On contact graphs of paths on a grid

03/09/2018 ∙ by Zakir Deniz, et al. ∙ DUZCE UNIVERSITESI University of Fribourg 0

In this paper we consider Contact graphs of Paths on a Grid (CPG graphs), i.e. graphs for which there exists a family of interiorly disjoint paths on a grid in one-to-one correspondance with their vertex set such that two vertices are adjacent if and only if the corresponding paths touch at a grid-point. We examine this class from a structural point of view which leads to constant upper bounds on the clique number, the chromatic number and the clique chromatic number. We further investigate the relation between planar graphs and CPG graphs and show that CPG graphs are not necessarily planar and not all planar graphs are CPG. Our class generalizes the well studied class of VCPG graphs.

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1 Introduction

Asinowski et al. [3] introduced the class of vertex intersection graphs of paths on a grid, referred to as VPG graphs. An undirected graph is called a VPG graph if one can associate a path on a grid with each vertex such that two vertices are adjacent if and only if the corresponding paths intersect on at least one grid-point. It is not difficult to see that the class of VPG graphs coincides with the class of string graphs, i.e. intersection graphs of curves in the plane (see [3]).

A natural restriction which was forthwith considered consists in limiting the number of bends (i.e. degrees turns at a grid-point) that the paths may have: an undirected graph is a -VPG graph, for some integer , if one can associate a path on a grid having at most bends with each vertex such that two vertices are adjacent if and only if the corresponding paths intersect on at least one grid-point. Since their introduction, -VPG have been extensively studied (see [2, 3, 5, 7, 8, 9, 13, 14, 17, 18, 19]).

A notion closely related to intersection graphs is that of contact graphs. Such graphs can be seen as a special type of intersection graphs of geometrical objects in which these objects are not allowed to have common interior points but only to touch each other. Contact graphs of various types of objects have been studied in the literature (see, e.g., [1, 10, 11, 20, 21, 22]). In this paper, we consider Contact graphs of Paths on a Grid (CPG graphs for short) which are defined as follows. A graph is a CPG graph if the vertices of can be represented by a family of interiorly disjoint paths on a grid, two vertices being adjacent in if and only if the corresponding paths touch, i.e. share a grid-point which is an endpoint of at least one of the two paths (see Fig. 1). Note that this class is hereditary, i.e. closed under vertex deletion. Similarly to VPG, a -CPG graph is a CPG graph admitting a representation in which each path has at most bends. Clearly, any -CPG graph is also a -VPG graph.

(a) Allowed contacts.

(b) Forbidden contact.
Figure 1: Examples of types of contact between two paths (the endpoints of a path are marked by an arrow).

Aerts and Felsner [1] considered a similar family of graphs, namely those admitting a Vertex Contact representation of Paths on a Grid (VCPG for short). The vertices of such graphs can be represented by a family of interiorly disjoint paths on a grid, but the adjacencies are defined slightly differently: two vertices are adjacent if and only if the endpoint of one of the corresponding paths touches an interior point of the other corresponding path (observe that this is equivalent to adding the constraint forbidding two paths from having a common endpoint, i.e. contacts as in Fig. 0(a) on the right). This class has been considered by other authors as well (see [6, 7, 13, 18, 24]).

It is not difficult to see that graphs admitting a VCPG are planar (see [1]) and it immediately follows from the definition that those graphs are CPG graphs. This containment is in fact strict even when restricted to planar CPG graphs, as there exist, in addition to nonplanar CPG graphs, planar graphs which are CPG but do not admit a VCPG.

To the best of our knowledge, the class of CPG graphs has never been studied in itself and our present intention is to provide some structural properties (see Section 3). By considering a specific weight function on the vertices, we provide upper bounds on the number of edges in CPG graphs as well as on the clique number and the chromatic number (see Section 3). In particular, we show that -CPG graphs are 4-colorable and that 3-colorability restricted to -CPG is -complete (see Section 5). We further prove that recognizing -CPG graphs is -complete. Additionally, we show that the classes of CPG graphs and planar graphs are incomparable (see Section 4).

2 Preliminaries

Throughout this paper, all considered graphs are undirected, finite and simple. For any graph theoretical notion not defined here, we refer the reader to [12].

Let be a graph with vertex set and edge set . The degree of a vertex , denoted by , is the number of neighbors of in . A graph is -regular if the degree of every vertex in is . A clique (resp. stable set) in is a set of pairwise adjacent (resp. nonadjacent) vertices. The graph obtained from by deleting a vertex is denoted by . For a given graph , is -free if it contains no induced subgraph isomorphic to .

As usual, (resp. ) denotes the complete graph (resp. chordless cycle) on vertices and denotes the complete bipartite graph with bipartition such that and . Given a graph , the line graph of , denoted by , is the graph such that each vertex in corresponds to an edge in and two vertices are adjacent in if and only if their corresponding edges in have a common endvertex.

A graph is planar if it can be drawn in the plane without crossing edges; such a drawing is then called a planar embedding of . A planar embedding divides the plane into several regions referred to as faces. A planar graph is maximally planar if adding any edge renders it nonplanar. A maximally planar graph has exactly faces, where is the number of vertices in the graph. A graph is a minor of a graph , if can be obtained from by deleting edges and vertices and by contracting edges. It is well-known that a graph is planar if and only if it does not contain or as a minor [12].

A coloring of a graph is a mapping associating with every vertex an integer , called a color, such that for every edge . If at most distinct colors are used, is called a -coloring. The smallest integer such that admits a -coloring is called the chromatic number of , denoted by .

Consider a rectangular grid where the horizontal lines are referred to as rows and the vertical lines as columns. The grid-point lying on row and column is denoted by . An interior point of a path on is a point belonging to and different from its endpoints; the interior of is the set of all its interior points. A graph is CPG if there exists a collection of interiorly disjoint paths on a grid such that is in one-to-one correspondence with and two vertices are adjacent in if and only if the corresponding paths touch; if every path in has at most bends, is -CPG. The pair is a CPG representation of , and more specifically a -bend CPG representation if every path in has at most bends. In the following, the path representing some vertex in a CPG representation of a graph is denoted by , or simply if it is clear from the context.

Let be a CPG graph and be a CPG representation of . A grid-point is of type I if it corresponds to an endpoint of four paths in (see Fig. 1(a)), and of type II if it corresponds to an endpoint of two paths in and an interior point of a third path in (see Fig. 1(b)).

(a) Type I.

Subtype a

Subtype b
(b) Type II.
Figure 2: Two types of grid-points.

For any grid-point , we denote by the number of edges in the subgraph induced by the vertices whose corresponding paths contain or have as an endpoint. Note that this subgraph is a clique and so if paths touch at grid-point .

For any path , we denote by (resp. ) the interior (resp. endpoints) of . For a vertex , we define the weight of with respect to , denoted by or simply if it is clear from the context, as follows. Let () be the endpoints of the corresponding path in and consider, for ,

Then .

Observation 1

Let be a CPG graph and be a CPG representation of . For any vertex and , where equality holds if and only if is a grid-point of type I or II.

Indeed, the contribution of to is maximal if all four grid-edges containing are used by paths of , which may only happen when is a grid-point of type I or II.

Remark. In fact, we have for any vertex and .

Observation 2

Let be a CPG graph and be a CPG representation of . Then

where equality holds if and only if all paths of pairwise touch at most once.

Indeed, if , we may assume that either an endpoint of touches the interior of , or and have a common endpoint. In the first case, the edge is fully accounted for in the weight of , and in the second case, the edge is accounted for in both and by one half. The characterization of equality then easily follows.

3 Structural Properties of CPG Graphs

In this section, we investigate CPG graphs from a structural point of view and present some useful properties which we will further exploit.

Lemma 1

A CPG graph is either 6-regular or has a vertex of degree at most 5.

Proof

If is a CPG graph and is a CPG representation of , by combining Observations 1 and 2, we obtain

Remark. We can show that there exists an infinite family of 6-regular CPG graphs. Due to lack of space, this proof is here omitted but can be found in Section 7.1 of the Appendix.

For -CPG graphs, we can strengthen Lemma 1 as follows.

Proposition 1

Every -CPG graph has a vertex of degree at most 5.

Proof

Let be a -CPG graph and be a -bend CPG representation of . Denote by the upper-most endpoint of a path among the left-most endpoints in , and by (with ) an arbitrary path having as an endpoint. Since is a -bend CPG representation, no path uses the grid-edge on the left of , for otherwise would not be a left-most endpoint. Therefore, contributes to the weight of with respect to by at most and, by Observations 1 and 2, we have

which implies the existence of a vertex of degree at most 5. ∎

A natural question that arises when considering CPG graphs is whether they may contain large cliques. It immediately follows from Observation 2 that CPG graphs cannot contain , for . This can be further improved as shown in the next result.

Theorem 3.1

CPG graphs are -free.

Proof

Since the class of CPG graphs is hereditary, it is sufficient to show that is not a CPG graph. Suppose, to the contrary, that is a CPG graph and consider a CPG representation of . Observe first that the weight of every vertex with respect to must be exactly , as otherwise by Observation 1, we would have which contradicts Observation 2. This implies in particular that every grid-point corresponding to an endpoint of a path is either of type I or II. Furthermore, any two paths must touch at most once, for otherwise by Observation 2, . Hence, if we denote by (resp. ) the set of grid-points of type I (resp. type II), then since for all and for all , we have that , which implies . Suppose that there exists a path having one endpoint corresponding to a grid-point of type I and the other corresponding to a grid-point of type II. Since the corresponding vertex has degree 6, must then properly contain an endpoint of another path which, as first observed, necessarily corresponds to a grid-point of type II. But vertex would then have degree as no two paths touch more than once, a contradiction. Hence, every path has both its endpoints of the same type. But then, ; indeed, if there exists a path having both its endpoints of type I, since no two paths touch more than once, this implies that every path has both its endpoints of type I, i.e. , a contradiction. Now, if we consider each grid-point of type II as a vertex and connect any two such vertices when the corresponding grid-points belong to a same path, then we obtain a planar embedding of a 4-regular graph on 7 vertices. But this contradicts the fact that every 4-regular graph on 7 vertices contains as a minor (a proof of this result can be found in Section 7.2 of the Appendix). ∎

However, CPG graphs may contain cliques on 6 vertices as shown in Proposition 2. Due to lack of space, its proof is omitted here and can be found in Section 7.4 of the Appendix.

Proposition 2

is in -CPG -CPG.

We conclude this section with a complexity result pointing towards the fact that there may not be a polynomial characterization of -CPG graphs. Let us first introduce rectilinear planar graphs: a graph is rectilinear planar if it admits a rectilinear planar drawing, i.e. a drawing mapping each edge to a horizontal or vertical segment.

Theorem 3.2

Recognition is -complete for -CPG graphs.

Proof

We show that a graph is rectilinear planar if and only if its line graph is -CPG. As Recognition for rectilinear planar graphs was shown to be -complete in [16], this concludes the proof. Suppose is a rectilinear planar graph and let be the collection of horizontal and vertical segments in a rectilinear planar drawing of . It is not difficult to see that the contact graph of is isomorphic to . Conversely, assume that is a -CPG graph and consider a 0-bend CPG representation of . Since is -free [4], every path in has at most two contact points. Thus, by eventually shortening paths, we may assume that contacts only happen at endpoints of paths. Therefore, induces a rectilinear planar drawing of , where each vertex corresponds to a contact point in and each edge is mapped to its corresponding path in . ∎

4 Planar CPG Graphs

In this section, we focus on planar graphs and their relation with CPG graphs. In particular, we show that not every planar graph is CPG and not all CPG graphs are planar.111We can further show that not all CPG graphs are 1-planar as is CPG but not 1-planar [25].

Lemma 2

If is a CPG graph for which there exists a CPG representation containing no grid-point of type I or II.a, then is planar. In particular, if is a triangle-free CPG graph, then is planar.

Proof

Let be a CPG graph for which there exists a CPG representation containing no grid-point of type I or II.a. By considering each path of as a curve in the plane, it follows that is a curve contact graph having a representation (namely ) in which any point in the plane belongs to at most three curves. Furthermore, whenever a point in the plane belongs to the interior of a curve and corresponds to an endpoint of two other curves, then those two curves lie on the same side of (recall that there is no grid-point of type II.a). Hence, it follows from Proposition 2.1 in [20] that is planar.

If is a triangle-free CPG graph, then no CPG representation of contains grid-points of type I or II.a. Hence, is planar. ∎

Remark. Since is a triangle-free nonplanar graph, it follows from Lemma 2 that is not CPG. Therefore, CPG graphs are -free. Observe however that for any , -CPG is not a subclass of planar graphs as there exist -CPG graphs which are not planar (see Fig. 3).

(a) A nonplanar graph .

(b) A -bend CPG representation of .
Figure 3: A -CPG graph containing as a minor (contract the edge ).

It immediatly follows from [7] that all triangle-free planar graphs are -CPG; hence, we have the following corollary.

Corollary 1

If a graph is triangle-free, then is planar if and only if is -CPG.

The next result allows us to detect planar graphs that are not CPG.

Lemma 3

Let be a planar graph. If is a CPG graph, then has at most vertices of degree at most 3, where denotes the number of faces of . In particular, if is maximally planar, then has at most 12 vertices of degree at most 3.

Proof

Let be a planar CPG graph and a CPG representation of . Denote by the subset of vertices in of degree at most 3. If a path , with , touches every other path in at most once, then, since at least one endpoint of is then not a grid-point of type I or II, the weight of with respect to is at most . Thus, if we assume that this is the case for all paths whose corresponding vertex is in , we have by Observation 2

On the other hand, if there exists such that touches some path more than once, then the above inequality still holds as the corresponding edge is already accounted for. Using the fact that (Euler’s formula), we obtain the desired upper bound. Moreover, if is maximally planar, then and so . ∎

Remark. In Fig. 3(a), we give an example of a maximally planar graph which is not CPG due to Lemma 3. It is constructed by iteratively adding a vertex in a triangular face, starting from the triangle, so that it has exactly 13 vertices of degree 3. There exist however maximally planar graphs which are CPG (see Fig. 3(b)). Note that maximally planar graphs do not admit a VCPG [1].

(a) A non CPG maximally planar graph.

1

2

3

4

5

6

7

(b) A maximally planar CPG graph.
Figure 4: Two maximally planar graphs.

5 Coloring CPG Graphs

In this section, we provide tight upper bounds on the chromatic number of -CPG graphs for different values of and investigate the 3-Colorability problem for CPG graphs. The proof of the following result is an easy exercise left to the reader (see Section 7.3 of the Appendix).

Theorem 5.1

CPG graphs are 6-colorable.

Remark. Since is -CPG, this bound is tight for -CPG graphs with . We leave as an open problem whether this bound is also tight for -CPG graphs (note that it is at least 5 since is -CPG).

Theorem 5.2

-CPG graphs are 4-colorable. Moreover, is a 4-chromatic -CPG graph.

Proof

Let be a -CPG graph and a -bend CPG representation of . Denote by (resp. ) the set of rows (resp. columns) of on which lies at least one path of . Since the representation contains no bend, if is a row in (resp. column in ), then the set of vertices having their corresponding path on induces a collection of disjoint paths in . If is another row in (resp. column in ), then no path in touches a path in . Hence, it suffices to use two colors to color the vertices having their corresponding path in a row of and two other colors to color the vertices having their corresponding path in a column of to obtain a proper coloring of . ∎

It immediately follows from a result in [21] that the 3-colorability problem is -complete in CPG, even if the graph admits a representation in which each grid-point belongs to at most two paths. We conclude this section by a strenghtening of this result.

Theorem 5.3

3-Colorability is -complete in -CPG.

Proof

We exhibit a polynomial reduction from 3-Colorability restricted to planar graphs of maximum degree 4, which was shown to be -complete in [15].

Let be a planar graph of maximum degree 4. It follows from [26] that admits a grid embedding where each vertex is mapped to a grid-point and each edge is mapped to a grid-path with at most 4 bends, in such a way that all paths are interiorly disjoint (such an embedding can be obtained in linear time). Denote by such an embedding, where is the set of grid-points in one-to-one correspondence with and is the set of grid-paths in one-to-one correspondence with . For any vertex , we denote by the grid-point in corresponding to and by (resp. ) the path of , if any, having as an endpoint and using the grid-edge above (resp. below) . For any edge , we denote by the path in corresponding to . We construct from a 0-bend CPG representation in such a way that the corresponding graph is 3-colorable if and only if is 3-colorable.

By eventually adding rows and columns to the grid, we may assume that the interior of each path in is surrounded by an empty region, i.e. no path or grid-point of lies in the interior of this region. In the following, we denote this region by (delimited by red dashed lines in every subsequent figure) and assume, without loss of generality, that it is always large enough for the following operations.

We first associate with every vertex a vertical path containing the grid-point as follows. If (resp. ) is not defined, the top (resp. lower) endpoint of is (resp. ) for a small enough so that the segment (resp. ) touches no path of . If has at least one bend, then the top endpoint of lies at the border of on column (see Fig. 4(a)). If has no bend, then the top endpoint of lies at the middle of (see Fig. 4(b)). Similarly, we define the lower endpoint of according to : if has at least one bend, then the lower endpoint of lies at the border of on column , otherwise it lies at the middle of .

(a) contains at least one bend.

(b) contains no bend.
Figure 5: Constructing the path corresponding to vertex (in blue).

For any path of with at least two bends, an interior vertical segment of is a vertical segment of containing none of its endpoints (note that since every path in has at most 4 bends, it may contain at most two interior vertical segments). We next replace every interior segment of by a slightly longer vertical path touching the border of (see Fig. 6).

(a) A path containing one interior vertical segment.

(b) A path containing two interior vertical segments.
Figure 6: Replacing interior vertical segments by 0-bend paths (in blue).

We finally introduce two gadgets (see Fig. 7) and , where is the subgraph of induced by , as follows. Denote by the set of vertical paths introduced so far and by the set of vertices of the contact graph of . Observe that contains a copy of and that two vertices are adjacent in the contact graph of if and only if they are both copies of vertices in and the path of corresponding to the edge between these two copies is a vertical path with no bend. Now, along each path of such that the vertical paths and of do not touch, we add gadgets and as follows. Let be the vertical paths of encountered in order when going along from to and let be the vertex of corresponding to , for . Note that (resp. ) is the path corresponding to vertex (resp. ) and that , for , is a path corresponding to an interior vertical segment of (this implies in particular that ). We add the gadget in between and by identifying with and with . Moreover, for any , we add the gadget in between and by identifying with and with (see Fig. 8 where and each box labeled (resp. ) means that gadget (resp. ) has been added by identifying the vertex lying to the left of the box to and the vertex lying on the right of the box to (resp. )).

Figure 7: The gadget (left) and a -bend CPG representation of it (right).

Figure 8: Adding gadgets and .

The resulting graph remains -CPG. Indeed, we may add -bend CPG representations of the gadgets and inside and at different heights so that they do not touch any other such gadget, as shown in Fig. 9. In Section 7.5 of the Appendix, we give a local example of the resulting -bend CPG representation .

(a) Adding gadget .

(b) Adding gadget .
Figure 9: Locally adding gadgets to control the color of the vertices.

We now show that is 3-colorable if and only if is. To this end, we prove the following.

Claim 1
  • [leftmargin=*]

  • In any 3-coloring of , we have .

  • In any 3-coloring of , we have and .

Proof. Let be a 3-coloring of and assume without loss of generality that . Clearly, at least two vertices among , and have the same color. If vertices , and all have the same color, say , then either and , or and . Therefore, and since is adjacent to all three colors, we then obtain a contradiction. Now if vertices and have the same color, say , then vertex has color and both and have color , a contradiction. Hence, either or . By symmetry, we may assume that vertices and have the same color, say , and that vertex has color . This implies that vertex has color , vertices and have color and vertex has color ; but then, . This proves the first point of the claim. Observe that each coloring of and with distinct colors can be extended to a 3-coloring of and .

As for the second point, since vertices and have color , both and must have color , and since vertex has color , vertex must have color . Consequently, .

We finally conclude the proof of Theorem 5.3. By Claim 1, if is a 3-coloring of then, for any path of , we have and for all . Hence, induces a 3-coloring of . Conversely, it is easy to see that any 3-coloring of can be extended to a 3-coloring of . ∎

6 Conclusion

We conclude by stating the following open questions:

  1. Are -CPG graphs 5-colorable?

  2. Can we characterize those planar graphs which are CPG?

  3. Is Recognition -complete for -CPG graphs with ?

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7 Appendix

7.1 Proof of the existence of an infinite family of 6-regular CPG graphs

It is clear that there exists an infinite family of CPG graphs having a vertex of degree at most 5. On the other hand, the existence of an infinite family of 6-regular CPG graphs is a priori not guaranteed. We can however show that it is the case. Indeed, consider a 4-regular planar graph . From [26], it follows that admits an embedding on the grid where each vertex is mapped to a distinct grid-point and each edge is mapped to a path on the grid whose endpoints are the grid-points corresponding to the endvertices of , in such a manner that all paths are interiorly disjoint. We derive therefrom a CPG representation of the line graph of which is 6-regular: each edge in the embedding of on the grid corresponds to the path associated with vertex of and each vertex in the embedding of on the grid is the grid-point where the four corresponding paths pairwise touch. The existence of an infinite family of 6-regular CPG graphs then follows from the existence of an infinite family of 4-regular planar graphs [23].

7.2 4-regular graphs on 7 vertices are non-planar

We show that every 4-regular graph on 7 vertices contains as a minor. Let and . Then and have at least 3 neighbors in . If , then clearly contains as a minor. Hence, has a neighbor which is non-adjacent to ; by symmetry, also has a neighbor which is non-adjacent to . But then and must be adjacent as well as and (recall that the graph is 4-regular), and again contains as a minor.

7.3 Proof of Theorem 5.1

If is a CPG graph, by Lemma 1 is either 6-regular or has a vertex of degree at most 5. In the first case, the result follows from Theorem 3.1 and Brooks’ Theorem (any graph

, apart from the complete graph and the cycle of odd length, may be colored using

colors where is the maximum degree of ). Otherwise, contains a vertex of degree at most 5 and we conclude by induction.

7.4 Proof of Proposition 2

Before turning to the proof of Proposition 2, we first make several observations regarding -CPG graphs. All following statements remain true up to reflection across a vertical or horizontal line and by inverting the role of rows and columns.

Observation 3

Let be a -CPG graph and be a -bend CPG representation of . Assume there exist two distinct grid-points in , and , with , such that (resp. ) is an endpoint or the bend-point of a path (resp. ) in . Then, if (resp. ) uses the grid-edge on the left of (resp. on the right of ), and can not touch (see Fig. 10).

Figure 10: Examples where and can not touch.
Observation 4

Let be a -CPG graph and be a -bend CPG representation of . Assume there exist three distinct grid-points in , , and , with , such that (resp. ) is an endpoint of a path (resp. ) in and is an endpoint or the bend-point of a path in . Then, if (resp. , ) uses the grid-edge on the left of (resp. below , below ), all three paths can not pairwise touch (see Fig. 11).

Figure 11: Examples where , and can not pairwise touch.

Proof of Proposition 2. is in -CPG as shown in Fig. 12. Assume by contradiction that is a -CPG graph and consider a -bend CPG representation of . Since every vertex is of degree 5, we can assume, without loss of generality, that every endpoint of a path belongs to another path.

Figure 12: A -bend CPG representation of .

In the following, let with , be a path in and denote by and its two endpoints.

Claim 2

contains no grid-point of type I.

Proof. Assume, without loss of generality, that is a grid-point of type I. Since , there exists a path either touching in its interior or having as an endpoint. If has no bend, it follows from Observation 3 that can not touch one of the paths touching at . If has a bend, we may assume without loss of generality that the horizontal segment of contains and lies to the left and below its vertical segment (see Fig. 13). Then, either we conclude similarly by Observation 3 (see Fig. 12(a) where colors are used to specify which paths cannot touch depending on the position of ); or, denote by the other path not touching at . If has as an endpoint, then by Observation 3, must use the grid-edge on the left of and must touch the interior of the vertical segment of and lie to its left; and we conclude by Observation 4 that , and the path with endpoint using the grid-edge below can then not pairwise touch (see Fig. 12(b) where colors are used to specify which paths cannot pairwise touch).

(a) Different positions for .

(b) Three paths that can not pairwise touch.
Figure 13: has a bend.

Otherwise, touches the interior of the vertical segment of while lying to its left, and contains ; but then, has an endpoint lying to the right of the vertical segment of and we distinguish three cases depending on the row on which is . First, if lies on the same row as , we conclude by Observation 4 that three , and the path with endpoint using the grid-edge below cannot pairwise touch (see Fig. 13(a) where colors are used to specify which paths cannot pairwise touch). Second, if has a bend and lies on a row above the row of , then cannot touch the path with endpoint using the grid-edge below (see Fig. 13(b) where colors are used to specify which paths cannot touch). Finally, if has a bend and lies on a row below the row of , then , and the path with endpoint using the grid-edge on the left of cannot pairwise touch (see Fig. 13(c) where colors are used to specify which paths cannot pairwise touch), which concludes the proof.

(a) , lie on the same row.

(b) lies above .

(c) lies below .
Figure 14: An endpoint of lies to the right of the vertical segment of .
Claim 3

contains no grid-point of type II.

Proof. We first show that both endpoints of any path in cannot be grid-points of type II. For the sake of contradiction, assume, without loss of generality, that both and are grid-points of type II. If has no bend, then by Observation 3, one path touching at and one path touching at cannot touch (see Fig. 15 where colors are used to specify which paths cannot touch).

Figure 15: Examples where two paths cannot touch when has no bend.

If has a bend, assume without loss of generality that the horizontal segment of contains and lies to the left and below of its vertical segment (see Fig. 16). If is a grid-point of type II.b, then is the bend-point or an endpoint of a path using the grid-edge below . But then, since is assumed to be grid-point of type II, it is similarly an endpoint or the bend-point of a path using the grid-edge above ; and we conclude by Observation 3 that those two paths cannot touch. We conclude by symmetry that can neither be a grid-point of type II.b and assume henceforth that both and are grid-points of type II.a. Now since , there exists a path touching in its interior. If touches the vertical (resp. horizontal) segment of and lies to its right (resp. below) then, by Observation 3, cannot touch the other path having (resp. ) as an endpoint (see Fig. 15(a) where colors are used to specify which paths cannot touch); and, if touches the vertical segment of and lies to its left, or if touches the horizontal segment of and lies above it, then by Observation 4, and the other paths having respectively and as an endpoint can then not pairwise touch (see Fig. 15(b) where colors are used to specify which paths can not pairwise touch), which concludes the first part of this proof.

(a) touches on the right.

(b) touches on the left.
Figure 16: Examples where two paths cannot touch when has a bend.

Now suppose that contains a grid-point of type II and assume without loss of generality that . Since is not a grid-point of type I nor a grid-point of type II, there must exist a path touching in its interior and a path either also touching in its interior or having as an endpoint.

Let us first assume that is of type II.b. Then, if has no bend, we conclude by Observation 3 that cannot touch one of the paths touching at as the latter is the bend-point and an endpoint of two distinct paths using grid-edges orthogonal to . Hence, must have a bend and as previously, we may assume without loss of generality that the horizontal segment of contains and lies to the left and below of its vertical segment. By Observation 3, we know that cannot touch the vertical segment of from the right nor can it touch the horizontal segment of (see Fig. 16(a)); and since the same holds for , we conclude by Observation 4 that , and the path with endpoint using the grid-edge below cannot pairwise touch (see Fig. 16(b) where colors are used to specify which paths cannot pairwise touch).

(a) The unique possibility for to touch .

(b) Example where three paths cannot pairwise touch.
Figure 17: has a bend and is of type II.b.

Assume henceforth that is of type II.a and denote by the other path having as an endpoint. If has no bend, by Observation 3, must then touch orthogonally as it would otherwise not be able to touch . If lies on the opposite side of than , it must then have by Observation 3, a common endpoint with belonging to the interior of i.e. is a grid-point of type II.a; but then, it is clear that