Circular permutation graphs

Abstract

AbstractA new class of intersection graphs called circular permutation graphs is introduced and characterized. A circular permutation diagram for a permutation P(1),…,P(n) consists of two circles C1 and C2; the numbers 1′, 2′,…,n′ and P(1),…,P(n) on C1 and C2, respectively; and a set of n chords 1, 2,…, n connecting i to i′ such that two chords intersect each other at most once. A graph G represents a circular permutation diagram if there is a labeling of V(G) with {1,…, n} such that i is adjacement to j iff i and j intersect. Graphs which represent at least one permutation diagram are called circular permutation graphs. Circular permutation graphs generalize permutation graphs [2], [8] and are embedded in the set of comparability graphs [4]. The characterization leads to a recognition algorithm which requires O(δ|E|) steps where δ is the maximum degree of a vertex.