Hamiltonian path and cycle, and path and cycle toughness are some of the hard problems on permutation graphs. However, the tools provided by the Euclidean representation make it easier to handle these as well as other distance related problems like the clustering problem. Hamiltonian paths and cycles can be constructed in traceable and Hamiltonian permutation graphs respectively which visually traverse the Euclidean representation diagonally along layers of vertices. Path and cycle toughness can be established by identifying specific vertices in the Euclidean representation whose removal divides the graph into connected components. Graphs with specified diameter exhibit distinctive closed geometric shapes which help in the study of clustering problems. We believe that the Euclidean representation provides a powerful tool for revealing the structure of permutation graphs. The Euclidean representation presents an excellent framework for visually exploring a permutation graph. It is expected that the Euclidean representation will help solve a variety of distance and adjacency related problems on permutation graphs.
Subject Area
Computer science|Mathematics
Recommended Citation
Riedesel, Charles P, "The structure of permutation graphs" (1995). ETD collection for University of Nebraska-Lincoln. AAI9611067.
https://digitalcommons.unl.edu/dissertations/AAI9611067