Transitive Orientation of Graphs and Identification of Permutation Graphs

Abstract

The graphs considered in this paper are assumed to be finite, with no edge joining a vertex to itself and with no two distinct edges joining the same pair of vertices. An undirected graph will be denoted by G or ( V, E ), where V is the set of vertices and E is the set of edges. An edge joining the vertices i,j ∊ V will be denoted by the unordered pair ( i,j ). An orientation of G = ( V, E ) is an assignment of a unique direction i → j or j → i to every edge ( i,j ) ∊ E . The resulting directed image of G will be denoted by G → or ( V , E→ ), where E→ is now a set of ordered pairs E→ = {[ i,j ]| ( i,j ) ∊ E and i → j }. Notice the difference in notation (brackets versus parentheses) for ordered and unordered pairs.