- Email: [email protected]
Special asteroidal quadruple on directed path graph non rooted path graph
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Electronic Notes in Discrete Mathematics 44 (2013) 47–52 www.elsevier.com/locate/endm
Special asteroidal quadruple on directed path graph non rooted path graph. Marisa Gutierrez
1
Conicet, Dto de Matem´ atica FCE-UNLP La Plata, Argentina
Silvia B. Tondato
2
Dto de Matem´ atica FCE-UNLP La Plata, Argentina
Abstract An asteroidal triple in a graph G is a set of three non-adjacent vertices such that for any two of them there exists a path between them that does not intersect the neighborhood of the third. A special asteroidal triple in a graph G is an asteroidal triple such that each pair is linked by a special connection. A special asteroidal triple play a central role in a characterization of directed path graphs by Cameron, Ho´ ang and L´evˆeque. They also introduce a related notion of asteroidal quadruple and conjecture a characterization of rooted path graphs. In this original form this conjecture is not complete, still in leafage four, as was showed in [5] but as suggested by the conjecture, a characterization by forbidding particular type of asteroidal quadruples may holds. We prove that the conjecture in the original form is true on directed path graphs with leafage four having two minimal separators with multiplicity two. Thus we build the family of forbidden for this case. Keywords: Clique trees, rooted path graphs, asteroidal quadruples.
1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.endm.2013.10.008
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M. Gutierrez, S.B. Tondato / Electronic Notes in Discrete Mathematics 44 (2013) 47–52
Introduction
A classical result [4] states that a graph is chordal if and only if it has a clique tree, i.e a tree T whose vertices are the cliques of the graph and for each vertex x, the cliques of G that contain x induce a subtree: Tx of T . Natural subclasses of chordal graphs are path graphs (U V ), directed path graphs (DV ) and rooted directed path graphs (RDV ). A graph is UV if it admits a UV model, i.e. a clique tree T such that Tx is a subpath of T for every x ∈ V (G). A graph is DV (resp. RDV ) if it admits a DV model (resp. RDV model ) , i.e. a UV model (resp. DV model) that can be oriented (oriented and rooted) such that Tx is a directed subpath of T . Panda [7] found the characterization of DV by forbidden induced subgraphs and then Cameron, Ho´ang and L´evˆeque [2] gave a characterization of this class in terms of forbidden asteroidal triples. Characterizing RDV by forbidden induced subgraphs or forbidden asteroids are open problems. It is certainly too difficult to characterizing RDV by forbidden induced subgraphs as there are too many (families of) graphs to exclude but Cameron, Ho´ang and L´evˆeque [1] conjecture that a DV graph is a RDV graph if and only if it contains no special asteroidal quadruple defined through special type of connections between these vertices. Although they defined other special connections, in the present work we will need only two types of these connections: Type 1: vertices a1 , w, a2 and edges a1 w, a2 w; Type 2: vertices a1 , a, b, c, d, a2 and edges a1 a, a1 b, ab, bc, cd, da, ac, a2 c, a2 d. It was proved [2] if G is a DV graph and a1 , a2 are two nonadjacent vertices that are linked by a special connection of Type 1 or Type 2 then for every DV model T of G, the subpath T (a1 , a2 ) is a directed path. An asteroidal quadruple is a set of four non-adjacent vertices such that any three of them is an asteroidal triple. A special asteroidal quadruple is an asteroidal quadruple such that there are two pairs disjoint of vertices linked by special connection of Type 1 or Type 2. Gutierrez, L´evˆeque and Tondato [5] proved also that every DV graph that is not RDV graph has an asteroidal quadruple. Therefore any DV non RDV graph must be leafage at least four, we will study only that of leafage four. We show some properties of DV models in DV minimal non RDV graph that allow us to prove the conjecture on these graphs having two minimal separators of multiplicity two. The multiplicity of a minimal separator S in a graph G is 1 2
Email: [email protected] Email: [email protected]
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c − 1 where c is the number of connected components of G \ S having a vertex complete to S. Finally, we build the family of forbidden for this case.
2
Definitions, notation and previous results.
Let T be a clique tree. We often use capital letters to the vertices of V (T ) and write X ∈ T instead of X ∈ V (T ), and e ∈ T instead of e ∈ E(T ). If T is a subtree of T , then GT denotes the subgraph of G that is induced by the vertices of ∪X∈V (T ) X. If G is a graph and V ⊆ V (G), then G \ V denotes the subgraph of G induced by V (G) \ V . If E ⊆ E(G), then G − E denotes the subgraph of G induced by E(G) \ E . If G, G are two graphs, then G + G denotes the graph whose vertices are V (G) ∪ V (G ) and edges are E(G) ∪ E(G ). Note that if T, T are two trees such that |V (T ) ∩ V (T )| ≤ 1, then T + T is a forest. Let T be a tree. For V ⊆ V (T ), let T [V ] be the minimal subtree of T containing V . Then for X, Y ∈ V (T ), T [X, Y ] is the subpath of T between X and Y . Let T [X, Y ) = T [X, Y ] \ Y , T (X, Y ] = T [X, Y ] \ X and T (X, Y ) = T [X, Y ] \ {X, Y }. A vertex X ∈ V (T (Y, Z)) has a vertex crossing in T [Y, Z] if X ∩ X = ∅ where X and X are the two neighbors of X in T [Y, Z]. Let G be a chordal graph and T a clique tree. The label of an edge AB of T is defined as lab(AB) = A ∩ B. We say that X ∈ V (T ) dominates e ∈ E(T ) if lab(e) ⊆ X. We say that e ∈ E(T ) dominates e ∈ E(T ) if lab(e) ⊆ lab(e ). Recall that the labels of edges correspond to minimal separators of G. The multiplicity of e is the number of edges having its label and the multiplicity of a minimal separator S in a graph G is c − 1 where c is the number of connected components of G \ S having a vertex complete to S. It is clear that both multiplicity share. If e has multiplicity two we will say that it has a twin edge. Clearly, for each edge e of a clique tree, in every clique tree there is e such that lab(e) = lab( e), we will say that e and e are equivalents. The leafage of a chordal graph G is a minimum integer such that G admits a clique tree T with leaves. UV-leafage and DV-leafage are defined analogously with T being a UV model and DV model respectively. For any DV graph we have DV-leafage = UV-leafage = leafage [3] and [6]. As was mentioned, every DV graph minimally non RDV graph has an asteroidal quadruple then has leafage at least four. In the following results assume that: T is a DV model with four leaves H1 , H2 , H3 , H4 and two vertices of degree three, C1 , C2 being C1 the only vertex of degree 3 in T [H1 , H2 ]. We will say that T [Hi , C1 ] for i = 1, 2 and T [Hj , C2 ] for j = 3, 4 are the T -branches corresponding to Hi , i = 1, ..., 4 respectively. Similar notation will be used in
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case that T is a star where the vertices C1 and C2 are the same. Lemma 2.1 Let e ∈ T [H1 , C1 ] then: i)|lab(e)| > 1; ii)There are at least two vertices in lab(e) with different ends towards C1 ; iii) If e ∈ T (H1 , C1 ], it has a dominated edge outside T [H1 , C1 ]. Proof. By way of contradiction. Let e = AB with B ∈ T [A, C1 ]. i) Suppose A ∩ B = {x} for x ∈ V (G). Let T = T − E(T [H1 , B]). By minimality of G, GT is a RDV graph. Let T be a RDV model of GT rooted on a vertex R. Let Z, W ∈ T such that Tx = T [Z, W ] and W ∈ T [Z, R]. The DV model of G, T1 = T + ZA + T [A, H1 ] is rooted on R. ii) Suppose that every vertex x ∈ lab(e) satisfies Tx ∩ (T − E(T [H1 , A])) = T [A, X] for some X in T . Let T = T − E(T [H1 , B]). Let T be a RDV model of GT rooted on a vertex R. Observe that all vertices of lab(e) are twins in GT . Hence for every x ∈ lab(e) Tx = T [Z, W ] with W ∈ T [Z, R]. As in i), T1 is rooted on R. iii) Suppose that none edge outside T [H1 , C1 ] is dominated by e. Take e dominated by e in T (H1 , C1 ] nearest C1 (it could be e). Let e = A B with B ∈ T [A , C1 ] and T = T − E(T [H1 , A ]). By the election of e , A is a leaf in any clique tree of GT . Let T be a RDV model of GT rooted on a vertex R. Clearly, T has only one edge equivalent to e and this edge must be incident in A . Hence the DV model of G T + T [H1 , A ] can be rooted on R or H1 . 2 Lemma 2.2 There is a vertex crossing C1 in T [H1 , H2 ] if and only if there is a vertex crossing C2 in T [H3 , H4 ]. Lemma 2.3 Two twin edges must be in different T -branches. Moreover they are the unique twin edges in these branches. Theorem 2.4 Let G be a DV minimally non RDV graph with leafage four and T a DV model of G with four leaves. If e and e are twin edges in T -branches corresponding to leaves Hi and Hk respectively i = k ∈ {1, ..., 4}. Then G \ Hi and G \ Hk are connected graphs. Moreover, the simplicial vertices of Hi and Hk are linked by a special connection of Type 1 or Type 2. Proof. Without loss of generality let e ∈ T [H1 , C1 ], e ∈ T [H3 , C2 ] and let a1 , a3 be simplicial vertices of H1 and H3 respectively. Suppose by contradiction, that G \ H1 is not connected. Hence H1 has a dominated edge e0 not incident in H1 . By Lemma 2.1, e = e0 and e = e0 . Since T is a DV model e0 ∈ / T [H2 , C1 ] + T [H4 , C2 ]. Then it is in T [H1 , H3 ]. If the edges e, e0 , e appear in this order along T [H1 , H3 ] then lab(e) = lab(e ) = lab(e0 ) contra-
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dicting Lemma 2.3. Hence e, e , e0 must appear in this order, contradicting Lemma 2.1. Thus G \ H1 is connected. Analogously, G \ H3 is connected. Therefore neither e nor e can be incident in H1 and H3 respectively. Hence |E(T [H1 , C1 ])| > 1 and there is e1 = A1 B1 ∈ T [H1 , C1 ] being A1 the neighbor of H1 . By Lemma 2.1, there exist dominated edges of e1 . Take e1 that maximally farthest from e1 . Analogously, there are e3 = A3 B3 ∈ T [H3 , C2 ] and e3 . As lab(e) = lab(e ) and T is a DV model then e1 , e3 ∈ T [H1 , H3 ]. In case that e1 is incident in H3 , by Lemma 2.1 there is x ∈ lab(e1 ) such that x ∈ H1 ∩ H3 . Therefore, a1 and a3 are linked by a special connection of Type 1. Analogously if e3 is incident in H1 . Observe that if e1 is between C1 and e , e1 = e , T [H1 , H3 ] had three edges with the same label which contradicts Lemma 2.3. Hence e1 is between e and A3 . Analogously e3 is between e and A1 . Therefore lab(e1 ) = lab(e3 ). By Lemma 2.3, e1 = e and e3 = e. Clearly, every vertex in lab(e1 ) = lab(e3 ) is in A1 ∩ A3 and exist x, y ∈ A1 ∩ A3 being x ∈ H1 and y ∈ H3 . Among all x ∈ H1 ∩ A3 chose one that maximizes |Cx |. Analogously for y. If x ∈ H3 or y ∈ H1 then a1 and a3 are linked by a special connection of Type 1. Suppose x ∈ H1 \ H3 and y ∈ H3 \ H1 . By Lemma 2.1, there are x1 = x and x3 = y with x1 ∈ H1 ∩ A1 and x3 ∈ H3 ∩ A3 respectively. Among all the x1 ∈ H1 ∩ A1 we chose one minimizing |Cx1 |, analogously for x3 . If x1 and x3 were adjacent some of them would be in lab(e1 ). Suppose x1 ∈ lab(e1 ) then x1 ∈ lab(e3 ) and x1 ∈ H1 ∩ A3 . As x1 was chosen minimizing |Cx1 | then Cx = Cx1 and x and x1 are twins in G, a contradiction. Therefore x1 and x3 are not adjacent. So a1 and a3 are linked by a special connection of Type 2.2 Theorem 2.5 If G is DV minimally non RDV with leafage four and two minimal separators with multiplicity two then: i) G has a special asteroidal quadruple; ii) G is one of the graph G1 , G3 , G5 or one of the families G2 , G4 , G6 in Figure 1. Proof. Let T be a DV model with four leaves of G. Since G has separators in the conditions of the Theorem, it is has four edges e, e , d, d being e and e (resp. d and d ) twin edges. By Lemma 2.3, we can assume e ∈ T [H1 , C1 ], e ∈ T [H3 , C2 ], d ∈ T [H2 , C1 ] and d ∈ T [H4 , C2 ]. By Theorem 2.4, there is a special connection of Type 1 or Type 2 between a1 , a3 (resp. a2 , a4 ) being ai the simplicial vertex of Hi and a1 , a2 , a3 , a4 is an asteroidal quadruple of G. As in the proof of Theorem 2.4, there exist edges e1 , e3 and e = e3 and e = e1 . First suppose that there is a special connection of Type 1 and no Type 2 between a1 and a3 , then there is a vertex x ∈ H1 ∩ H3 . By Lemma 2.1, there is a vertex w in lab(e1 ) − {x} not in H1 . Take w minimizing |Cw |. Since
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M. Gutierrez, S.B. Tondato / Electronic Notes in Discrete Mathematics 44 (2013) 47–52 G1
G2
G3
G4
G5
G6
Fig. 1. Graphs and its DV models
there is not a special connection of Type 2 between a1 and a3 then w ∈ / H3 . By Lemma 2.1, there is x1 ∈ H1 ∩ A1 (resp. x3 ∈ H3 ∩ A3 ) minimizing |Cx1 | (resp. |Cx3 |). If Cx1 ∩ Cx3 = ∅ then x1 or x3 is in lab(e1 ). Suppose that x1 ∈ lab(e1 ) then there is a vertex in A1 − B1 and by the election of x1 it is a simplicial vertex: s. Observe that G\s also has a special asteroidal quadruple, a contradiction. Hence Cx1 ∩ Cx3 = ∅. Therefore a special connection of Type 1 implies the existence of two paths a1 , x, a3 and a1 , x1 , w, x3 , a3 between a1 and a3 being x adjacent to x1 , w and x3 . Graphs G3 and family G4 appear in case that the both connection are Type 1. Graphs G5 and family G6 appear in case that one connection is Type 1 and the other is Type 2. Finally, graphs G1 and family G2 appear in case that the both are Type 2. 2
References [1] K. Cameron, C. T. Ho´ang, B. L´evˆeque, Asteroids in rooted and directed path graphs, Electronic Notes in Discrete Mathematics 32 (2009), 67-74. [2] K. Cameron, C. T. Ho´ang, B. L´evˆeque, Characterizing directed path graphs by forbidden asteroids, Journal of Graph Theory 68 (2011), 103–112. [3] S. Chaplick, J. Stacho, The vertex leafage of chordal graphs, arXiv:1104.2524v2, manuscript 2012. [4] F. Gavril, The intersection graphs of subtrees in trees are exactly the chordal graphs, J. Combin. Theory B 16 (1974), 47–56. [5] M. Gutierrez, B. L´evˆeque, S. B. Tondato, Asteroidal quadruples in non rooted path graphs, manuscript 2012. [6] M. Gutierrez, S. B. Tondato, On path models of path graphs, manuscript 2011. [7] B.S. Panda, The forbidden subgraph characterization of directed vertex graphs, Discrete Mathematics 196 (1999), 239–256.
Electronic Notes in Discrete Mathematics 44 (2013) 47–52 www.elsevier.com/locate/endm
Special asteroidal quadruple on directed path graph non rooted path graph. Marisa Gutierrez
1
Conicet, Dto de Matem´ atica FCE-UNLP La Plata, Argentina
Silvia B. Tondato
2
Dto de Matem´ atica FCE-UNLP La Plata, Argentina
Abstract An asteroidal triple in a graph G is a set of three non-adjacent vertices such that for any two of them there exists a path between them that does not intersect the neighborhood of the third. A special asteroidal triple in a graph G is an asteroidal triple such that each pair is linked by a special connection. A special asteroidal triple play a central role in a characterization of directed path graphs by Cameron, Ho´ ang and L´evˆeque. They also introduce a related notion of asteroidal quadruple and conjecture a characterization of rooted path graphs. In this original form this conjecture is not complete, still in leafage four, as was showed in [5] but as suggested by the conjecture, a characterization by forbidding particular type of asteroidal quadruples may holds. We prove that the conjecture in the original form is true on directed path graphs with leafage four having two minimal separators with multiplicity two. Thus we build the family of forbidden for this case. Keywords: Clique trees, rooted path graphs, asteroidal quadruples.
1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.endm.2013.10.008
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M. Gutierrez, S.B. Tondato / Electronic Notes in Discrete Mathematics 44 (2013) 47–52
Introduction
A classical result [4] states that a graph is chordal if and only if it has a clique tree, i.e a tree T whose vertices are the cliques of the graph and for each vertex x, the cliques of G that contain x induce a subtree: Tx of T . Natural subclasses of chordal graphs are path graphs (U V ), directed path graphs (DV ) and rooted directed path graphs (RDV ). A graph is UV if it admits a UV model, i.e. a clique tree T such that Tx is a subpath of T for every x ∈ V (G). A graph is DV (resp. RDV ) if it admits a DV model (resp. RDV model ) , i.e. a UV model (resp. DV model) that can be oriented (oriented and rooted) such that Tx is a directed subpath of T . Panda [7] found the characterization of DV by forbidden induced subgraphs and then Cameron, Ho´ang and L´evˆeque [2] gave a characterization of this class in terms of forbidden asteroidal triples. Characterizing RDV by forbidden induced subgraphs or forbidden asteroids are open problems. It is certainly too difficult to characterizing RDV by forbidden induced subgraphs as there are too many (families of) graphs to exclude but Cameron, Ho´ang and L´evˆeque [1] conjecture that a DV graph is a RDV graph if and only if it contains no special asteroidal quadruple defined through special type of connections between these vertices. Although they defined other special connections, in the present work we will need only two types of these connections: Type 1: vertices a1 , w, a2 and edges a1 w, a2 w; Type 2: vertices a1 , a, b, c, d, a2 and edges a1 a, a1 b, ab, bc, cd, da, ac, a2 c, a2 d. It was proved [2] if G is a DV graph and a1 , a2 are two nonadjacent vertices that are linked by a special connection of Type 1 or Type 2 then for every DV model T of G, the subpath T (a1 , a2 ) is a directed path. An asteroidal quadruple is a set of four non-adjacent vertices such that any three of them is an asteroidal triple. A special asteroidal quadruple is an asteroidal quadruple such that there are two pairs disjoint of vertices linked by special connection of Type 1 or Type 2. Gutierrez, L´evˆeque and Tondato [5] proved also that every DV graph that is not RDV graph has an asteroidal quadruple. Therefore any DV non RDV graph must be leafage at least four, we will study only that of leafage four. We show some properties of DV models in DV minimal non RDV graph that allow us to prove the conjecture on these graphs having two minimal separators of multiplicity two. The multiplicity of a minimal separator S in a graph G is 1 2
Email: [email protected] Email: [email protected]
M. Gutierrez, S.B. Tondato / Electronic Notes in Discrete Mathematics 44 (2013) 47–52
49
c − 1 where c is the number of connected components of G \ S having a vertex complete to S. Finally, we build the family of forbidden for this case.
2
Definitions, notation and previous results.
Let T be a clique tree. We often use capital letters to the vertices of V (T ) and write X ∈ T instead of X ∈ V (T ), and e ∈ T instead of e ∈ E(T ). If T is a subtree of T , then GT denotes the subgraph of G that is induced by the vertices of ∪X∈V (T ) X. If G is a graph and V ⊆ V (G), then G \ V denotes the subgraph of G induced by V (G) \ V . If E ⊆ E(G), then G − E denotes the subgraph of G induced by E(G) \ E . If G, G are two graphs, then G + G denotes the graph whose vertices are V (G) ∪ V (G ) and edges are E(G) ∪ E(G ). Note that if T, T are two trees such that |V (T ) ∩ V (T )| ≤ 1, then T + T is a forest. Let T be a tree. For V ⊆ V (T ), let T [V ] be the minimal subtree of T containing V . Then for X, Y ∈ V (T ), T [X, Y ] is the subpath of T between X and Y . Let T [X, Y ) = T [X, Y ] \ Y , T (X, Y ] = T [X, Y ] \ X and T (X, Y ) = T [X, Y ] \ {X, Y }. A vertex X ∈ V (T (Y, Z)) has a vertex crossing in T [Y, Z] if X ∩ X = ∅ where X and X are the two neighbors of X in T [Y, Z]. Let G be a chordal graph and T a clique tree. The label of an edge AB of T is defined as lab(AB) = A ∩ B. We say that X ∈ V (T ) dominates e ∈ E(T ) if lab(e) ⊆ X. We say that e ∈ E(T ) dominates e ∈ E(T ) if lab(e) ⊆ lab(e ). Recall that the labels of edges correspond to minimal separators of G. The multiplicity of e is the number of edges having its label and the multiplicity of a minimal separator S in a graph G is c − 1 where c is the number of connected components of G \ S having a vertex complete to S. It is clear that both multiplicity share. If e has multiplicity two we will say that it has a twin edge. Clearly, for each edge e of a clique tree, in every clique tree there is e such that lab(e) = lab( e), we will say that e and e are equivalents. The leafage of a chordal graph G is a minimum integer such that G admits a clique tree T with leaves. UV-leafage and DV-leafage are defined analogously with T being a UV model and DV model respectively. For any DV graph we have DV-leafage = UV-leafage = leafage [3] and [6]. As was mentioned, every DV graph minimally non RDV graph has an asteroidal quadruple then has leafage at least four. In the following results assume that: T is a DV model with four leaves H1 , H2 , H3 , H4 and two vertices of degree three, C1 , C2 being C1 the only vertex of degree 3 in T [H1 , H2 ]. We will say that T [Hi , C1 ] for i = 1, 2 and T [Hj , C2 ] for j = 3, 4 are the T -branches corresponding to Hi , i = 1, ..., 4 respectively. Similar notation will be used in
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case that T is a star where the vertices C1 and C2 are the same. Lemma 2.1 Let e ∈ T [H1 , C1 ] then: i)|lab(e)| > 1; ii)There are at least two vertices in lab(e) with different ends towards C1 ; iii) If e ∈ T (H1 , C1 ], it has a dominated edge outside T [H1 , C1 ]. Proof. By way of contradiction. Let e = AB with B ∈ T [A, C1 ]. i) Suppose A ∩ B = {x} for x ∈ V (G). Let T = T − E(T [H1 , B]). By minimality of G, GT is a RDV graph. Let T be a RDV model of GT rooted on a vertex R. Let Z, W ∈ T such that Tx = T [Z, W ] and W ∈ T [Z, R]. The DV model of G, T1 = T + ZA + T [A, H1 ] is rooted on R. ii) Suppose that every vertex x ∈ lab(e) satisfies Tx ∩ (T − E(T [H1 , A])) = T [A, X] for some X in T . Let T = T − E(T [H1 , B]). Let T be a RDV model of GT rooted on a vertex R. Observe that all vertices of lab(e) are twins in GT . Hence for every x ∈ lab(e) Tx = T [Z, W ] with W ∈ T [Z, R]. As in i), T1 is rooted on R. iii) Suppose that none edge outside T [H1 , C1 ] is dominated by e. Take e dominated by e in T (H1 , C1 ] nearest C1 (it could be e). Let e = A B with B ∈ T [A , C1 ] and T = T − E(T [H1 , A ]). By the election of e , A is a leaf in any clique tree of GT . Let T be a RDV model of GT rooted on a vertex R. Clearly, T has only one edge equivalent to e and this edge must be incident in A . Hence the DV model of G T + T [H1 , A ] can be rooted on R or H1 . 2 Lemma 2.2 There is a vertex crossing C1 in T [H1 , H2 ] if and only if there is a vertex crossing C2 in T [H3 , H4 ]. Lemma 2.3 Two twin edges must be in different T -branches. Moreover they are the unique twin edges in these branches. Theorem 2.4 Let G be a DV minimally non RDV graph with leafage four and T a DV model of G with four leaves. If e and e are twin edges in T -branches corresponding to leaves Hi and Hk respectively i = k ∈ {1, ..., 4}. Then G \ Hi and G \ Hk are connected graphs. Moreover, the simplicial vertices of Hi and Hk are linked by a special connection of Type 1 or Type 2. Proof. Without loss of generality let e ∈ T [H1 , C1 ], e ∈ T [H3 , C2 ] and let a1 , a3 be simplicial vertices of H1 and H3 respectively. Suppose by contradiction, that G \ H1 is not connected. Hence H1 has a dominated edge e0 not incident in H1 . By Lemma 2.1, e = e0 and e = e0 . Since T is a DV model e0 ∈ / T [H2 , C1 ] + T [H4 , C2 ]. Then it is in T [H1 , H3 ]. If the edges e, e0 , e appear in this order along T [H1 , H3 ] then lab(e) = lab(e ) = lab(e0 ) contra-
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dicting Lemma 2.3. Hence e, e , e0 must appear in this order, contradicting Lemma 2.1. Thus G \ H1 is connected. Analogously, G \ H3 is connected. Therefore neither e nor e can be incident in H1 and H3 respectively. Hence |E(T [H1 , C1 ])| > 1 and there is e1 = A1 B1 ∈ T [H1 , C1 ] being A1 the neighbor of H1 . By Lemma 2.1, there exist dominated edges of e1 . Take e1 that maximally farthest from e1 . Analogously, there are e3 = A3 B3 ∈ T [H3 , C2 ] and e3 . As lab(e) = lab(e ) and T is a DV model then e1 , e3 ∈ T [H1 , H3 ]. In case that e1 is incident in H3 , by Lemma 2.1 there is x ∈ lab(e1 ) such that x ∈ H1 ∩ H3 . Therefore, a1 and a3 are linked by a special connection of Type 1. Analogously if e3 is incident in H1 . Observe that if e1 is between C1 and e , e1 = e , T [H1 , H3 ] had three edges with the same label which contradicts Lemma 2.3. Hence e1 is between e and A3 . Analogously e3 is between e and A1 . Therefore lab(e1 ) = lab(e3 ). By Lemma 2.3, e1 = e and e3 = e. Clearly, every vertex in lab(e1 ) = lab(e3 ) is in A1 ∩ A3 and exist x, y ∈ A1 ∩ A3 being x ∈ H1 and y ∈ H3 . Among all x ∈ H1 ∩ A3 chose one that maximizes |Cx |. Analogously for y. If x ∈ H3 or y ∈ H1 then a1 and a3 are linked by a special connection of Type 1. Suppose x ∈ H1 \ H3 and y ∈ H3 \ H1 . By Lemma 2.1, there are x1 = x and x3 = y with x1 ∈ H1 ∩ A1 and x3 ∈ H3 ∩ A3 respectively. Among all the x1 ∈ H1 ∩ A1 we chose one minimizing |Cx1 |, analogously for x3 . If x1 and x3 were adjacent some of them would be in lab(e1 ). Suppose x1 ∈ lab(e1 ) then x1 ∈ lab(e3 ) and x1 ∈ H1 ∩ A3 . As x1 was chosen minimizing |Cx1 | then Cx = Cx1 and x and x1 are twins in G, a contradiction. Therefore x1 and x3 are not adjacent. So a1 and a3 are linked by a special connection of Type 2.2 Theorem 2.5 If G is DV minimally non RDV with leafage four and two minimal separators with multiplicity two then: i) G has a special asteroidal quadruple; ii) G is one of the graph G1 , G3 , G5 or one of the families G2 , G4 , G6 in Figure 1. Proof. Let T be a DV model with four leaves of G. Since G has separators in the conditions of the Theorem, it is has four edges e, e , d, d being e and e (resp. d and d ) twin edges. By Lemma 2.3, we can assume e ∈ T [H1 , C1 ], e ∈ T [H3 , C2 ], d ∈ T [H2 , C1 ] and d ∈ T [H4 , C2 ]. By Theorem 2.4, there is a special connection of Type 1 or Type 2 between a1 , a3 (resp. a2 , a4 ) being ai the simplicial vertex of Hi and a1 , a2 , a3 , a4 is an asteroidal quadruple of G. As in the proof of Theorem 2.4, there exist edges e1 , e3 and e = e3 and e = e1 . First suppose that there is a special connection of Type 1 and no Type 2 between a1 and a3 , then there is a vertex x ∈ H1 ∩ H3 . By Lemma 2.1, there is a vertex w in lab(e1 ) − {x} not in H1 . Take w minimizing |Cw |. Since
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M. Gutierrez, S.B. Tondato / Electronic Notes in Discrete Mathematics 44 (2013) 47–52 G1
G2
G3
G4
G5
G6
Fig. 1. Graphs and its DV models
there is not a special connection of Type 2 between a1 and a3 then w ∈ / H3 . By Lemma 2.1, there is x1 ∈ H1 ∩ A1 (resp. x3 ∈ H3 ∩ A3 ) minimizing |Cx1 | (resp. |Cx3 |). If Cx1 ∩ Cx3 = ∅ then x1 or x3 is in lab(e1 ). Suppose that x1 ∈ lab(e1 ) then there is a vertex in A1 − B1 and by the election of x1 it is a simplicial vertex: s. Observe that G\s also has a special asteroidal quadruple, a contradiction. Hence Cx1 ∩ Cx3 = ∅. Therefore a special connection of Type 1 implies the existence of two paths a1 , x, a3 and a1 , x1 , w, x3 , a3 between a1 and a3 being x adjacent to x1 , w and x3 . Graphs G3 and family G4 appear in case that the both connection are Type 1. Graphs G5 and family G6 appear in case that one connection is Type 1 and the other is Type 2. Finally, graphs G1 and family G2 appear in case that the both are Type 2. 2
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