On polygon numbers of circle graphs and distance hereditary graphs

Circle graphs are intersection graphs of chords in a circle and kpolygon graphs are the intersection graphs of chords in a convex ksided polygon where each chord has its endpoints on distinct sides. Every k-polygon graph is a circle graph and every circle graph is a kpolygon graph for some k. The polygon number ψ(G) of a circle graph G is the minimum k such that G is a k-polygon graph and the polygon number of a circle representation is the minimum number of corners that must be added to the circle to produce a polygon representation. Given a circle graph G and an integer k, determining whether ψ(G) ≤ k is NP-complete, while the problem is solvable in polynomial time for fixed k, and the polygon number of a circle representation can be computed in polynomial time [10]. In this paper, we give bounds on ψ(G) when G is an arbitrary circle graph – upper bounds in terms of the independence number, the clique cover number, and the number of vertices of G, and a lower bound in terms of the asteroidal number of G – and show that ψ(G) is equal to the asteroidal number of G when G is a connected nonclique distance hereditary graph. We then use our results to develop characterizations of distance hereditary permutation graphs. Finally, we ∗[email protected] †[email protected] ‡Edmonton, Alberta, Canada T6G 2E8 1 ar X iv :1 40 1. 15 41 v1 [ cs .D M ] 7 J an 2 01 4 describe linear time algorithms for finding the polygon number of a circle representation and for finding the asteroidal number of a distance hereditary graph, improvements over previously known algorithms for