Characterization and Construction of Permutation Graphs

Severino V. Gervacio,Phoebe Chloe F. Ramos,Teofina A. Rapanut

doi:10.4236/ojdm.2013.31007

Severino V. Gervacio, Phoebe Chloe F. Ramos + Show 1 more

Open Access

https://doi.org/10.4236/ojdm.2013.31007

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Abstract

If is a permutation of , the graph has vertices where xy is an edge of if and only if (x, y) or (y, x) is an inversion of . Any graph isomorphic to is called a permutation graph. In 1967 Gallai characterized permutation graphs in terms of forbidden induced subgraphs. In 1971 Pnueli, Lempel, and Even showed that a graph is a permutation graph if and only if both the graph and its complement have transitive orientations. In 2010 Limouzy characterized permutation graphs in terms of forbidden Seidel minors. In this paper, we characterize permutation graphs in terms of a cohesive order of its vertices. We show that only the caterpillars are permutation graphs among the trees. A simple method of constructing permutation graphs is also presented here.

Highlights

IntroductionFor our purpose in this paper, any graph isomorphic to G for some permutation will be called a permutation graph
The term graph of inversions was used by Ramos in [1]
We show that only the caterpillars are permutation graphs among the trees

Summary

Introduction

For our purpose in this paper, any graph isomorphic to G for some permutation will be called a permutation graph. There is an implementation PermutationGraph[p] in the Combinatorica package of Mathematica [2] that creates the permutation graph Gp. In 1967 Gallai [3] characterized permutation graphs in terms of forbidden induced subgraphs. In 1971 Pnueli, Lempel, and Even [4] showed that a graph G is a permutation graph if and only if both G and its complement G have transitive orientations. In 2010 Limouzy [5] gave a characterization of permutation graphs in terms of forbidden Seidel minors. A characterization of permutation graphs in terms of cohesive vertex-set order is established in this paper. We prove that the only permutation graphs among the trees are the caterpillars. We describe a simple method of constructing permutation graphs

Cohesive Vertex-Set Order

Construction and Examples of Permutation Graphs

Concluding Remarks