An intersection representation of a graph is a map that assigns to every vertex a set in such a way that two vertices and are adjacent if and only if the two corresponding sets and intersect. The graph is then the intersection graph of the set system . Many natural graph classes can be defined as intersection graphs of sets of a special type.
One of the most general classes of this type is the class of string graphs, denoted String. A string graph is an intersection graph of strings, which are bounded continuous curves in the plane. All the graph classes we consider in this paper are subclasses of string graphs.
A natural way of restricting a string representation is to impose geometric restrictions on the strings we consider. This leads, for instance, to segment graphs, which are intersection graphs of straight line segments, or to L-graphs, which are intersection graphs of L-shapes, where an L-shape is a union of a vertical segment and a horizontal segment, in which the bottom endpoint of the vertical segment coincides with the left endpoint of the horizontal one. Apart from L-shapes, we shall also consider L -shapes, which are obtained by reflecting an L-shape along a vertical axis.
Apart from restricting the geometry of the strings, one may also restrict a string representation by imposing conditions on the placement of their endpoints. Following the terminology of Cardinal et al. , we will say that a representation is grounded if all the strings have one endpoint on a common line (called grounding line) and the remaining points of the strings are confined to a single open halfplane with respect to the grounding line. We will usually assume that the grounding line is the -axis, and the strings extend below the line. The endpoint belonging to the grounding line is the anchor of the string.
Similarly, a string representation is an outer representation, if all the strings are confined to a disk, and each string has one endpoint on the boundary of the disk. The endpoint on the boundary is again called the anchor of the string. One may easily see that a graph admits a grounded string representation if and only if it admits an outer-string representation. Such graphs are known as outer-string graphs, and we denote their class Outer-string.
Our first main result, Theorem 2.1 in Section 2, is a characterisation of the class of grounded L-graphs by vertex orderings avoiding a pair of forbidden patterns. Our next main result, presented in Section 3, is a collection of constructions providing separations between the classes in Figure 1, showing that there are no nontrivial previously unknown inclusions among them.
Let us now formally introduce the graph classes we are interested in, and briefly review some relevant previously known results.
1-string graphs are the graphs that admit a string representation in which any two distinct strings intersect at most once. The class of 1-string graphs is denoted 1-String.
Outer-1-string graphs (denoted Outer-1-string) are the graphs that have a string intersection representation which is simultaneously a 1-string representation and an outer-string representation. Note that not every graph from 1-stringOuter-string is necessarily in Outer-1-string, as we shall see in Section 3.
L-graphs (L) are the intersection graphs of L-shapes. This type of representation has received significant amount of interest lately. A notable recent result is a theorem of Gonçalves, Isenmann and Pennarun  showing that every planar graph is an L-graph. Since it is known that L-graphs are a subclass of segment graphs , this result strengthens an earlier result of Chalopin and Gonçalves  showing that all planar graphs are segment graphs.
Max point-tolerance graphs (Mpt), also known as monotone L-graphs, are the graphs with an L-representation in which all the bends of the L-shapes belong to a common downward-sloping line. This class was independently introduced by Soto and Thraves Caro , by Catanzaro et al.  and by Ahmed et al. . Apart from the above intersection representation by L-shapes, it admits several other equivalent characterisations. Notably, Mpt graphs can be characterised as graphs that admit a vertex ordering that avoids a certain forbidden pattern [1, 6, 15]. This graph class is known to be a superclass of several important graph classes, such as outerplanar graphs and interval graphs, among others [1, 6, 15].
Grounded segment graphs (Grounded-seg) are the intersection graphs admitting a grounded segment representation. Cardinal et al.  proved that these are also precisely the intersection graphs of downward rays in the plane. Note that any grounded segment graph also admits an outer-segment representation, but the converse does not hold, as shown by Cardinal et al. . Cardinal et al. also showed that outer-segment graphs are a proper subclass of outer-1-string graphs. This strengthens an earlier result of Cabello and Jejčič , who showed that outer-segment graphs are a proper subclass of outer-string graphs.
Grounded L-graphs (Grounded-L) are the intersection graphs of grounded L-shapes, that is, L-shapes with top endpoint on the -axis. This class of graphs has been first considered by McGuiness , who represented them as intersections of upward-infinite L-shapes. These graphs can also equivalently be represented as intersections of L-shapes inside a disk, with the top endpoint of each L-shape anchored to the boundary of the disk. McGuiness has shown that this class is -bounded, i.e., these graphs have chromatic number bounded from above by a function of their clique number. The -boundedness result has been later generalized to all outer-string graphs by Rok and Walczak .
Grounded -graphs (Grounded-) are analogous to grounded L-graphs, but their representation may use both L-shapes and L -shapes. An argument of Middendorf and Pfeiffer  shows that Grounded- is a subclass of Grounded-Seg.
Circle graphs (Circle) are the intersection graphs of chords inside a circle, or equivalently, the intersection graphs of L-shapes drawn inside a circle, so that both endpoints of each L-shape touch the circle. Circle graphs include all outerplanar graphs .
Interval graphs (Int) are the intersection graphs of intervals on the real line. Equivalently, we may easily observe that these are exactly the graphs with an intersection representation which is simultaneously an Mpt-representation and a Grounded-L-representation. But note that not every graph from the intersection of Mpt and Grounded-L is an interval graph, as witnessed, e.g., by any cycle of length .
Permutation graphs (Per) are the intersection graphs of segments between a pair of parallel lines, with each segment having one endpoint on each of the two lines. Equivalently, we may observe that these are exactly the graphs admitting an L-representation in which the top endpoints of all the L-shapes are on a common horizontal line and the right endpoints are on a common vertical line.
We will always assume implicitly that the intersection representations we deal with satisfy certain non-degeneracy assumptions. In particular, we will assume that the strings have no self-intersections, that any two strings intersect in at most finitely many points (except for interval representations), and that any intersection of two strings is a proper crossing. In particular, an endpoint of a string does not belong to another string. Moreover, we will assume that every segment in a segment representation is non-degenerate, i.e., it has distinct endpoints. This also applies to horizontal and vertical segments forming an L-shape or L -shape. These assumptions imply, in particular, that in any -representation, each intersection is realized as a crossing of a horizontal segment with a vertical one.
Note that in any grounded representation with a horizontal grounding line, the left-to-right 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. Similarly, for an Mpt representation, we can define the induced order by following the top-left to bottom-right order of the bends along their common supporting line. Induced vertex orders play an important part both in characterising graphs in a given class and in separating different classes.
2 Vertex orders with forbidden patterns
Our main result is a characterisation of grounded L-graphs as graphs that admit vertex orderings avoiding a pair of four-vertex patterns. Let us begin by formalising the key notions.
An ordered graph is a pair , where is a graph and is a linear order on . A pattern of order is a triple where is the set while and are two disjoint subsets of . The set is the vertex set of the pattern , is the set of compulsory edges of , and is the set of forbidden edges.
For an ordered graph with , we say that contains a pattern of order if contains distinct vertices such that for every the vertices and are adjacent in , while for every , and are non-adjacent in . If does not contain , we say that it avoids . For simplicity, we will often write an edge as .
Many important graph classes can be characterised in terms of vertex orderings with forbidden patterns, that is, for a class there is a pattern such that a graph is in if and only if it admits a linear order such that avoids ; see Figure 2 for examples of classes with their forbidden patterns. The forbidden pattern characterisation of Mpt was found independently by at least three groups of authors [1, 6, 15].
As our first main result, we show that Grounded-L is characterised by a pair of forbidden patterns.
Consider the patterns and ; see Figure 3. A graph is a grounded L-graph if and only if it has a vertex ordering that avoids both and . In fact, a linear order on avoids the two patterns and if and only if has a grounded L-representation which induces the linear order .
Suppose first that has a grounded L-representation. Let be the L-shapes used in the representation, ordered left to right according to the positions of their anchors. Let and denote, respectively, the horizontal and vertical segment of . Let be the vertex represented by . We will show that the vertex ordering avoids the two patterns and .
Assume that contains , and let be the four vertices forming a copy of . Since is an edge, the two L-shapes and intersect. Let be the rectangle whose vertices are the anchors of and , the bend of and the intersection of and . Since neither nor is adjacent to , we see that is completely outside of , while is inside . It follows that and are disjoint, and fail to represent the compulsory edge of .
Suppose now that contains , and let now be the four vertices forming a copy . Since is an edge, the segment intersects . Moreover, intersects , while does not intersect , and in particular, and fail to represent the compulsory edge of . We conclude that any grounded L-representation of induces a vertex order that avoids and .
To prove the converse, assume that is a graph with a vertex ordering which avoids both and . We will construct a grounded L-representation of inducing the order , with being the L-shape representing the vertex .
We fix the anchor of to be the point on the horizontal axis. Next, we process the vertices left to right, and for a vertex we define the representing shape , assuming have already been defined, and assuming further that for any such that is an edge of , the horizontal segment of reaches to the right of the point .
To define , we first describe its vertical segment . Let be the set of vertices such that and . If is empty, choose the vertical segment to be shorter than any of . In particular, will not intersect any of the L-shapes constructed in previous steps. If is nonempty, let be a vertex from chosen so that is as long as possible (and therefore is as low as possible). Then define to be slightly longer than , so that intersects (recall that reaches to the right of ) but does not intersect any L-shape whose horizontal segment is below . This choice of guarantees that intersects for any .
It remains to define the segment . Let be the largest index such that and . If no such exists, set . The horizontal segment then has length , and in particular, its right endpoint has horizontal coordinate .
Having defined the L-shapes as above, let us verify that their intersection graph is . If is an edge of with , then the definition of guarantees that intersects , and therefore the two L-shapes and intersect.
To prove the converse, suppose for contradiction that for some the two L-shapes and intersect while is not an edge of . Choose such a pair so that is the smallest possible. There must be an index such that is an edge of , otherwise would be too short to intersect . Similarly, there must be an index such that is an edge of , and is longer than , otherwise would not be long enough to intersect .
We now distinguish two cases depending on the relative position of and . If , then and are disjoint (recall that is longer than ) and hence is not an edge of . It follows that the four vertices form the pattern , a contradiction. Suppose now that . It follows that intersects , and therefore is an edge of , by the minimality of . Thus, the four vertices form the pattern , which is again a contradiction. ∎
3 Separations between classes
Consider again the classes in Figure 1. The inclusions indicated by arrows are either easy to observe or follow from known results that we have pointed out in the introduction. Our goal now is to argue that there are no other inclusions among these classes except those that follow by transitivity from the depicted arrows. In particular, the classes are all distinct.
As our main tool, we will use a lemma which is a slight modification of the ‘Cycle Lemma’ of Cardinal et al. . The lemma allows to prescribe the cyclic order of a subset of vertices in an outer-1-string representation of a graph. Let be a graph on vertices , and let be the linear order . The cyclic shift of is the linear order defined as . The reversal of , denoted , is defined as with . We say that two linear orders of are equivalent if one can be obtained from the other by a sequence of cyclic shifts and reversals.
A cycle extension of the ordered graph is an (unordered) graph with these properties (see Figure 4):
is the disjoint union of the sets and . The vertices induce the graph (in particular, ), and induce a cycle of length with edges .
For each vertex , either is adjacent to and has no other neighbors in , or is adjacent to and and has no other neighbors in .
For the classes of graphs we consider, an intersection representation of a graph inducing an order can always be extended into a representation of a cycle extension of , without modifying the curves representing . This is formalised by the next lemma.
Given a graph class , for every -representation of a graph inducing an order on there is a cycle extension of such that a -representation of can be constructed by adding into the given -representation of the curves representing the vertices of .
Suppose we are given a -representation of . It is easy to see that we can add the curves representing the cycle close enough to the grounding line; see Figure 5. Note that for Mpt-representations, each original L-shape may have to be intersected by two consecutive L-shapes from the added cycle. In all the other types of representations, each vertex of will have a unique neighbor among the . ∎
Recall that two linear orders are equivalent if one can be obtained from the other by a sequence of cyclic shifts and reversals. The key property of cycle extensions of is that they restrict the possible vertex orders of the -part to an order equivalent to , as shown by the next lemma.
If is an ordered graph with a cycle extension , then in every grounded 1-string representation of , the order of the vertices of induced by the representation is equivalent to the order .
The proof follows the same ideas as the proof of the Cycle Lemma of Cardinal et al. [4, Lemma 6].
Suppose is an ordered graph with vertices and edge-set , and is its cycle extension, with vertices as in the definition of cycle extension and . When working with the indices of the vertices in , we will assume that arithmetic operations are performed modulo , so , etc.
Suppose that has a grounded 1-string representation. We may transform this representation into an outer-1-string representation, while preserving the induced vertex order up to equivalence. Suppose then that an outer-1-string representation of is given, inside a disk whose boundary is a circle . Let be the string representing , and let be the intersection point of and . The subcurve of between the two intersection points and is the central part of , denoted . The part of between the anchor and the first point of is the initial part of , denoted . Let be the common endpoint of and . Note that is equal to or to . The sequence of curves forms a closed Jordan curve, denoted by . Note that contains all the points . Let be the interior region of .
Consider now a vertex , represented by a string . Note that can only intersect the curve in a point of or possibly . Let be the planar region bounded by the union of the following four curves: , , the arc of between and that contains , and the arc of between the anchors of and that contains the anchors of and .
Note that is the only string among the strings representing that can intersect the boundary of . Note also that the string cannot intersect the boundary of for , and therefore is contained in . Since intersects , and since also cannot cross the boundary of for , it follows that is also contained in , and in particular, the anchor of is in . Therefore, the anchors of appear on in the order which, up to equivalence, corresponds to the order on . ∎
We will now use Lemmas 3.1 and 3.2 to construct graphs that have no representation in a given intersection class. Our goal is to show that there are no inclusions missing in Figure 1. The classes Int, Circle, Outerplanar and Per are well studied , and simple examples show that there are no inclusions among them other than those depicted in Figure 1.
Catanzaro et al. [6, Observation 6.9] observed that the graph (the octahedron) is a permutation graph not in Mpt, and therefore neither Per nor any superclass of Per is contained in Mpt. Cardinal et al.  showed that Grounded-seg is a proper subclass of Outer-1-string. To complete the hierarchy, we only need the following separations.
The following properties hold.
The class Grounded- is not a subclass of Grounded-L.
The class Grounded-seg is not a subclass of Grounded-.
The class Mpt is not a subclass of Outer-1-string.
We first prove part (i) of the theorem. Consider the graph with and . Figure 6 (left) shows a grounded -representation of which induces the order defined as on . Note that there is no grounded L-representation of that would induce the vertex order , because contains the pattern of Theorem 2.1.
Let be the ordered graph obtained by putting and the mirror image of side by side. Formally, has vertex set , edge set and vertex order . Finally, let be the ordered graph obtained by placing two disjoint copies of side by side. Clearly has a grounded -representation which induces the vertex order . However, has no grounded L-representation inducing a vertex order equivalent with , since in any vertex order equivalent with there are four consecutive vertices forming a copy of .
By Lemma 3.1, the ordered graph has a cycle extension that admits a grounded -representation. By Lemma 3.2, any grounded 1-string representation (and therefore any grounded L-representation) of induces on an order which is equivalent with . It follows that has no grounded L-representation, and therefore Grounded- is not a subclass of Grounded-L, as claimed.
For the other two parts of the theorem, the argument is analogous, the main difference is in the choice of the initial ordered graph . To prove part (ii), consider the graph on six vertices whose Grounded-seg representation is in the middle of Figure 6, and let be the vertex order induced by the depicted representation.
Let us argue that has no grounded -representation inducing the vertex order . For contradiction, suppose that such a representation exists, and let denote the L-shape or L -shape representing in this representation. Let and be the horizontal and vertical segment of , respectively. Suppose, without loss of generality, that is longer than . Since and intersect, must intersect , and is a L -shape. Since intersects both and , must be longer than , and intersects . But this means that must intersect either or in order to intersect , a contradiction.
Note that the graph is isomorphic to its reversal. Consider the ordered graph obtained by placing two copies of side by side: note that in any vertex order equivalent to , contains a copy of , and therefore there is no grounded -representation of inducing a vertex order equivalent to . We apply Lemmas 3.1 and 3.2 to and obtain its cycle extension , which is in Grounded-seg but not in Grounded-.
To prove part (iii), consider the graph whose Mpt-representation is depicted in the right part of Figure 6, and let be the vertex order induced by the representation. We claim that there is no grounded 1-string representation of inducing the order . For contradiction, suppose that such a representation exists, and let be the string representing the vertex . Additionally, let denote the anchor of , and for a pair of intersecting strings , let denote their intersection.
Suppose, without loss of generality, that when we follow starting in , we encounter before we encounter . Let be the closed Jordan curve obtained as the union of the subcurve of between and , the subcurve of between and , the subcurve of between and , and the segment of the grounding line. Note that is inside (except , which lies on ), and both and are outside . Therefore, must intersect at least twice: once between and , and once between and . However, can only intersect in the point , a contradiction.
To complete the proof, we first observe that has no grounded 1-string representation inducing a vertex order equivalent with , since such a representation could be trivially transformed into a grounded 1-string representation inducing . We apply Lemmas 3.1 and 3.2 to , to obtain a graph which is in Mpt but not in Outer-1-string. ∎
Note that these results imply that Outer-string is a proper superclass of both Mpt and Outer-1-string.
We remark that Mpt is clearly a subclass of 1-string and of Outer-string, but it is not a subclass of Outer-1-string, as we just saw.
4 Concluding remarks
We have seen that the vertex orders induced by grounded L-representations can be characterised by a pair of forbidden patterns. Previously, a characterisation by a single forbidden pattern has been found for vertex orders induced by Mpt representations [1, 6, 15]. It is an open problem whether such a characterisation can be obtained for other similar grounded intersection classes, such as the class Grounded-.
Another problem concerns the recognition complexity of the graph classes we considered. Recognition of max point-tolerance graphs is mentioned as a prominent open problem by Ahmed et al. , by Catanzaro et al. , as well as by Soto and Thraves Caro . For the classes Grounded-L and Grounded-, recognition is open as well. On the other hand, the recognition problem for Grounded-seg is known to be -complete, as shown by Cardinal et al. . In particular, Grounded-seg cannot be characterised by finitely many forbidden vertex order patterns, unless is equal to NP.
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