Circular permutation graph family with applications

Abstract

In a circular permutation diagram, there are two sets of terminals on two concentric circles: C in and C out. Given a permutation Π = [π 1, π 2, …, π n ], terminal i on C in and terminal π i on C out are connected by a wire. The intersection graph G c of a circular permutation diagram D c is called a circular permutation graph of a permutation Π corresponding to the diagram D c . The set of all circular permutation graphs of a permutation Π is called the circular permutation graph family of permutation Π. In this paper, we propose the following: (1) an O(∣ V∣ + ∣ E∣) time algorithm to check if a labeled graph G = ( V, E) is a labeled circular permutation graph. (2) An O( n log n + nt) time algorithm to find a maximum independent set of a family, where n = ∣Π∣ and t is the cardinality of the output. (Number t in the worst case is O( n). However, if Π is uniformly distributed (and independent from i), its expected value is O(√ n).) (3) An O(min(δ∣ V c ∣log log∣ V c ∣,∣ V c ∣log∣ V c ∣) + ∣ E c ∣) time algorithm for finding a maximum independent set of a circular permutation diagram D c , where δ is the minimum degree of vertices in the intersection graph G c = ( V c , E c ) of D c . (4) An O( n log log n) time algorithm for finding a maximum clique and the chromatic number of a circular permutation diagram, where n is the number of wires in the diagram.