A linear ordering of the vertices of a graph Gseparates 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}\documentclass[12pt]{minimal}
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\begin{document}$$\{a,b\}$$\end{document} and {c,d}\documentclass[12pt]{minimal}
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\begin{document}$$\{c,d\}$$\end{document} of Gstrongly independent if the set of endpoints {a,b,c,d}\documentclass[12pt]{minimal}
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\begin{document}$$\{a,b,c,d\}$$\end{document} induces a 2K2\documentclass[12pt]{minimal}
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\begin{document}$$2K_2$$\end{document} in G. The induced separation dimension of a graph G is the smallest cardinality of a family L\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {L}$$\end{document} 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\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {L}$$\end{document}. For each k∈N\documentclass[12pt]{minimal}
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\begin{document}$$k \in \mathbb {N}$$\end{document}, the family of graphs with induced separation dimension at most k is denoted by ISD(k)\documentclass[12pt]{minimal}
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\begin{document}$${\text {ISD}}(k)$$\end{document}. In this article, we initiate a study of this new dimensional parameter. The class ISD(1)\documentclass[12pt]{minimal}
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\begin{document}$${\text {ISD}}(1)$$\end{document} 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)\documentclass[12pt]{minimal}
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\begin{document}$${\text {ISD}}(1)$$\end{document} 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)\documentclass[12pt]{minimal}
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\begin{document}$${\text {ISD}}(1)$$\end{document} is NP-complete for general graphs. Nevertheless, we show that the maximum induced matching problem can be solved efficiently in ISD(1)\documentclass[12pt]{minimal}
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\begin{document}$${\text {ISD}}(1)$$\end{document}. We then briefly study ISD(2)\documentclass[12pt]{minimal}
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\begin{document}$${\text {ISD}}(2)$$\end{document} 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(lgn)\documentclass[12pt]{minimal}
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\begin{document}$$\Theta (\lg n)$$\end{document} 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≥2\documentclass[12pt]{minimal}
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\begin{document}$$k \ge 2$$\end{document}, the recognition problem for ISD(k)\documentclass[12pt]{minimal}
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\begin{document}$${\text {ISD}}(k)$$\end{document} is NP-complete even on AT-free graphs.
