DDiscussiones MathematicaeGraph Theory xx ( xxxx ) ON B -EPG AND EPT GRAPHS Liliana Alc´on
Universidad Nacional de La Plata, La Plata, Argentina.CONICET e-mail: [email protected]
Mar´ıa P´ıa Mazzoleni
Universidad Nacional de La Plata, La Plata, Argentina. e-mail: [email protected] andTanilson Dias dos Santos
Federal University of Tocantins, Palmas, Brazil e-mail: [email protected]
Abstract
This research contains as a main result the prove that every Chordal B -EPG graph is simultaneously in the graph classes VPT and EPT. Inaddition, we describe structures that must be present in any B -EPG graphwhich does not admit a Helly- B -EPG representation. In particular, thispaper presents some features of non-trivial families of graphs properly con-tained in Helly- B EPG, namely Bipartite, Block, Cactus and Line of Bi-partite graphs.
Keywords:
Edge-intersection of paths on a grid, Edge-intersection graphof paths in a tree, Helly property, Intersection graphs, Single bend paths,Vertex-intersection graph of paths in a tree.. a r X i v : . [ c s . D M ] J u l L. Alc´on, M. P. Mazzoleni and T. D. Santos1. Introduction
Models based on paths intersection may consider intersections by vertices orintersections by edges. Cases where the paths are hosted on a tree appear first inthe literature, see for instance [9, 10, 11]. Representations using paths on a gridwere considered later, see [12, 13, 15].Let P be a family of paths on a host tree T . Two types of intersection graphsfrom the pair < P, T > are defined, namely VPT and EPT graphs. The edgeintersection graph of P , EPT(P), has vertices which correspond to the membersof P , and two vertices are adjacent in EPT(P) if and only if the correspondingpaths in P share at least one edge in T. Similarly, the vertex intersection graph of P , VPT(P), has vertices which correspond to the members of P , and two verticesare adjacent in VPT(P) if and only if the corresponding paths in P share at leastone vertex in T . VPT and EPT graphs are incomparable families of graphs.However, when the maximum degree of the host tree is restricted to three thefamily of VPT graphs coincides with the family of EPT graphs [10]. Also it isknown that any Chordal EPT graph is VPT (see [19]). Recall that it was shownthat Chordal graphs are the vertex intersection graphs of subtrees of a tree [8].Edge intersection graphs of paths on a grid are called EPG graphs .In [12], the authors proved that every graph is EPG, and started the studyof the subclasses defined by bounding the number of times any path used in therepresentation can bend. Graphs admitting a representation where paths haveat most k changes of direction (bends) were called B k -EPG. In particular, whenthe paths have at most one bend we have the B -EPG graphs or a single bendEPG graphs .A pertinent question in the context of path intersection graphs is as follows:given two classes of path intersection graphs, the first whose host is a tree and thesecond whose host is a grid, is there an intersection or containment relationshipamong these classes? What do we know about it?In the present paper we will explore B -EPG graphs, in particular diamond-free graphs and Chordal graphs. We will work on the question about the con-tainment relation between VPT, EPT and B -EPG graph classes.A collection of sets satisfies the Helly property when every pair-wise inter-secting sub-collection has at least one common element. When this property issatisfied by the set of vertices (edges) of the paths used in a representation, weget a Helly representation. Helly- B -EPG graphs were studied in [5]. It is knownthat not every B -EPG graph admits a Helly- B -EPG representation. We areinterested in determining the subgraphs that make B -EPG graphs do not admita Helly representation. In the present work, we describe some structures thatwill be present in any such subgraph, and, in addition, we present new Helly- B EPG subclasses. Moreover, we describe new Helly- B EPG subclasses and we n B -EPG and EPT graphs
2. Definitions and Technical Results
The vertex set and the edge set of a graph G are denoted by V ( G ) and E ( G ),respectively. Given a vertex v ∈ V ( G ), N ( v ) represents the open neighborhood of v in G . For a subset S ⊆ V ( G ), G [ S ] is the subgraph of G induced by S . If F is any family of graphs, we say that G is F -free if G has no induced subgraphisomorphic to a member of F . A cycle , denoted by C n , is a sequence of distinctvertices v , . . . , v n , v where v i (cid:54) = v j for i (cid:54) = j and ( v i , v i + 1) ∈ E ( G ), such that n ≥
3. A chord is an edge that is between two non-consecutive vertices in asequence of vertices of a cycle. An induced cycle or chordless cycles is a cyclethat has no chord, in this paper an induce cycle will simply be called cycle . Agraph G formed by an induced cycle H plus a single universal vertex v connectedto all vertices of H is called wheel graph . If the wheel has n vertices, it is denotedby n -wheel.The k -sun graph S k , k ≥
3, consists of 2 k vertices, an independent set X = { x , . . . , x k } and a clique Y = { y , . . . , y k } , and edges set E ∪ E , where E = { ( x , y ); ( y , x ); ( x , y ); ( y , x ); . . . , ( x k , y k ); ( y k , x ) } forms the outer cycle and E = { ( y i , y j ) | i (cid:54) = j } forms the inner clique.A graph is a B k -EPG graph if it admits an EPG representation in whicheach path has at most k bends. When k = 1 we say that this is a single bendEPG representation or simply a B -EPG representation. A clique is a set ofpairwise adjacent vertices and an independent set is a set of pairwise non adjacentvertices. Given an EPG representation of a graph G , we will identify each vertex v of G with the corresponding path P v of the grid used in the representation.Accordingly, for instance, we will say that a vertex of G covers or contains someedge of the grid (meaning that the corresponding path does), or that a set of pathsof the representation induces a subgraph of G (meaning that the correspondingset of vertices does).In a B -EPG representation, a clique K is said to be an edge-clique if all thevertices of K share a common edge of the grid (see Figure 1(a)). A claw of thegrid is a set of three edges of the grid incident into a same point of the grid, whichis called the center of the claw . The two edges of the claw that have the samedirection form the base of the claw . If K is not an edge-clique, then there existsa claw of the grid (and only one) such that the vertices of K are those containingexactly two of the three edges of the claw; such a clique is called claw-clique [12](see Figure 1(b)).Notice that if three vertices induce a claw-clique, then exactly two of themturn at the center of the corresponding claw of the grid, and the third one contains L. Alc´on, M. P. Mazzoleni and T. D. Santos (a) Representation of aclique as edge-clique. (b) Representation of aclique as claw-clique.
Figure 1. Examples of clique representations. the base of the claw. Furthermore, any other vertex adjacent to the three mustcontain two of the edges of that claw, then the following lemma holds.
Lemma 1.
If three vertices are together in more than one maximal clique of agraph G , then in any B -EPG representation of G the three vertices do not forma claw-clique. In [3] Asinowski et al. proved the following lemma for C -free graphs. Lemma 2. [3] Let G be a B -EPG graph. If G is C -free, then there exists a B -EPG representation of G such that every maximal claw-clique K is representedon a claw of the grid whose base is covered only by vertices of K . We have obtained the following similar result for diamond-free graphs. A diamond is a graph G with vertex set V ( G ) = { a, b, c, d } and edge set E ( G ) = { ab, ac, bc, bd, cd } . Lemma 3.
Let G be a B -EPG graph. If G is diamond-free, then in any B -EPG representation of G , every maximal claw-clique K is represented on a clawof the grid whose edges are covered only by vertices of K . Proof.
Let K be a maximal clique which is a claw-clique in a given B -EPGrepresentation of G . Then there exist three vertices of K which induce a claw-clique K (cid:48) on the same claw of the grid than K . Assume, in order to derive acontradiction, that a vertex v / ∈ K covers some edge of the claw. Clearly, v must cover only one of such edges. Therefore v and the vertices of K (cid:48) induce adiamond, a contradiction.Let Q be a grid and let ( a , b ) , ( a , b ) , ( a , b ) , ( a , b ) be a 4-star centered at b as depicted in Figure 2(a). Let P = { P , . . . , P } be a collection of four pathseach containing a different pair of edges of the 4-star. Following [12], we say thatthe four paths form • a true pie when each one has a bend at b , Figure 2(b); and n B -EPG and EPT graphs (a) 4-star in grid. (b) True pie. (c) False pie. Figure 2. B -EPG representation of the induced cycle of size 4 as pies with emphasis incenter b . • a false pie when exactly two of the paths bend at b and they do not sharean edge of the 4-star, Figure 2(c).Clearly if four paths of a B -EPG representation of G form a pie, then thecorresponding vertices induce a 4-cycle in G . The following result can be easilyproved. We say that a set of paths form a claw when each pair of edges of theclaw is covered by some of the paths. Lemma 4.
In any B -EPG representation of a graph G , a set of paths formingtwo different claws centered at the same point of the grid contains four pathsforming either a true pie or a false pie. Therefore, in any B -EPG representationof a chordal graph G , no two maximal claw-cliques of G are centered at the samepoint of the grid. Lemma 5.
Let G be a graph whose vertex set can be partitioned into a non trivialclique K and an independent set I = { w , w , w } , such that each vertex of K isadjacent to each vertex of I . Then, in any B -EPG representation of G , at leastone of the cliques K i = K ∪ { w i } , with ≤ i ≤ , is an edge-clique. Proof.
Assume, in order to derive a contradiction, that the three cliques areclaw-cliques. By Lemma 4, they have different centers, say the points q , q , q of the grid, respectively. Since at least two paths have a bend at the center ofa claw, for each i ∈ { , , } , there must exist a vertex v i of K such that thecorresponding path P v i turns at the point q i of the grid. Notice that each one ofthe three paths P v i must contain the three grid points q , q and q . To provethat this is not possible, we will consider, without loss of generality, two cases.First, q is between q and q in P v . Then, P v cannot turn at q and contain q and q . And second, q is between q and q in P v . In this case, P v cannotturn at q and contain q and q ; thus the proof is completed.Three vertices u, v, w of a graph G form an asteroidal triple (AT) of G if forevery pair of them there exists a path connecting the two vertices and such that L. Alc´on, M. P. Mazzoleni and T. D. Santos the path avoids the neighborhood of the remaining vertex [4]. A graph withoutan asteroidal triple is called
AT-free . Lemma 6 [3] . Let v be any vertex of a B -EPG graph G . Then G [ N ( v )] isAT-free. Let C be any subset of the vertices of a graph G . The branch graph B ( G | C ),see [12], of G over C has a vertex set, V ( B ), consisting of all the vertices of G not in C but adjacent to some member of C , i.e. V ( B ) = N ( C ) − C . Adjacencyin B ( G | C ) is defined as follows: we join two vertices x and y by an edge in E ( B )if and only if in G occurs:1. x and y are not adjacent;2. x and y have a common neighbor u ∈ C ;3. the sets N ( x ) ∩ C and N ( y ) ∩ C are not comparable, i.e. there exist pri-vate neighbors w, z ∈ C such that w is adjacent to x but not to y , and z is adjacent to y but not to x ; we say that x and y are neighborhoodincomparable.We let χ ( G ) denote the chromatic number of G . Lemma 7 [12] . Let C be any maximal clique of a B -EPG graph G . Then, thebranch graph B ( G | C ) is { P , C n for n ≥ } -free, and χ ( B ( G/C )) ≤ .
3. Subclasses of Helly- B -EPG Graphs In this section, we delimit some subclasses of B -EPG graphs that admit a Helly- B -EPG representation. It is known that B -EPG and Helly- B EPG are heredi-tary classes, so they can be characterized by forbidden structures. In both cases,finding the list of minimal forbidden induced subgraphs are challenging openproblems. Taking a step towards solving those problems, we describe a fewstructures at least one of which will necessarily be present in any B -EPG graphthat does not admit a Helly representation. In addition, we show that the wellknown families of Block graphs, Cactus and Line of Bipartite graphs are totallycontained in the class Helly- B EPG.Let S , S (cid:48) , S (cid:48)(cid:48) and C be the graphs depicted in Figure 4. Theorem 8.
Let G be a B -EPG graph. If G is { S , S (cid:48) , S (cid:48)(cid:48) , C } -free then G isa Helly- B -EPG graph. Proof. If G is not a Helly- B -EPG graph, then in each B -EPG representationof G , there is at least one clique that is represented as claw-clique and no as n B -EPG and EPT graphs (a) Claw with paths. (b) Subgraph induced by paths. Figure 3. Reconstruction of the intersection model. edge-clique. Consider any B -EPG representation of G and let K be a maximalclique which is represented as a claw-clique. Assume, w.l.o.g, K is on a claw ofthe grid with base [ x , x ] × { y } and center C = ( x , y ). Denote by P K theset of paths corresponding to the vertices of K . By Lemma 2, the grid segment[ x , x ] × { y } is covered only by vertices of K .For every (cid:121) -path (resp. (cid:120) -path ) belonging to P K , we do the following: ifthe path does not intersect any path P t / ∈ P K on column x , then we delete itsvertical segment and add the grid segment [ x , x ] × { y } (resp. [ x , x ] × { y } ).If after this transformation there is no more (cid:121) -paths (resp. (cid:120) -paths) in P K , thenwe are done since we have obtained an edge-clique. So we may assume thatevery (cid:121) -path and every (cid:120) -path in P K intersects some path P t / ∈ P K on column x (notice that we can assume is the same path P t for all the vertices).Now, if none of the (cid:121) -paths belonging to P K intersects a path non in P K onthe line y , then we can replace the horizontal part of those paths by the segment[ x , x ] × { y } , getting an edge representation of the clique K . Thus, we canassume there exists at least one (cid:121) -path P v ∈ P K intersecting some path P t (cid:48) / ∈ P K on line y . Analogously, there exists at least one (cid:120) -path P v (cid:48) ∈ P K intersectingsome path P t (cid:48)(cid:48) / ∈ K on line y , as depicted in Figure 3. Notice that vertex t (cid:48) cannot be adjacent to any of the vertices t , v (cid:48) or t (cid:48)(cid:48) ; and, in addition, vertex t (cid:48)(cid:48) cannot be adjacent to t , or v .Finally, since K is claw-clique, there is a path P u ∈ P K covering the base ofthe claw. Depending on the possibles adjacencies between u and t (cid:48) or t (cid:48)(cid:48) , one ofthe graphs S , S (cid:48) or S (cid:48)(cid:48) is obtained.Notice that any bull-free graph is { S , S (cid:48) , S (cid:48)(cid:48) } -free, so our previous resultimplies Lemma 5 of [3].Next theorem has as consequence the identification of several graph classeswhere the existence of a B -EPG representation ensures the existence of a Helly- B -EPG representation. L. Alc´on, M. P. Mazzoleni and T. D. Santos (a) Graph S . (b) Graph S (cid:48) .(c) Graph S (cid:48)(cid:48) . (b) Graph C . Figure 4. Graphs on the statement of Theorem 8.
Theorem 9. If G is a B -EPG and diamond-free graph then G is a Helly- B -EPG graph. Proof. If G is not a Helly- B -EPG graph, then in each B -EPG representationof G , there is at least one clique that is represented as claw-clique and no asedge-clique. Consider any B -EPG representation of G and let K be a maximalclique which is represented as a claw-clique. Assume, w.l.o.g, K is on a clawof the grid with base [ x , x ] × { y } and center C = ( x , y ). Denote by P K the set of paths corresponding to the vertices of K . By Lemma 3, the gridsegment [ x , x ] × { y } is covered only by vertices of K . For every (cid:121) -path (resp. (cid:120) -path ) belonging to P K , we do the following: if the path does not intersect anypath P t / ∈ P K on column x , then we delete its vertical segment and add thegrid segment [ x , x ] × { y } (resp. [ x , x ] × { y } ). If after this transformationthere is no more (cid:121) -paths (resp. (cid:120) -paths) in P K , then we are done since we haveobtained an edge-clique. So we may assume that every (cid:121) -path and every (cid:120) -pathin P K intersects some path P t / ∈ P K on column x (notice that we can assumeis the same path P t for all the vertices). Since K is claw-clique, there is a path P u ∈ P K covering the base of the claw. Thus, G [ v, v (cid:48) , u, t ] induces a diamond, acontradiction.An independent set of vertices is a set of vertices no two of which are adjacent.A graph G is said to be Bipartite if its set of vertices can be partitioned into twodistinct independent sets. There are Bipartite graphs that are non B -EPG, forinstance K , and K , (see [7]). Clearly , since bipartite graphs are triangle-free, any B -EPG representation of a bipartite graph is also a Helly- B -EPGrepresentation. A similar result (but a bit weaker) is obtained as corollary of the n B -EPG and EPT graphs Corollary 10. If G is a Bipartite B -EPG graph then G is a Helly- B -EPGgraph. Proof.
The Bipartite graphs are diamond-free, thus by Theorem 9 these graphsare Helly- B -EPG graphs.A Block graph or Clique Tree is a type of graph in which every biconnectedcomponent (block) is a clique.
Corollary 11.
Block graphs are Helly- B EPG.
Proof.
Block graphs are known to be exactly the Chordal diamond-free graphs,so by Theorem 19 of [3], all Block graphs are B -EPG. If follows from Theorem 9that all Block graphs are Helly- B EPG.A
Cactus (sometimes called a Cactus Tree) graph is a connected graph inwhich any two cycles have at most one vertex in common. Equivalently, it isa connected graph in which every edge belongs to at most one cycle, or (fornontrivial Cactus) in which every block (maximal subgraph without a cut-vertex)is an edge or a cycle. The family of graphs in which each component is a Cactusis closed under graph minor operations. This graph family may be characterizedby a single forbidden minor, the diamond graph.
Corollary 12.
Cactus graphs are Helly- B EPG.
Proof.
In [6], it is proved that every Cactus graph is a monotonic B -EPGgraph (there is a B -EPG representation where all paths are ascending in rowsand columns). Thus, Cactus graphs are B -EPG graphs.Since Cactus are diamond-free, by Theorem 9, the proof follows.Given a graph G , its Line graph L ( G ) is a graph such that each vertex of L ( G ) represents an edge of G and two vertices of L ( G ) are adjacent if and onlyif their corresponding edges share a common endpoint (i.e. “are incident”) in G .A graph G is a Line graph of a Bipartite graph (or simply
Line of Bipartite ) ifand only if it contains no claw, no odd cycle (with more than 3 vertices), and nodiamond as induced subgraph, [16].In [17] was proved that every Line graph has a representation with at most2 bends. We proved in the following corollary that when restricted to the Lineof Bipartite we can obtain a representation Helly and one-bended.
Corollary 13.
Line of Bipartite graphs are Helly- B EPG. L. Alc´on, M. P. Mazzoleni and T. D. Santos
Proof.
Line of Bipartite graphs were proved to be B -EPG in [14]. Since theyare diamond-free, the proof follows from Theorem 9.The diagram of Figure 5 illustrates the containment relationship betweenthe graph classes studied so far in this work. We list in Figure 6 examples ofgraphs in each numbered region of the diagram. The numbers of each item belowcorrespond to the regions of the same number in the diagram depicted in Figure 5.(1) ( B -EPG) - (Helly- B -EPG) graphs, depicted in Figure 6(a), graph E ;(2) (Line of Bipartite) - (Cactus) - (Block) - (Bipartite) graphs, depicted inFigure 6(b), graph E ;(3) (Helly- B EPG) - (Line of Bipartite) - (Block) - (Cactus) - (Bipartite)graphs, depicted in Figure 6(c), graph E ;(4) (Block) ∩ (Line of Bipartite) - (Cactus) - (Bipartite), depicted in Fig-ure 6(d), graph E ;(5) (Block) ∩ (Line of Bipartite) ∩ (Cactus) - (Bipartite), depicted in Fig-ure 6(e), graph E ;(6) (Cactus) ∩ (Line of Bipartite) - (Block) - (Bipartite). This intersection isempty. Let G be a graph that is Cactus and Line of Bipartite then G is { claw, odd cycle, diamond } -free. But G is not a Bipartite graph, then G has odd cycle, absurd with the hypothesis of G is Line of Bipartite;(7) (Bipartite) ∩ (Line of Bipartite) - (Cactus) - (Block) graphs, depicted inFigure 6(f), graph E ;(8) (Bipartite) ∩ (Line of Bipartite) ∩ (Cactus) - (Block) graphs, depicted inFigure 6(g), graph E ;(9) (Bipartite) ∩ (Line of Bipartite) ∩ (Cactus) ∩ (Block) graphs, depicted inFigure 6(h), graph E ;(10) (Bipartite) ∩ (Cactus) ∩ (Block) - (Line of Bipartite) graphs, depicted inFigure 6(i), graph E ;(11) (Bipartite) ∩ (Cactus) - (Block) - (Line of Bipartite) graphs, depicted inFigure 6(j), graph E ;(12) (Bipartite) ∩ (Helly- B EPG) - (Cactus) - (Block) - (Line of Bipartite)graphs, depicted in Figure 6(k), graph E ; n B -EPG and EPT graphs B -EPG) graphs, depicted in Figure 6(l), graph E ;(14) (Block) - (Bipartite) - (Line of Bipartite) - (Cactus) graphs, depicted inFigure 6(m), graph E ;(15) (Block) ∩ (Cactus) - (Line of Bipartite) - (Bipartite) graphs, depicted inFigure 6(n), graph E ;(16) (Cactus) - (Block) - (Line of Bipartite) - (Bipartite) graphs, depicted inFigure 6(o), graph E , the odd cycles C n +1 , n ≥ B -EPG) - (Bipartite) graphs, depicted in Figure 6(p),graph E ; Figure 5. Diagram of some graph classes.
In next section we explore the Chordal B -EPG graphs through of a subsetof forbidden graphs and we will proof that this class is in the strict intersectionof VPT and EPT graphs.
4. Containment relationship among Chordal B -EPG, VPT andEPT graphs Any graph that admits a B -EPG representation whose paths do not cover all theedges of a polygon of the grid (i.e. the subjacent grid subgraph is a tree) is alsoan EPT graph: the same representation is both B -EPG and EP T . However, itis easily verifiable that the subjacent grid subgraph of any B -EPG representa-tion of a cycle C n with n ≥ C n is an EPT graph. Ourlong-rage goal is understanding the B -EPG graphs that are also EPT graphs.2 L. Alc´on, M. P. Mazzoleni and T. D. Santos (a) Graph E . (b) Graph E . (c) Graph E . (d) Graph E .(e) Graph E . (f) Graph E . (g) Graph E . (h) Graph E .(i) Graph E . (j) Graph E . (k) Graph E . (l) Graph E .(m) Graph E . (n) Graph E . (o) Graph E , C n +1 , n ≥
2. (p) Graph E . Figure 6. The set of instances for the Venn Diagram on Figure 5.
When can a B -EPG representation be reorganized into an EPT representation?In this section, we answer that question for Chordal B -EPG graphs, in fact weprove that every Chordal B -EPG graph is EPT. We made several unsuccessfulattempts to prove this result by considering for a graph G , a B -EPG represen-tation whose paths cover all the edges of some polygon on the grid, and tryingto show that if none of the paths could be modified in order to avoid an edgeof the polygon, then G had some chordless cycle (i.e. G is not chordal). Thesurprise was that the only way we found to demonstrate our main Theorem 23was through V P T graphs. We will prove the following theorem.
Theorem 14.
Chordal B -EPG (cid:40) VPT.
In L´evˆeque et al. [18] apud [2], VPT graphs were characterized by a familyof minimal forbidden induced subgraphs, the ones depicted in Figure 7 plus theinduced cycles C n for n ≥
4. Therefore, in order to prove that Chordal B -EPGgraphs are VPT is enough to show that none of the graphs in Figure 7 is B -EPG.First notice that in each one of the graphs F , F , F , F and F ( Figures 7(a),(b), (c), (d), (e), respectively), the neighborhood of the universal vertex (the one n B -EPG and EPT graphs B -EPG.Now, in each one of the graphs F , F , F , F , F and F (Figures 7(k),(l), (m), (n), (o), (p), respectively), let C be the maximal clique in bold. It iseasy to check that, in all cases, the branch graph B ( G | C ) contains an inducedcycle C n , for some n ≥
4, or an induced path P ; thus, by Lemma 7, graphs F , F , F , F , F and F are not B -EPG. Observation 15.
Let e (cid:96) , e m and e r be three distinct edges of a one-bend path P ,and assume that e m is between e (cid:96) and e r on P . If P (cid:96) and P r are one-bend pathssuch that: P (cid:96) contains e (cid:96) , P r contains e r , and P (cid:96) and P r intersect in at least oneedge, then P (cid:96) or P r contains e m . Observation 16.
Let e and q be an edge and a point of a one-bend path P ,respectively. If a one-bend path P (cid:48) contains both e and q , then P (cid:48) contains thewhole segment of P between q and e . Lemma 17.
Let G be a graph whose vertex set can be partitioned into a clique K = { a, b } and an independent set I = { x, y, z } , such that each vertex of K isadjacent to each vertex of I . If in a given B -EPG representation of G , P a ∩ P y is between P a ∩ P x and P a ∩ P z , then { a, b, y } is an edge-clique, and P a ∩ P y ⊂ P b .Even more, any vertex adjacent to both a and y , but not to b (or to b and y , butnot to a ) has to be adjacent to x or to z . Proof.
Assume in order to obtain a contradiction that { a, b, y } is not an edge-clique. Then, by Lemma 5, we can assume, w.l.o.g., that { a, b, x } is an edge-clique. It implies that there is an edge e (cid:96) of P a ∩ P x covered by P b . Since everyedge of P a ∩ P z is covered by P z , z and b are adjacent, and z and y are nonadjacent, we have by Observation 15, that every edge of P a ∩ P y is covered by P b ,which implies that { a, b, y } is an edge-clique, contrary to the assumption.Thus, { a, b, y } is an edge-clique. By Observation 16, we have that the wholeinterval of P a between P a ∩ P x and P a ∩ P z is contained in P b , and so, in particular, P a ∩ P y ⊂ P b . Observe that this implies that if q is an end vertex of the interval P a ∩ P y , and e is the edge of P a incident on q that do not belong to P y , then e belongs to P b or to P x or to P z .Now, assume there exists a vertex v adjacent to both a and y , but not to b . Then, the clique { a, y, v } has to be a claw-clique. Let q be the center of theclaw, notice that q has to be an end vertex of the interval P a ∩ P y . Since v is notadjacent to b , it follows from the observation at the end of the paragraph above,that v has to be adjacent to x or to z . Lemma 18.
The graph F on Figure 7(f ) is not B -EPG. L. Alc´on, M. P. Mazzoleni and T. D. Santos
Proof.
Let K = { , } and I = { , , } . If there exists a B -EPG representationof F , by Lemma 17, because of the existence of the vertices 6, 7 and 8, none ofthe vertices 3, 4 and 5 may intersect 1 between the remaining two, thus such arepresentation does not exist. Lemma 19.
The graph F on Figure 7(g) is not B -EPG. Proof.
Let K = { , } and I = { , , } . If there exists a B -EPG representationof F , by Lemma 17, because of the existence of the vertices 7 and 8, the vertex6 must intersect vertex 1 between 3 and 4. But considering K (cid:48) = { , } , becauseof the existence of the vertices 5 and 6, vertex 4 must intersect vertex 1 between5 and 6. This contradiction implies that such a representation does not exist. Lemma 20.
The graphs F , F and F (8) on Figures 7(h), (i) and (j), respec-tively, are not B -EPG. Proof.
Let K = { , } and I = { , , } . If there exists a B -EPG representationof any one of those graphs, by Lemma 17, because of the existence of the vertices4 and 5, the vertex 1 must intersect vertex 2 between 6 and 7. In addition, since { , , } is a clique, 8 intersects 2 in an edge of P ∩ P (edge-clique) or in an edgeincident to P ∩ P (claw-clique). Analogously, because of the clique { , , } , 8intersects 2 in an edge of P ∩ P (edge-clique) or in an edge incident to P ∩ P (claw-clique). In any case, it implies that 8 intersects 2 on two different edges,each one in a different side of P ∩ P , thus, by Observation 16, P contains theinterval P ∩ P , in contradiction with the fact that 1 and 8 are not adjacent. Lemma 21.
The graphs F ( n ) for n ≥ on Figure 7(j) are not B -EPG. Proof.
The case n = 8 was considered in the previous Lemma 20, so assume n ≥
9. Let K = { , } and I = { , , } . If there exists a B -EPG representationof any one of those graphs, by Lemma 17, because of the existence of the vertices4 and 5, the vertex 1 must intersect vertex 2 between 6 and 7. In addition,since { , , } is a clique, 8 intersects 2 in an edge of P ∩ P (edge-clique) orin an edge incident to P ∩ P (claw-clique). Analogously, because of the clique { , , n } , n intersects 2 in an edge of P ∩ P (edge-clique) or in an edge incidentto P ∩ P (claw-clique). In any case, it implies that 8 and n intersect 2 on twodifferent edges, each one in a different side of P ∩ P . Therefore, there exist twoconsecutive vertices of the path 8 , , . . . , n , say the vertices j and j + 1, such thateach one intersects P on a different side of P ∩ P . Thus, by Observation 15, P j or P j +1 must contain the interval P ∩ P , in contradiction with the fact thatneither j nor j + 1 is adjacent to 1. n B -EPG and EPT graphs ( a ) G r a ph F . ( b ) G r a ph F . ( c ) G r a ph F . ( d ) G r a ph F . ( e ) G r a ph F ( n ) , n ≥ . ( f ) G r a ph F . ( g ) G r a ph F . ( h ) G r a ph F . ( i ) G r a ph F . ( j ) G r a ph F ( n ) , n ≥ . ( k ) F ( k ) , k ≥ . ( l ) F ( k ) , k ≥ . ( m ) F ( k + ) , k ≥ . ( n ) F ( k + ) , k ≥ . ( o ) F ( k + ) , k ≥ . ( p ) F ( k + ) , k ≥ . F i g u r e . T h e C h o r d a li ndu ce d s ub g r a ph s f o r b i dd e n t o V P T (t h e v e r t i ce s i n t h ec y c l e m a r k e db y b o l d e d g e s f o r m a c li q u e ) . L. Alc´on, M. P. Mazzoleni and T. D. Santos
We have proved that every minimal forbidden induced subgraph for VPTis also a forbidden induced subgraph for Chordal B -EPG. Moreover, there aregraphs in VPT that do not belong to B -EPG, for instance the graph 4-sun S is not in B -EPG, see [12], but it has a VPT representation, see Figures 8(a)and 8(b). Thus, VPT graphs properly contain Chordal B -EPG graphs. Thisends the proof of Theorem 14. Corollary 22.
Each one of the graphs depicted on Figure 7 is a forbidden inducedsubgraph for the class B -EPG. (a) Graph S . (b) A VPT and EPT representation of S . Figure 8. Graph S and one of its possible VPT and EPT representations. Theorem 23.
Chordal B -EPG (cid:40) EPT.
Proof.
Let G be a Chordal B -EPG graph. By the previous Theorem 14, G isVPT. And, by Lemma 7, χ ( B ( G/C )) ≤ C of G . In [1](see Theorem 10), it was proved that if the chromatic number of the branch graphof a VPT graph is at most h for every maximal clique, then the graph admits aVPT representation on a host tree with maximum degree h . Therefore, G admitsa VPT representation on a host tree with maximum degree 3. Finally, in [10](see Theorem 2), it was prove that any VPT graph that admits a representationon a host tree with maximum degree 3 is also an EPT graph. Consequently, G is EPT.The same graph S used in the proof of the previous theorem (see Figure 8(b))shows that there are EPT graphs that are not B -EPG. n B -EPG and EPT graphs
5. Conclusion and Open Questions
In this paper, we have considered three different path-intersection graph classes: B -EPG, VPT and EPT graphs. We showed that { S , S (cid:48) , S (cid:48)(cid:48) , C } -free graphsand others non-trivial subclasses of B -EPG graphs have are Helly- B -EPG,namely by instance Bipartite, Block, Cactus and Line of Bipartite graphs.We presented an infinite family of forbidden induced subgraphs for the class B -EPG and in particular we proved that Chordal B -EPG ⊂ VPT ∩ EPT.In [3], Asinowski and Ries described the Split graphs that are B -EPGgraphs in case the the stable set or the central size have size three. The graphs F , F , F , F and F , given in Figure 7 are Split, we have used a different ap-proach to prove that they are not B -EPG graphs. So one question is pertinent:Can we characterize Split graphs in general based in results of this paper?Finally, another interesting research would be to explore families of Helly-EPG graphs more deeply. We would like to understand the behavior of othergraph classes inside B -EPG graph class, i.e. if given an input graph G that isan instance of (for example) Weakly Chordal B -EPG. What is the relationshipof G with the EPT/VPT graph class? What happens when we demand that therepresentations be Helly- B EPG? Does recognizing problem remains hard foreach one of these classes?
Acknowledgement
The present work was done while the third author was a doctoral research fellowat National University of La Plata - UNLP, Math Department. The support ofthis institution is gratefully acknowledged.The third author (Tanilson) would like to thank the partial financing of thisstudy by the Coordena¸c˜ao de Aperfei¸coamento de Pessoal de N´ıvel Superior -Brasil (CAPES) - Finance Code 001.
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