1 Introduction
Intersection graphs of geometrical objects in the plane are among the most studied graph classes and have applications in various domains such as for instance biology, statistics, psychology and computing (see [13]). We define the intersection graph of a family of non empty sets as the graph whose vertices correspond to the elements of , and two vertices are adjacent in if and only if the corresponding elements in have a non-empty intersection.
Golumbic et al. introduced in [7] the class of edge intersection graphs of paths on a grid (EPG graphs), i.e. graphs for which there exists a collection of nontrivial paths on a rectangular grid in one-to-one correspondance with their vertex set, such that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid, and showed that every graph is in fact an EPG graph. A natural restriction which was thereupon considered, suggests to limit the number of bends (i.e. 90 degrees turns at a grid-point) that a path may have; for , the class -EPG consists of those EPG graphs admitting a representation in which each path has at most bends.
Since their introduction, -EPG graphs have been extensively studied from several points of view (see for instance [1, 2, 3, 5, 6, 7, 8, 9, 14, 15]). One major interest is the so-called bend number; for a graph class , the bend number of is the minimum integer such that every graph is a -EPG graph. The problem of determining the bend number of graph classes has been widely investigated (see for instance [3, 6, 7, 8] for planar graphs, Halin graphs, line graphs, outerplanar graphs).
Since -EPG graphs are equivalent to the well-studied class of interval graphs, a particular attention has been paid to -EPG graphs. The authors in [9] showed that recognising -EPG graphs is an NP-complete problem, a result which was further extended to -EPG graphs in [14]. Therefore, special graph classes were considered. For instance, the authors in [2] provided characterisations of some subclasses of chordal graphs which are -EPG by families of minimal forbidden induced subgraphs; in [5], the authors presented a characterisation of cographs that are -EPG and provided a linear time recognition algorithm.
In this paper, we are interested in a subclass of circular arc graphs (CA for short), namely proper circular arc graphs. In [1], the authors showed that CA graphs are -EPG and further proved that normal circular arc graphs have bend number equal to 2, a result from which we can easily deduce that the bend number of proper circular arc graphs is 2 (see Section 2). They also considered additional constraints on the EPG representations by demanding that the union of the paths lies on the boundary of a rectangle of the grid (EPR graphs). Similarly to EPG graphs, they defined for the class -EPR and proved that not all circular arc graphs are -EPR (it is easily seen that CA = -EPR = EPR). With the intent of pursuing the work done in [1], we here provide a characterisation of proper circular arc graphs that are -EPG by a family of minimal forbidden induced subgraphs (see Section 3) which is a first step towards characterising the minimal graphs in (CA -EPG) (CA -EPG). We conclude Section 3 by noting that a characterisation by a family of minimal forbidden induced subgraphs of proper circular arc graphs which are -EPR easily follows from [1] and [11].
2 Preliminaries
Throughout this paper, all considered graphs are connected, finite and simple. For all graph theoretical terms and notations not defined here, we refer the reader to [4].
Let be an undirected graph with vertex set and edge set . A clique (resp. independent set) is a subset of vertices that are pairwise adjacent (resp. nonadjacent). If and are two disjoint subsets of vertices, we say that is complete to (resp. is anti-complete to) , which we denote by (resp. ), if every vertex in is adjacent (resp. nonadjacent) to every vertex in . A dominating set in is a subset of vertices such that every vertex not in is adjacent to at least one vertex in .
We denote by , , the chordless cycle on vertices and by , , the complete graph on vertices. A k-wheel, , denoted by , is a chordless cycle on vertices with an additional vertex, referred to as the center of the wheel, adjacent to every vertex of the cycle. The 3-sun, denoted by , consists of an independent set and a clique such that is adjacent to and , , where indices are taken modulo 3. Given a graph and an integer , the power graph of has the same vertex set as with two vertices being adjacent in if and only if their distance (i.e. the length of a shortest path between the two vertices) in is at most .
If is a graph and is a subset of vertices, we denote by the graph obtained from by deleting all vertices in . Equivalently, is the subgraph of induced by , denoted by . If consists of a single vertex, say , we simply write . The complement graph of is the graph having the same vertex set as with two vertices being adjacent in if and only if they are nonadjacent in . The disjoint union of and is denoted by .
Let be a collection of graphs. For , we say that contains no induced if contains no induced subgraph isomorphic to . A graph is -free if it contains no induced subgraph isomorphic to some graph belonging to .
Recall that an interval graph is an intersection graph of intervals on the real line. A graph is said to be chordal if it does not contain any chordless cycle of length at least four as an induced subgraph. An independent set of three vertices such that each pair is joined by a path that avoids the neighborhood of the third is called an asteroidal triple. The following is a well-known characterisation of interval graphs.
Theorem 1 ([10]).
A graph is an interval graph if and only if it is chordal and contains no asteroidal triple.
A circular arc graph (CA graph) is an intersection graph of open arcs on a circle, i.e. a graph is a circular arc graph if one can associate an open arc on a circle with each vertex such that two vertices are adjacent if and only if their corresponding arcs intersect. If denotes the corresponding circle and the corresponding set of arcs, then is called a circular arc representation of . A circular arc graph having a circular arc representation where no two arcs cover the circle is called a normal circular arc graph (NCA graph). A circular arc graph having a circular arc representation where no arc properly contains another is called a proper circular arc graph (PCA graph). It is well known that every PCA graph admits a representation which is simultaneously proper and normal (see [16]); in particular, every PCA graph is a NCA graph. The following theorem provides a minimal forbidden induced subgraph characterisation for PCA graphs (see Fig. 1).
Theorem 2 ([17]).
A graph is a PCA graph if and only if it is -free.
A graph is a Helly circular arc graph (HCA graph) if it has a circular arc representation in which any subset of pairwise intersecting arcs has a common point on the circle. A graph that admits a circular arc representation which is simultaneously normal and Helly, i.e. no three arcs or less cover the circle, is called a normal Helly circular arc graph (NHCA graph). Similarly, one can define the class of proper Helly circular arc graphs (PHCA graphs) corresponding to those graphs that admit a circular arc representation in which no three arcs cover the circle and no arc properly contains another. It was shown in [12] that a PCA graph is PHCA if it admits a proper circular arc representation in which no two or three arcs cover the circle; in particular, every PHCA graph is a NHCA graph.
Consider a rectangular grid where the horizontal lines are referred to as rows and the vertical lines as columns. A grid-point lying on row and column is referred to as . If is a collection of nontrivial simple paths on the grid, the edge intersection graph of is the graph whose vertex set is in one-to-one correspondance with and two vertices are adjacent if and only if the corresponding paths share at least one grid-edge. The path representing some vertex will be denoted by . Then is referred to as an EPG representation of or a k-bend EPG representation of if every path of has at most -bends (i.e. 90 degrees turns at a grid-point) with . The class of graphs admitting a -bend EPG representation is called -EPG.
A graph is said to be an edge intersection graph of paths on a rectangle (EPR graph) if there exists a set of paths on a rectangle of the grid in one-to-one correspondance with the vertex set of , where two vertices are adjacent in if and only if their corresponding paths share at least one grid-edge; is then referred to as an EPR representation of . For , we denote by -EPR the class of graphs for which there exists an EPR representation where every path has at most bends.
The authors in [1] proved that NCA graphs have a bend number of 2 and presented an infinite family of NCA graphs, namely , which are not -EPG. Since any is in fact a PCA graph, we deduce the following corollary from the fact that PCA NCA.
Corollary 3.
PCA graphs have a bend number of 2.
3 Proper circular arc -EPG graphs
As we have seen in Section 2, the bend number of proper circular arc graphs is 2. In this section, we provide a characterisation, by a family of minimal forbidden induced subgraphs (see Fig. 2), for PCA graphs which are -EPG.
Theorem 4.
Let be a PCA graph. Then is -EPG if and only if is -free.
Proof.
Necessary condition. Let us show that for all , is not -EPG. Observe that all these graphs contain an induced 4-wheel; we denote by , , and the four vertices of the 4-cycle of the 4-wheel, and (one of) its center(s). As shown in [2], this 4-cycle can only be represented by either a true pie or a false pie as should intersect all four corresponding paths of the cycle (see Fig. 2(a)).
Assume henceforth that the 4-wheel is represented by a true pie using column and row of the grid (a similar reasoning applies if it is represented by a false pie). Then, lies either on column or row and strictly contains the grid-point , since it must intersect every path of the 4-cycle (see Fig. 2(c)). Consequently, can not be -EPG.
If a vertex is adjacent to three vertices of the 4-cycle, say , and , its associated path must contain row and/or column , since it has to intersect , and . However, can not lie entirely on column or row as it would otherwise intersect . Hence, it lies on both similarly to ; but then necessarily intersects , which implies that is not -EPG (see Fig. 2(d)).
If a second vertex intersects three different vertices of the 4-cycle, we distinguish two cases. Either those vertices are , and (the case where they are , and is symmetric), then as shown previously, lies on column and row similarly to , and therefore intersects . Or they are , and in which case lies on column and row similarly to , and consequently shares only the grid-point with . Hence, neither nor are -EPG.
If is a vertex adjacent to exactly two consecutive vertices of the 4-cycle, say and , then uses row , where and intersect, without strictly containing the grid-point as otherwise it would intersect other vertices of the 4-cycle (note that may then only intersect centers of the 4-wheel which are adjacent and lie on row ). Hence, and are not -EPG (see Fig. 2(b)).
Finally, since a vertex adjacent to only and would use column , where and intersect, without strictly containing the grid-point , and cannot be -EPG. We conclude this part of the proof by noticing that -EPG for , as shown in [1].
Sufficient condition. Let be a PCA graph which is -free. Consider a normal proper representation of , where is a circle and is a set of open arcs of in one-to-one correspondance with the vertices of (notice that such a representation exists due to [16]). Before turning to the proof, let us first make the following observation.
Observation 1.
Any set of arcs in covering the circle corresponds to a dominating set in . In particular, if contains a 4-wheel as an induced subgraph, then has a dominating triangle.
Proof. It is clear that such a set of arcs corresponds to a dominating set in . If contains a 4-wheel, then the arcs corresponding to 4-cycle of the 4-wheel cover the circle . But then the arc representing the center of the 4-wheel together with two arcs corresponding to two vertices of must also cover , i.e. has a dominating triangle.
If we assume that no three arcs in cover , then is PHCA and the result follows from the fact that PHCA -free NHCA -free -EPR (see [1]). Hence, we may assume that there exist three arcs in covering . If contains a 4-wheel, let denote the dominating triangle following from Observation 1, with being the center of the 4-wheel. Otherwise, let be any triangle whose corresponding arcs cover . In both cases, each vertex is adjacent to at least two vertices of . Indeed, if contains a -wheel, then this follows from the fact that is claw-free (see Theorem 2); if contains no -wheel, this follows from the fact that is a proper representation. For , denote by , where indices are taken modulo 3, the subset of vertices adjacent to only and . Note that each is a clique as would otherwise contain an induced claw, namely for any two such that . Similarly, consider the subset of vertices adjacent to all three vertices of . We now distinguish cases depending on whether contains a 4-wheel as an induced subgraph or not.
Case 1 ( contains an induced 4-wheel).
According to the above, and are not anticomplete. Thus, there exist and such that , which together with and form the 4-cycle of the 4-wheel. Since is dominating and has no induced claw, each remaining vertex of is adjacent to at least two vertices of . Consider accordingly the subset of vertices (resp. , ) adjacent to only and (resp. and , and ), the subset of vertices (resp. ; ; ) adjacent to only , and (resp. , and ; , and ; , and ) and the subset of vertices adjacent to all vertices of (note that ). Since contains no induced claw, each is a clique, as well as each . Furthermore, since contains no induced:
Now, if we assume that both and are nonempty, then both are complete to as would otherwise contain either or as an induced subgraph (see Fig. 1). But then is a clique since does not contain as an induced subgraph. Consequently, and can not both be nonempty; if it were indeed the case, both and would also be complete to , and would then contain an induced claw, namely and , with , and . Hence, we may assume, without loss of generality, that (the same reasoning applies if we assume that ). But then and must be anti-complete as would otherwise contain an induced claw (the same as before), and is consequently -EPG (see Fig. 4).
Now, assume without loss of generality, that and . We know from the above that is anti-complete to . Also, for all , must be either complete to and anti-complete to or, conversely, anti-complete to and complete to , as would otherwise contain an induced claw; indeed, if there exist and such that is nonadjacent (resp. adjacent) to both and , then and (resp. and ) induce a claw. Thus, we can partition into subsets for , and, since is -free, both of these subsets are cliques. Assuming and are non empty, there cannot exist and such that since would otherwise contain an induced (see Fig. 1); indeed, and would form a 4-wheel, with one vertex of adjacent to only and , and one vertex of adjacent to only and . Hence, is a clique; but then, contains as an induced subgraph. Thus, we have to assume that exactly one of and is empty, and is then -EPG as an induced subgraph of the previous case.
Suppose now that only one of the is non empty, for instance . If is adjacent to some vertex , then is complete to as would otherwise contain an induced (see Fig. 1). We can therefore partition into two subsets and . Since does not contain an induced , must be a clique; and, since does not contain an induced claw, must also be a clique. If one of and is empty, then is -EPG as an induced subgraph of the first case. If both are non empty, then is a clique, as would otherwise contain an induced (see Fig. 1), and is consequently -EPG (see Fig. 5).
Finally, if we assume that every is empty, since does not contain or a claw as an induced subgraph, we can partition into two cliques, and , which are anti-complete, and again is -EPG (see Fig. 6).
Case 2 ( contains no induced 4-wheel).
Assume henceforth that and are pairwise anti-complete. If all three subsets are non empty, then for all , there must exist such that is complete to both and , and anti-complete to (otherwise would contain an induced claw). Hence, we can partition into three subsets () which must be cliques since does not contain an induced claw. But then either is a clique, in which case is -EPG (see Fig. 7), or there exists and such that , and contains a 4-wheel with (for some ) as its 4-cycle and as its center, which is contrary to our assumption.
If we now assume that at least one of the subsets is empty, then at least one vertex of is adjacent to every vertex of and is consequently an interval graph. Indeed, assume that contains an induced cycle with . Then, together with , it would form an -wheel. But since is proper, it cannot contain a -wheel with . Hence and would contain a 4-wheel which is contrary to our assumption. Therefore, has no induced cycle of length larger than 3, i.e. is chordal. Furthermore, if there existed three pairwise nonadjacent vertices in , then together with they would induce a claw. Hence, contains no asteroidal triple, i.e. is an interval graph (see Theorem 1), and therefore -EPG. ∎
From the characterisation of -EPR graphs, i.e. intersection graphs of paths on a rectangle of a grid where each path has at most one bend, given in [1], we deduce the following characterisation by a family of minimal forbidden induced subgraphs of PCA graphs which are -EPR. It is easily seen that the class of circular arc graphs is exactly the class -EPR. The authors of [1] further proved that NCA graphs have a bend number, with respect to EPR representations, of 2; hence, since PCA NCA, PCA graphs also have a bend number, with respect to EPR representations, of at most 2.
Corollary 5.
Let be a PCA graph. Then is -EPR if and only if is -free.
4 Conclusion
In this paper, we present characterisations by (infinite) families of minimal forbidden induced subgraphs for -EPG PCA and -EPR PCA. This is a first step towards finding a characterisation of the minimal graphs in (CA -EPG) (CA -EPG), a question left open in [1].
Acknowledgments
This research was carried out when Dr M.P. Mazzoleni was visiting the University of Fribourg. The support of this institution is gratefully acknowledged.
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