Induced separation dimension

A linear ordering of the vertices of a graph G separates two edges of G if both the endpoints of one precede both the endpoints of the other in the order. We call two edges {a, b} and {c, d} of G strongly independent if the set of endpoints {a, b, c, d} induces a 2K₂ in G. The induced separation dimension of a graph G is the smallest cardinality of a family L of linear orders of V (G) such that every pair of strongly independent edges in G are separated in at least one of the linear orders in L. For each k ∈ ℕ, the family of graphs with induced separation dimension at most k is denoted by ISD(k). In this article, we initiate a study of this new dimensional parameter. The class ISD(1) or, equivalently, the family of graphs which can be embedded on a line so that every pair of strongly independent edges are disjoint line segments, is already an interesting case. On the positive side, we give characterizations for chordal graphs in ISD(1) which immediately lead to a polynomial time algorithm which determines the induced separation dimension of chordal graphs. On the negative side, we show that the recognition problem for ISD(1) is NP-complete for general graphs. We then briefly study ISD(2) and show that it contains many important graph classes like outerplanar graphs, chordal graphs, circular arc graphs and polygon-circle graphs. Finally, we describe two techniques to construct graphs with large induced separation dimension. The first one is used to show that the maximum induced separation dimension of a graph on n vertices is Θ(lg n) and the second one is used to construct AT-free graphs with arbitrarily large induced separation dimension.