The Induced Separation Dimension of a Graph

Abstract

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 $$2K_2$$ in G. The induced separation dimension of a graph G is the smallest cardinality of a family $$\mathcal {L}$$ 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 $$\mathcal {L}$$ . For each $$k \in \mathbb {N}$$ , the family of graphs with induced separation dimension at most k is denoted by $${\text {ISD}}(k)$$ . In this article, we initiate a study of this new dimensional parameter. The class $${\text {ISD}}(1)$$ 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 $${\text {ISD}}(1)$$ 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 $${\text {ISD}}(1)$$ is NP-complete for general graphs. Nevertheless, we show that the maximum induced matching problem can be solved efficiently in $${\text {ISD}}(1)$$ . We then briefly study $${\text {ISD}}(2)$$ 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 $$\Theta (\lg n)$$ and the second one is used to construct AT-free graphs with arbitrarily large induced separation dimension. The second construction is also used to show that, for every $$k \ge 2$$ , the recognition problem for $${\text {ISD}}(k)$$ is NP-complete even on AT-free graphs.