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Algorithmica

, Volume 80, Issue 10, pp 2834–2848 | Cite as

The Induced Separation Dimension of a Graph

  • Emile Ziedan
  • Deepak Rajendraprasad
  • Rogers Mathew
  • Martin Charles Golumbic
  • Jérémie Dusart
Article
  • 318 Downloads

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.

Keywords

Induced separation dimension Vertex ordering Acyclic orientation Asteroidal triples 

Mathematics Subject Classification

05C75 05C62 05C85 

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Computer Science, The Caesarea Rothschild InstituteUniversity of HaifaHaifaIsrael
  2. 2.Department of Computer ScienceIndian Institute of TechnologyKharagpurIndia

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