CPG graphs: Some structural and hardness results

03/05/2019 ∙ by Esther Galby, et al. ∙ 0

In this paper we continue the systematic study of Contact graphs of Paths on a Grid (CPG graphs) initiated in [11]. A CPG graph is a graph for which there exists a collection of pairwise interiorly disjoint paths on a grid in one-to-one correspondence with its vertex set such that two vertices are adjacent if and only if the corresponding paths touch at a grid-point. If every such path has at most k bends for some k ≥ 0, the graph is said to be B_k-CPG. We show that for any k ≥ 0, the class of B_k-CPG graphs is strictly contained in the class of B_k+1-CPG graphs even within the class of planar graphs, thus implying that there exists no k ≥ 0 such that every planar CPG graph is B_k-CPG. Additionally, we examine the computational complexity of several graph problems restricted to CPG graphs. In particular, we show that Independent Set and Clique Cover remain NP-hard for B_0-CPG graphs. Finally, we consider the related classes B_k-EPG of edge-intersection graphs of paths with at most k bends on a grid. Although it is possible to optimally color a B_0-EPG graph in polynomial time, as this class coincides with that of interval graphs, we show that, in contrast, 3-Colorability is NP-complete for B_1-EPG graphs.

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1 Introduction

Golumbic et al. [19] introduced the class of Edge intersection graphs of Paths on a Grid (EPG for short) that is, graphs for which there exists a collection of paths on a grid in one-to-one correspondence with their vertex set such that two vertices are adjacent if and only if the corresponding paths intersect on at least one grid-edge. Since every graph is EPG [19], a natural restriction which was forthwith considered consists in limiting the number of bends (90 degree turns at a grid-point) that the paths may have: a graph is -EPG, for some integer , if it has an EPG-representation in which each path on the grid corresponding to a vertex has at most bends.

Later on, Asinowski et al. [4] considered the closely related class of Vertex intersection graphs of Paths on a Grid (VPG for short). Likewise, the vertices of such graphs may be represented by paths on a grid but two vertices are adjacent if and only if the corresponding paths intersect on at least one grid-point. Similarly to EPG, they then defined the class -VPG, for , consisting of those VPG graphs admitting a representation where each path has at most bends.

Since their introduction, EPG and VPG graphs have been extensively studied (see, e.g., [2, 3, 4, 5, 6, 10, 13, 14, 15, 21, 22, 27]). One of the main considered problems consists in determining the bend-number with respect to EPG (resp. VPG) representations of a given graph class : the bend number of with respect to EPG (resp. VPG) representations is the minimum integer such that every graph in is -EPG (resp. -VPG). For example, Gonçalves et al. [20] showed that planar graphs have a bend number with respect to VPG representations of ; the bend number of planar graphs with respect to EPG representations has yet to be determined (it is either or by [21]).

In this paper, we are interested in the contact counterpart of these graph classes, namely the class of Contact graphs of Paths on a Grid (CPG for short). Although CPG graphs have been considered in the literature (see, e.g., [7, 8, 14, 16, 20, 23]), a systematic study of this class was first initiated in [11]. A graph is a CPG graph if there exists a collection of pairwise interiorly disjoint paths on a grid in one-to-one correspondence with such that two vertices are adjacent if and only if the corresponding paths touch. If every such path has at most bends, for some , then the graph is -CPG. The pair is a CPG representation of and, more specifically, a -bend CPG representation of if every path in has at most bends. The bend number with respect to CPG representations is defined mutatis mutandis. Clearly, any -CPG graph is also a -VPG graph and a -EPG graph.

Aerts and Felsner [1] considered the family of graphs admitting a Vertex Contact representation of Paths on a Grid, which constitutes a subclass of CPG graphs as paths in such representations are not allowed to share a common endpoint. Such graphs are easily seen to be planar and in fact, this class is strictly contained in that of planar CPG graphs [11]. It is then natural to ask whether every planar graph is CPG and whether there exists such that every planar CPG graph is -CPG. The former was answered in the negative in [11]. The latter is settled in Section 3 by exhibiting for any , a planar graph which is -CPG but not -CPG. Note that since every planar graph is -VPG [20], it follows that for any , even within the class of planar graphs (and in fact, it is easy to see that this remains true for ).

Asinowski et al. [4] provided several complexity results for problems restricted to -VPG graphs. More specifically, they showed that Independent Set, Hamiltonian Cycle, Hamiltonian Path and Colorability are all -complete, whereas Clique is polynomial-time solvable. It is then natural to consider the complexity of such problems when restricted to -CPG graphs. The fact that every planar bipartite graph is -CPG [16] immediately implies that problems which are -complete for this class, such as Dominating Set, Feedback Vertex Set, Hamiltonian Cycle and Hamiltonian Path (see, e.g., [9, 26]), remain -complete in -CPG. It was furthermore shown in [11] that 3-Colorability is -complete in -CPG. In Section 4, we consider Independent Set and Clique Cover and show that they are -complete in -CPG.

It was shown in [13] that Colorability is -complete in -EPG. On the other hand, the problem is polynomial-time solvable in -EPG as this class coincides with that of interval graphs. In Section 5, we provide a complexity dichotomy for the related 3-Colorability problem restricted to EPG graphs. The problem is -complete in -EPG, for each , since it is -complete in -CPG [11]. We complete the picture by showing that 3-Colorability is -complete in -EPG.

We remark that all our complexity results hold even if a CPG (resp. EPG) representation of the graph is given.

2 Preliminaries

Throughout the paper, the 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. The degree of a vertex is the size of its neighborhood and a vertex is cubic if it has degree 3. For , the -subdivision of an edge consists in replacing it by a path , where are new vertices; the -subdivision of is the graph obtained by -subdividing every edge of . The line graph of has vertex set and two vertices and are adjacent if and only if they have a common endvertex in .

A graph is subcubic if every vertex has degree at most 3. A graph is planar if it admits an embedding in the plane where no two edges cross. Given a graph , we say that a graph is H-free if it contains no induced subgraph isomorphic to . The triangle is the complete graph on three vertices and the diamond is the graph obtained by removing an edge from the complete graph on four vertices.

A subset is an independent set of if any two vertices in are nonadjacent; the maximum size of an independent set of is denoted by . A subset is a vertex cover of if each edge of is incident to at least one vertex in ; the minimum size of a vertex cover of is denoted by . A subset is a clique if any two vertices in are adjacent. A clique cover of is a set of cliques such that each vertex of belongs to at least one of them; the minimum number of cliques in a clique cover of is denoted by .

Let be a CPG graph and be a CPG representation of . The path in representing some vertex is denoted by . An interior point of a path is a point belonging to and different from its endpoints; the interior of is the set of all its interior points. A grid-point is of type II.a if it is an endpoint of two paths and an interior point, different from a bend-point, of a third path; a grid-point is of type II.b if it is an endpoint of two paths and a bend-point of a third path (see [11]). A snake is a -bend CPG representation of a path.

3 Planar CPG graphs

Consider the planar graph , with , depicted in Figure 1. We refer to the vertices , for , as the secondary vertices, and to the vertices , for and a given , as the -sewing vertices. A -bend CPG representation of is given in Figure 2 (the blue paths correspond to sewing vertices and the red paths correspond to secondary vertices).

(a) The planar graph .

(b) The gadget .
Figure 1: The construction of .

Given a CPG representation of , a path in corresponding to a secondary vertex (resp. an -sewing vertex) is called a secondary path (resp. an -sewing path). A secondary path is pure if no endpoint of or belongs to . We now use the graph to prove the following.

(a) is even.

(b)

is odd.

Figure 2: A -bend CPG representation of (the endpoints are marked by an arrow).
Theorem 3.1

For any , there exists a planar graph which is -CPG but not -CPG.

Proof

We show that in any CPG representation of , there exists a path with at least bends. Since is -CPG (see Figure 2), this would conclude the proof. We first observe the following.

Observation 1

If a path is pure, then one endpoint of belongs to and the other endpoint belongs to .

We next prove the following two claims.

Claim 1

Let and be two pure paths and let and with be two -sewing vertices, for some . If a grid-point belongs to , then corresponds to an endpoint of both and , and a bend-point of either or .

It follows from Observation 1 and the fact that and are nonadjacent to both and , that no endpoint of or belongs to or . Consequently, one endpoint of (resp. ) belongs to and the other endpoint belongs to . By definition, corresponds to an endpoint of at least one of and , which implies that belongs to or . But then must be an endpoint of both and ; in particular, is a grid-point of type either II.a or II.b. Without loss of generality, we may assume that . We denote by (resp. ) the endpoint of (resp. ) belonging to . Suppose, to the contrary, that is of type II.a. The union of , and the portion of between and defines a closed curve dividing the plane into two regions. Since and touch neither , nor (recall that is pure), and lie entirely in one of these regions; and since and are adjacent, and in fact belong to the same region. On the other hand, since one endpoint of (resp. ) belongs to while the other endpoint belongs to , and both endpoints of are in , it follows that is the only contact point between (resp. ) and . But and are nonadjacent, which implies that crosses only once and has therefore one endpoint in each region. However, both endpoints of belong to , contradicting the fact that and lie in the same region. Hence, is of type II.b, thus concluding the proof. 

Claim 2

If two paths and are pure, then one of them contains at least bends and the other contains at least bends. Moreover, all these bend-points belong to (i,i+1)-sewing paths.

For each , consider a point . By Claim 1, is a bend-point of either or . Since and are the endpoints of , one of them belongs to while the other belongs to . Therefore, is a subset of one of the considered secondary path and is a subset of the other secondary path.

Finally, we claim that there exists an index such that , , and are all pure. Indeed, if it were not the case, there would be at least secondary paths which are not pure. But at most secondary paths can contain endpoints of or , a contradiction. It now follows from Claim 2 that has at least bends (which belong to -sewing paths) and that has at least bends (which belong to -sewing paths). Furthermore, one of and has at least bends which are endpoints of -sewing paths. Therefore, there is a path with at least bends. ∎

Corollary 1

For any , -CPG is strictly contained in -CPG, even within the class of planar graphs.

Although not every planar graph is CPG [11], we may still define the bend number, with respect to CPG representations, of the class of planar graphs which are CPG. From Theorem 3.1, we deduce the following.

Corollary 2

The class of planar CPG graphs has unbounded bend number with respect to CPG representations.

4 Complexity results for -CPG graphs

In this section, we show the -completeness of Independent Set and Clique Cover restricted to -CPG graphs. To this end, we first state the following two simple results.

Observation 2

For any and any triangle-free -CPG graph , the following holds:

  • There exists a -bend CPG representation of in which paths pairwise touch at most once.

  • If is subcubic, there exists a -bend CPG representation in which a path strictly contains one endpoint of another path if and only if its corresponding vertex is cubic.

Lemma 1 (Folklore)

If Independent Set is -complete for a graph class , then it is -complete for 2-subdivisions of graphs in .

Theorem 4.1

Independent Set is -complete for triangle-free subcubic -CPG graphs.

Proof

We exhibit a polynomial reduction from Independent Set restricted to 2-subdivisions of cubic planar graphs. Since Independent Set is -complete for cubic planar graphs [24], Lemma 1 implies that it remains -complete for the considered class.

Given a 2-subdivision of a cubic planar graph, we construct a -CPG graph as follows. Since is planar and triangle-free, it follows from [7] that we can obtain a -bend CPG representation of in linear time. By Observation 2(a) and 2(b), we can further obtain a -bend CPG representation of in which paths pairwise touch at most once and a path strictly contains one endpoint of another path if and only if its corresponding vertex has degree 3. Moreover, by eventually adding rows and columns, it is not difficult to see that we may assume that no path touches another path at its bend-point (recall that is triangle-free). For a path , let be the subset of paths touching . If has no bend, consider an arbitrary interior point of belonging only to . Then naturally divides into two segments and which partition into two subsets , for (see Figure 2(a)). Otherwise, has a bend and we denote by (resp. ) the subset of paths touching the horizontal (resp. vertical) segment of , referred to as (resp. ) (see Figure 2(b)). In this case also, is a partition of .

(a) has no bend.

(b) has a bend.
Figure 3: Partitioning into two subsets and .

By eventually adding new rows and columns, we now subdivide each path into five 0-bend paths , , , , , in such a way that (resp. ; ) touches only and (resp. and ; and ) as depicted in Figure 4. Let be the graph corresponding to the resulting 0-bend representation. Clearly, is a graph obtained from by replacing every vertex with a path , where (resp. ) corresponds to (resp. ), and apportioning the neighborhood of among and . We now show that , which would conclude the proof.

(a) has no bend.

(b) has a bend.
Figure 4: Subdiving a path into five -bend paths.

Given a maximum independent set of , we construct an independent set of as follows. If , then add , and to . Otherwise, add and to . Clearly, is an independent set of and .

Conversely, given a maximum independent set of , we construct an independent set of as follows. Observe that by maximality of , for any vertex , at least two vertices of are in . Indeed, if for some we have that contains at most one vertex , then is a strictly larger independent set of . Furthermore, we may assume that for any vertex , either both and are in or none of them is. Indeed, if is the only such vertex in , it suffices to consider the maximum independent set , for the unique . Also note that if both and are in , then is also in . We now add to if and only if both and are in . Clearly, is an independent set of and so . ∎

Remark. Since any triangle-free -CPG graph is planar [11], Theorem 4.1 implies that Independent Set is -complete for planar -CPG graphs.

Theorem 4.2

Clique Cover is -complete for -CPG line graphs.

Proof

We exhibit a polynomial reduction from Vertex Cover restricted to 2-subdivisions of triangle-free subcubic -CPG graphs, which is -complete by Theorem 4.1 and Lemma 1. Given a 2-subdivision of a triangle-free subcubic -CPG graph , we show that its line graph is -CPG (note that, by the proof of Theorem 4.1, we may assume to be a subdivision of a cubic graph). Since for any triangle-free graph we have (see, e.g., [25]), this would conclude the proof.

Consider a -bend CPG representation of in which a path strictly contains one endpoint of another path if and only if the corresponding vertex is cubic (see Observation 2(b)). We first show how to construct a -bend CPG representation of the -subdivision of . For every contact point of , arbitrarily choose one path of having as an endpoint. By eventually adding rows and columns, we may shorten so that is no longer an endpoint of and add two paths and , with having as endpoint and touching , and touching (see Figure 5).

Figure 5: 2-subdividing edges of .

Observe now that the -bend representation of thus obtained satisfies the following property: every maximal snake in the representation obtained from by removing every path whose corresponding vertex is cubic contains at least two paths lying either on the same row or on the same column. We now derive from a -bend CPG representation of as follows. If is a path strictly containing an endpoint of another path (i.e. corresponds to a cubic vertex), we split into two paths and so that is an endpoint of these two paths, shorten so that is no longer an endpoint of it and add a path touching and having as an endpoint. We refer to this operation as a triangle implant (see Figure 6). Note that since no two cubic vertices of are adjacent, no endpoint of is strictly contained in another path.

Figure 6: A triangle implant.

Next, we remove all the paths introduced by the triangle implants. The resulting representation is then a disjoint union of maximal snakes, each containing at least two paths lying either on the same row or on the same column. For every maximal snake, consider any two such paths and merge them into one path. By then reintroducing all the removed paths from the triangle implants, it is easy to see that we obtain a -bend CPG representation of . ∎

5 -coloring -EPG graphs

It was shown in [11] that 3-Colorability is -complete in -CPG. Since -CPG is a subclass of -EPG, it follows that this problem is -complete in -EPG. On the other hand, 3-Colorability is polynomial-time solvable in -EPG as this class coincides with that of interval graphs. We now settle the open case by showing that 3-Colorability is -complete in -EPG.

Theorem 5.1

3-Colorability is -complete for planar -EPG graphs.

Proof

We exhibit a polynomial reduction from 3-Colorability restricted to planar graphs of maximum degree 4, which was shown to be -complete in [18]. Given a planar graph of maximum degree 4, we construct a -EPG graph such that is 3-colorable if and only if is 3-colorable. It was shown in [28] that any planar graph of maximum degree 4 admits an embedding on the grid where vertices are mapped to grid-points and edges are mapped to pairwise interiorly disjoint grid-paths with at most 4 bends connecting the two grid-points corresponding to the endvertices. Moreover, such an embedding can be obtained in linear time. Therefore, let be such an embedding of , 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 for any edge , we denote by the path in corresponding to . The graph is then obtained as follows. For any edge , if the path contains bends, we replace the edge with a sequence of diamonds by identifying with the vertex of degree 2 in the first diamond of the sequence and connecting to the vertex of degree 2 in the last diamond of the sequence (see Figure 6(a) where has 3 bends and has no bend). We then construct from a 1-bend EPG representation of .

(a) The sequence of diamonds in .

(b) The corresponding 1-bend EPG representation.
Figure 7: Replacing an edge in with a sequence of diamonds.

By eventually adding rows and columns to the grid, we may assume that each path in is surrounded by an empty region i.e., no path distinct from and no grid-point in lie in the interior of this region, and that this region is always sufficiently large to allow the following operations. We first associate with every vertex a vertical path containing the grid-point as follows. If the grid-edge above (resp. below) is not used by any path in , the top (resp. lower) endpoint of is (resp. ) for a small enough so that the segment (resp. ) intersects no path in . Otherwise, the top (resp. lower) endpoint of lies on the vertical segment belonging to the path in using the grid-edge above (resp. below) (see Figure 8). Then, for any , we replace each segment of with a 0-bend representation of the diamond, connect the paths in two consecutive segments corresponding to the same vertex and add a bend, if necessary, to the path(s) corresponding to the vertex (resp. vertices) in the sequence of diamonds adjacent to (resp. ) so that it intersects (resp. ) without intersecting any other path (see Figure 6(b) for an example with ). This clearly gives a -bend CPG representation of (see the Appendix for an example). Since in any 3-coloring of the diamond the vertices of degree 2 have the same color, it is then easy to see that is -colorable if and only if is. ∎

Figure 8: Constructing the path (in blue) corresponding to the vertex when the grid-edge above is used by some path in . The empty region is delimited by red dashed lines.

6 Future Work

It was shown in [11] that Recognition is -complete for -CPG graphs, a result which we can extend to -CPG graphs for [17]. However, the case remains an open question.

7 Acknowledgement

We would like to thank Nicolas Champseix for his contribution to the proof of Theorem 3.1.

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Appendix 0.A The construction in the proof of Theorem 5.1

(a) Local embedding of .

(b) Local representation of .
Figure 9: An example of the transformation for a vertex .