Abstract
We consider the graph classes Grounded-L and Grounded-{𝖫,⅃} corresponding to graphs that admit an intersection representation by 𝖫-shaped curves (or 𝖫-shaped and ⅃-shaped curves, respectively), where additionally the topmost points of each curve are assumed to belong to a common horizontal line. We prove that Grounded-L graphs admit an equivalent characterisation in terms of vertex ordering with forbidden patterns.
 We also compare these classes to related intersection classes, such as the grounded segment graphs, the monotone 𝖫-graphs (a.k.a. max point-tolerance graphs), or the outer-1-string graphs. We give constructions showing that these classes are all distinct and satisfy only trivial or previously known inclusions.
Highlights
An intersection representation of a graph G = (V, E) is a map that assigns to every vertex x ∈ V a set sx in such a way that two vertices x and y are adjacent if and only if the two corresponding sets sx and sy intersect
We have seen that the vertex orders induced by grounded L-representations can be characterised by a pair of forbidden patterns
A characterisation by a single forbidden pattern has been found for vertex orders induced by Max point-tolerance graphs (Mpt) representations [1, 5, 15]
Summary
Graph admits an outer-segment representation, but the converse does not hold, as shown by Cardinal et al [4] They showed that outer-segment graphs form a proper subclass of the class of outer-1-string graphs. Note that in any grounded representation with a horizontal grounding line, the left-toright ordering of the anchors on the grounding line defines a linear order on the vertex set of the represented graph. We say that this linear order is induced by the representation. Induced vertex orders play an important role both in characterising graphs in a given class and in separating different classes
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