The Induced Separation Dimension of a Graph

The Induced Separation Dimension of a Graph 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 $$\{a,b\}$$ { a , b } and $$\{c,d\}$$ { c , d } of G strongly independent if the set of endpoints $$\{a,b,c,d\}$$ { a , b , c , d } induces a $$2K_2$$ 2 K 2 in G. The induced separation dimension of a graph G is the smallest cardinality of a family $$\mathcal {L}$$ 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 $$\mathcal {L}$$ L . For each $$k \in \mathbb {N}$$ k ∈ N , the family of graphs with induced separation dimension at most k is denoted by $${\text {ISD}}(k)$$ ISD ( k ) . In this article, we initiate a study of this new dimensional parameter. The class $${\text {ISD}}(1)$$ 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 $${\text {ISD}}(1)$$ 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 $${\text {ISD}}(1)$$ ISD ( 1 ) is NP-complete for general graphs. Nevertheless, we show that the maximum induced matching problem can be solved efficiently in $${\text {ISD}}(1)$$ ISD ( 1 ) . We then briefly study $${\text {ISD}}(2)$$ 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 $$\Theta (\lg n)$$ Θ ( 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 $$k \ge 2$$ k ≥ 2 , the recognition problem for $${\text {ISD}}(k)$$ ISD ( k ) is NP-complete even on AT-free graphs. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Algorithmica Springer Journals

The Induced Separation Dimension of a Graph

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Publisher
Springer US
Copyright
Copyright © 2017 by Springer Science+Business Media, LLC
Subject
Computer Science; Algorithm Analysis and Problem Complexity; Theory of Computation; Mathematics of Computing; Algorithms; Computer Systems Organization and Communication Networks; Data Structures, Cryptology and Information Theory
ISSN
0178-4617
eISSN
1432-0541
D.O.I.
10.1007/s00453-017-0353-x
Publisher site
See Article on Publisher Site

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 $$\{a,b\}$$ { a , b } and $$\{c,d\}$$ { c , d } of G strongly independent if the set of endpoints $$\{a,b,c,d\}$$ { a , b , c , d } induces a $$2K_2$$ 2 K 2 in G. The induced separation dimension of a graph G is the smallest cardinality of a family $$\mathcal {L}$$ 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 $$\mathcal {L}$$ L . For each $$k \in \mathbb {N}$$ k ∈ N , the family of graphs with induced separation dimension at most k is denoted by $${\text {ISD}}(k)$$ ISD ( k ) . In this article, we initiate a study of this new dimensional parameter. The class $${\text {ISD}}(1)$$ 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 $${\text {ISD}}(1)$$ 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 $${\text {ISD}}(1)$$ ISD ( 1 ) is NP-complete for general graphs. Nevertheless, we show that the maximum induced matching problem can be solved efficiently in $${\text {ISD}}(1)$$ ISD ( 1 ) . We then briefly study $${\text {ISD}}(2)$$ 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 $$\Theta (\lg n)$$ Θ ( 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 $$k \ge 2$$ k ≥ 2 , the recognition problem for $${\text {ISD}}(k)$$ ISD ( k ) is NP-complete even on AT-free graphs.

Journal

AlgorithmicaSpringer Journals

Published: Jul 31, 2017

References

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