Mortality of iterated Gallai graphs - Springer Link
Periodica Mathematiea Hungariea Vol. P7 (~), (1993), pp. 105-12~
MORTALITY OF ITERATED GALLAI GRAPHS VAN BANG LE (Berlin)
Abstract For a finite or infinite graph G, the Gallai graph F(G) of G is defined as the graph whose vertex set is the edge set E(G) of G; two distinct edges of G are
adjacent in F(G) if they are incident but do not span a triangle in G. For any positive integer t, the tth iterated Gallai graph F*(G) of G is defined by F(F~-I(G)), where
F~ := G. A graph is said to be Gallai-mortalif some of its iterated Gallal graphs finally equals the empty graph. In this paper we characterize Gallai-mortal graphs in several ways.
w 1. I n t r o d u c t i o n
For a finite or infinite graph G, the Gallai graph I,(G) of G is defined as the graph whose vertex set is the edge set E(G) of G; distinct edges e, e~ of G are adjacent in I,(G) if e = zy, e~ = yz for some vertices z, y and z of G such that z and z are non-adjacent in G. Thus I'(G) is a spanning subgraph of the well-known line graph L(G). Gallai graphs, first considered by T. GALLAI [2], play an important role in the investigation of comparability graphs. Moreover, Gallai graphs are also interesting for the theory of perfect graphs: A (finite) graph is called Gallai-perfect [8] if its Gallai graph does not contain a chordless cycle of odd length at least five; L. SUN [8] showed that Gallai-perfect graphs are perfect in the sense of C. BERGE. Gallai graphs are also used in V. CHV~TAL and N. SBIHI [1]. On the other hand, the strong perfect graph conjecture is open for the class of Gallai graphs. The existence of infinite graphs isomorphic to their Gallai is discussed by E. PRISNER in [5]. In [3] we showed that any graph H is an induced subgraph of some graph G ~ I'(G); moreover, if H is finite, then there exists some locally finite such G. So the I,-operator and the L-operator have quite different behaviour (cf. [4], [6]
also [7l). Mathematics subject classification numbers, 1991. Primary 05C99. Key words and phrases. Gailai graphs, iterated graph-valued functions. Akad~rniai Kiad6, Budapeat Kluwer Academic Publishers, Dordrecht
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For a graph G define r~ := G. Then for any positive integer t, the tth iterated Gallai graph V(G) of G is defined by r(r'-l(G)). A graph G is said to be Gallai-mortalifrt(G) is empty for some t > 0. Gallaimortal graphs play an important role in the investigation of Gallai-periodic graphs [3], and are interesting in their own right; they will be characterized in this paper. I am indebted to Professor H. A. JUNG for his helpful discussions; due to his suggestions I was finally able to characterize the infinite Gallai-mortal graphs. I thank also E. PRISNER and I. SATO for their reading of an earlier version of this paper. The following basic properties of Gallai graphs can be obtained easily from the definition. We use these properties often without reference. PROPOSITION 1.1. (i) If H is an induced subgraph of G, then r ( H ) is an induced subgraph o f r ( G ) . (ii) F(G 9 K1) ~ F(G) (J G for every graph G. Here, the complement of a graph G is denoted by G, and the graph G * K1 is obtained from G by adding a new vertex and all edges between vertices of G and that new vertex. Now we are going to describe a main result. If C is an induced cycle of length at least 4 in a graph G, then clearly r ( c ) is an induced cycle (of the same length) in r(G). Hence no Gallai-mortal graph can contain an induced cycle of length at least 4. Graphs without such cycles are called chordal. Thus, every Gallai-mortal graph G satisfies the CHORDALITY CONDITION (I). For any integer t >_ 0, F~(G) is chordal. (We shall see, as a consequence of the main results, that a finite graph is Gallai-mortal ff and only if it satisfies the Chordality Conditions (I).) Further it is also clear that ff P is an induced path of length 1 in a graph G, then r ( P ) is an induced path of length 1 - 1 in r(G). Thus, every Gallai-mortal graph G satisfies the INDUCED PATH CONDITION. G contains an induced path of maximum length. Notice that any finite graph satisfies the Induced Path Condition. The following example based on Proposition 1.1 (ii) gives a non Gallai-mortal graph which eatisfies the two conditions above. For an integer m >_ 0, a chordless path on m vertices is denoted by Pra, for any cardinal a, Ka denotes a complete graph on a vertices. Consider the graphs Gn, n >_ 1, which are recursively defined by G1 = K1 and Gn+l = Gn * K1. Then for any n > 1, it is easy to check by induction that Gn does not contains P4(~ ~
as an induced subgraph,
and
r"(G.) = r"(~-~.) = ~, but r " - l ( G . ) r $.
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Thus G = Un>I Gn satisfies the Chordality Condition (I) and the Induced Path Condition, whffe G cannot be Gallai-mortal. More precisely,
r(v) = U G.-1 u U r(GT-0 - Gu r(G). n>_2
n_>2
The feature in this situation is, that G contains "neighborhood chains" of any length (here realized in each Gn) in the following sense. For a vertex v of a graph G let Na(v) (or N(v), when the context is clear) be the set of neighbors of v in G, that is the set of all vertices in V(G)\{v) adjacent to v. An induced subgraph H of a graph G with 2 m - 1 vertices vl, x2,. 9 vm, wl, w2, 9.., w,n-1 is called a neighborhood chain of length m in G if
NH(vi) = V(H)\{vi,wl,...,wi_l}, for i = 1 , . . . , m , and
NH(wj) = { v l , . . . , W } , for j = 1 , . . . , m -
v2
1,
w!
a
Ym
w2
Win_ I
Figure 1 Note that in particular Vl, v 2 , . . . , vm are pairwise adjacent, and that w l , . . . , win-1 are pairwise non-adjacent. Thus the graph H is the comparability graph of the poset in Figure 1. Notice also that 1VH(vi) ~ NH(Vi+l), hence our notation. Clearly, i~\{vl, v,~ } is a neighborhood chain of length m - 1 in the complement
(m > 3).
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Now, the edges at vl of H induce a subgraph of F(G) which is isomorphic to H \ { v l } (see Proposition 1.1 (ii).) The latter contains the neighborhood chain [I\{vl, vm}. Consequently, every Gallai-mortal graph G has the following property NEIGHBORHOOD CHAIN CONDITION. G contains a neighborhood chain of mazimnm length. Notice that graphs of bounded degree satisfy the Neighborhood Chain Condition. We are now able to describe one of the main results:
A graph is Gallai-mortal if and only if it satisfies the three conditions above. In the proof we need a description in terms of forbidden induced subgraphs for graphs satisfying the Chorfality Condition (I). This will be done in the next section.
w 2. F o r b i d d e n i n d u c e d s u b g r a p h s for g r a p h s satisfying the C h o r d a l i t y C o n d i t i o n (I). ff G and H are isomorphic graphs, then we also write G = H. For any cardinal a, a G denotes a disjoint copies of the graph G, and for an integer m >_ 3, Cm denotes a chordless cycle on m vertices. A subgraph H of G is called homogeneous in G if every vertex of G outside H is adjacent to all or none vertices in H. Homogeneous cycles play an important role when we investigate the Gallai-periodic graphs of bounded degree. The following observations show that any graph listed in Figures 2, 3 and 4 yields an induced cycle of length at least 4 in some of its iterated Gallai graphs. The proofs of these observations are routine, hence we omitted them. OBSERVATION 2.1. F(G4) : G29 (3 3K1, F(G27 = G29 (3 K1, F(G29) = G1, r ( a l ) = c6. r(G3) = G22 u K~, r(G22) = C6 u K1. r(G30) = r2(G34) = G23, r(G2s) = G2s v K1, r(G~s) = a s u K~, r4(Gb) = G4 U 2K1. F(G2) --- Gs (9 2K1 r(G--~)= c9 and F4(~3) = C4 tJ 2 g l . F 4 ( ~ * g l ) has an induced C6, and F5(~55 9 g l ) an induced C4. OBSERVATION 2.2. There is a non-homogeneous
C4 in
rT(G6), rb(Gn),
r~(G2o), r~(a~6), ,rid r~(G3~). There is a non-homogeneous C6 in r4(GT), rS(Gs), r4(G9), r4(Glo),
r*(al~), r*(al~), r~(a.), rT(a~), r~(alT), rT(a~,), rS(a~5), rS(a~) and rS(a~2). There is a non-homogeneous C, in r(G16), r(G18), r(a21). There is a non-homogeneous C8, respectively, C9 in r(Ga2), respectively, in
r(a19).
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OBSERVATION 2.3. For any k >_ 2, r2(A2k_l) = A2k-3, F 2 ( B 2 k ) = B2k-2 I.J 2K1 and inductively, F2~-l(A2~_1) = F2k(B2~) = C4. Dk-1 is a proper induced subgraph of F(Dk), and inductively, G25 = F(D1) is an induced subgraph of Fk( Dk ). Thus, Fk+8(Dk) contains a non-homogeneous C6. [2(Es) = GsotJK1, so FI~ is a C4. For k >_ 2, E~k-1 is a proper induced subgraph of F2( E2k+l ). Thus the graph E3 is a proper induced subgraph of F2~(E2k+I), and so F2k+l~ contains a non-homogeneous Ca. F(F4) contains an induced Es and so Fll(F4) contains a non-homogeneous C4. For k >__2, F2k is a proper induced subgraph ofr2(F2k+2), hence F2k+ll(F2k+2 ) contains a non-homogeneous C4. Thus by the observations above, a graph G satisfying the Chordality Condition (I) has the following property. CHORDALITY CONDITION (II). G is chordal and does not contain G1, 99 G34, G1, G2, G3, G4 * K1, Gs * K1, and Azk-1, B2k. Dk, E2k+l, F2k+2, k > 1, as induced subgraphs (see Figures $, 3 and $). Notice that B2 is the complement G2 of the graph G2. A consequence of the main results is that the Chordality Conditions (I) and (II) are, in feint, equivalent; see Corollary 3.2.
w 3. C h a r a c t e r i z a t i o n s o f G a l l a i - m o r t a l g r a p h s If a graph G satisfies the Induced Path Condition or the Neighborhood Chain Condition, then we shall denote by p(G), v(G) the number of vertices of an induced path of maximum length in G, the maximum length of neighborhood chains in G, respectively. Graphs that do not contain an induced subgraph isomorphic to a given graph H are called H-free. THEOREM 3.1. For any graph G, the following statements are equivalent. (i) G is Gallai-mortal. (ii) G satisfies the Chordality Condition (I), the Induced Path Condition, and the Neighborhood Chain Condition. (iiii) G satisfies the Chordality Condition (II), the Induced Path Condition, and the Neighborhood Chain Condition. (iv) G satisfies the Induced Path Condition, the Neighborhood Chain Condition, and F'(G) = 0 for some t < p(G) + 2v(G) + 6. Moreover, if G is 194* Kl-free, then t _< p ( a ) + 2 v ( a ) - 1.
In the finite case, the theorem takes a simpler form. THEOREM 3.1'. For any finite graph G, the following statements are equiva. lent.
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G:
G2
G3
G4
05
G6
~
G:
1 and finite induced subgraph F in F*(G), there is some finite induced subgraph F ' in F*-I(G) such that F is an induced subgraph in r(F') (for example, F' is induced by the end vertices of edges in r'-l(G) which appear as vertices of F.) Now, if G does not satisfy the Chordality Condition (I), there is some t > 1 and some induced cycle C of length at last 4 in F*(G). Then, as noted above, there exists some finite induced subgraph H of G such that F*(H) contains C as an induced cycle. But H, as induced subgraph of G, satisfies the Chordality Condition (II) and, as a finite such graph, H satisfies the ChordMity Condition (I) by Theorem 3.1'. Thus we get a contradiction, hence G must satisfy the Chordality Condition (I). 9 The proof of Theorem 3.1 relies on a characterization of strongly Gallai-mortal graphs. Here, a graph G is called strongly Gallai-mortal if both G and G are Gallaimortal. The structure of Gallai-mortal graphs depends on that of strongly Gallaimortal graphs, as the following observation shows. Let H be the subgraph of a graph G induced by the neighborhood of a vertex of G. Since H 9 K1 is an induced subgraph of G, ~r is an induced subgraph of r(G) (see Proposition 1.1). Hence for any Gallai-mortal graph, the "neighborhood graphs" are strongly Gallai-mortal. For this reason we shall first characterize strongly Gallai-mortal graphs in Theorem 3.3 below, and then obtain a proof for Theorem 3.1. Before giving Theorem 3.3 we recall that a graph G satisfies the Neighborhood Chain Condition if and only if G does, as noted in the introduction; the same argument yields a little more: For graphs G satisfying the Neighborhood Chain Condition, v( G ) - 1 < v( G) t. If t is even, then by Claim 1.2, St is edgeless with at least two vertices, or St contains an induced P4. In any case, S / h a s two non-adjacent vertices w,, w~. Then vl, v3, v s , . . . , v , - i together with v2, v4,. 9 v,-2, w tt induce a neighborhood chain of length (t + 1)/2 of G. Again, 2v(G) - 1 >_ t. Let now G be disconnected. I f t = 1 then G is edgeless hence t = 1 < 2v(G) = 2. If t > 2 then, with H2 as in the construction, v(G) = v(H2). Since H2 is connected and has the decomposition {s/li = 2 , . . . , t}, as in the case above, we have t-l 2. (2) W e now show by induction on t that (2.1) if G is connected then the non-trivial components of rt-~(G) are that ofrk(s/) or ofrk(s-,-), fork = 0, I .... ,t- i.
and (2.2) If G is disconnected then the non-tirvial components o f r ' - 2 ( G ) are that ofrk(st)
or
fork = 0,1 .... ,t-
2.
Let namely Hi as in the construction (1). If G is connected, then the nontrivial components of r ( G ) are that of r(H2) and of I&l~. Thus i f t = 2 then we are done; if t > 3 then r(H2) = r ( H s ) and Ha is connected. It follows inductively from (1.4) and (1.5) (applied to Hz and H'~2) that the non-trivial components of and of are that of or for k = 0, 1,..., t - 2. Thus the non-trivial components of
r'-1(a) = rt-l(n2) u ISxlr'-2( ) are that of rk(s/) or rk(~t) for k = 0,1,...,t - i. That is (2.1).
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IF G is disconnected, t h e n / / 2 = G\S1 is connected and r(G) = r(H2); (2.1), the non-trivial components of r t - 2 ( G ) = r t - 2 ( H 2 ) are that of r k ( s t ) rk(s'~t) for k = 0, 1 , . . . , t - 2. That is (2.2). 9 (3) Now, by Claim 1.2, r 2 ( s t ) = r 2 ( ~ ) = 0 if G (hence St) is P4-free. other cases, rg(s,) = rg(s~,) - r (see also the proof of Lemma 4.2). Thus, by we have for connected G rt-l+9(G) = r
117
by or In (2)
and r'-l+2(G) = $ if G is P4-free,
and for disconnected G, rt-2+S(G) = r
and rt-2+2(G) = r ifa is Pa-free.
This and Claim 1.3 yield
r2~(a)+~(G) = O, and r2~(a)(G) = O if G is P4-free. Now Theorem 3.3 follows by symmetry and the fact v(G) l, or i = 2 . An even quasi.millepede (odd quasi-millepede} M* is a graph isomorphic to the Gallai graph of an even millepede (odd millepede) M. More precisely, a graph M* consisting of an induced path P = vlvz . . . vt of maximum length such that * d(Vl) = 1
is an even quasi-millepede of length l, if it has furthermore the following property 9 Any vertex outside P is adjacent tO exactly two vertices vi,vi+l for some even i, and (N(vi) N N ( v I + I ) ) \ V ( P ) induces a complete graph. It is an odd quasi-millepede of length l, if 9 Any vertez outside P is adjacent to exactly two vertices vi,vi+l for some odd i, and (N(vi) N N ( v i + I ) ) \ V ( P ) induces a complete graph. In any case, the vertex vl of M or M* is called the head of M or M*, respectively. Notice that the Gallai graph of a quasi-millepede has at most one non-trivial component which is a millepede. The proof of the following technical lemma is given in Section 7.
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LEMMA 5.1. Let G satisfy the Induced Path Condition and the Chordality Condition (II). If G is connected and p = p(G) > 5 then G is one of the graphs Y, Z, g(a, 8) listed in Figure 6, or G is an even (quasi-) millepede of length p, of G is a substitution of some graph satisfying the Chordality Condition (IlI) for the head of some odd (quasi-) millepede of length p.
u
/A
w
v
Z
w
w
Z(a, ~)
Figure 6 PROOF of Theorem 3.1. The implications "(i) ~ (ii) =# (iii)" are showed in Sections 1 and 2; The implication "(iv) => (i)" is trivial. Let G r 0 satisfy the Property (iii) of Theorem 3.1 we shall prove (iv). We first assume that G is connected and then consider two cases. (4) G has a central vertex. Then G satisfies the Property (iii) of Theorem 3.3, hence F2v(G)+7(G) = and r2~(G)(G) = $ whenever G is P4 * Kl-free. Thus Theorem 3.1 is proved. (5) G has no central vertex. By Lemma 4.1, p = p(G) > 4 and G is one of the graphs described in Lemma 4.2 (ii), or (iii) and in Lemma 5.1. If G is one of the graphs T , X , X , I ~ _ , I ~ + , J ~ , J ~ + (see Figure 5), or Y,Z, Z ( a , ~ ) (see Figure 6) then Fg(G) = 0, and we are done; see also the proofs of Lemmas 4.2 and 5.1. If G is an even (quasi-) millepede of length p then one shows easily by induction that FP+2(G) is empty. Thus we may assume that G is substitution of some graph H satisfying the Chordality Condition (III) for the head of some odd (quasi-) millepede of length p. In this case, the non-trivial components of FP-I(G) are that of rk(H) and of
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rk(_0) for k = 0 , 1 , . . . , p Theorem 3.3 yields
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1. Since v(H) = v ( G ) - 1 and v(~r) < v ( H ) + 1 = v(G),
rp-l+[2~+~l(G) = r p + ~ + 6 ( G ) = r and
r~-l+2~(G) = rp+2~-~(G) = r if H is P4-free, that is, if G is P4 * Kl-free. Thus Theorem 3.1 is proved for connected graphs G. If G is disconnected and Q is a connected component of G then p(Q) < p(G), v(Q) < v(G); Theorem 3.1 follows now by applying the result above for each component Q of G. 9
6. P r o o f o f L e m m a 4.2
An edge e of a connected graph G is a bridge if G\{e} is disconnected. Let P = VlV2... v,~ be an induced path of a graph G. A vertex v of G outside P is called a x-verte~ of Pi if v has x neighbors in P. If G is chordal, Np(v) must be a subpath of P. Then we shall can a x-vertex v of type (i), if Np(v) is the usbpath of P with endvertices vi and vi+~-l, that is, Np(v) = { v i , . . . , ~)i+~-1}. If furthermore G does not contain an induced A1 then P has no x-vertex for k > 5. Note that if P is maximal, P has also no 1-vertex of types (1) and (m). The following observation is used in the proof below. Let G satisfy the Chordality Condition (II). If H is a subgraph of G induced by a neighborhood of a vertex of G, then H satisfies the Chordality Condition (III). Let G be a graph satisfying the properties of Lemma 4.2. Assume that G has no central vertex. We have to show that the Statement (ii) or (iii) of Lemma 4.2 holds. We consider several cases. (6) There is an induced path P = VlV~V3V4 such that the middle edge v2v3 is a bridge of G. Let H be the subgraph of G induced by N(v2)\{v3}; K the subgraph induced by N(va)\{v2}. Then one of H and K must be complete (else G contains an induced B2), say K. Since G has no induced Ps, G is the substitution of H for the vertex Vl, and then of K for the vertex v4 of P. Since H is defined by a neighborhood of a vertex of G, H satisfies the Chordality Condition (III). Thus in this event, the Statement (ii) of the lemma occurs. Thus we may assume that G has no bridge which is a middle edge of'an induced P4. (7) There is a path P = V l V 2 V 3 ? ) 4 with two 3-vertices u, v. Then u and v are of distinct types (else G contains an induced G4; recall G is chordal), therefore u and v are the only 3-vertices. Furthermore, u and v are non-adjacent (else G contains an induced G2). Now, P has no 4-vertex (else G contains an induced A1), no 2-vertex (else G contains an induced Ps, or A1, or G2, or G12), no 1-vertex (else G contains an induced Ps, or G3, or A1), and no 0-vertex (else G contains an induced Ps; recall that G is connected). Thus G is the graph T in Figure 5. We have FS(G) = 0, v(G) = 2.
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Thus we may assume that any induced path P4 has at most one 3-vertex. (8) There is a path P = V l V 2 V 3 V 4 with a 4-vertez. Let M be the set of 4-vertices of P . Since G is chordal, M induc4s a complete subgraph in G. Moreover, for two vertices m l and m2 of M , we have
Na(ml) U {ml} _CNa(r 2) U
or Na(m2) U {m2} C_Na(ml) U {rod
(else G contains an induced G1, Gs, G4, or A1). Let m E M with m a x i m a l neighborhood. Since G has no central vertex, there is some vertex v such that v is non-adjacent to m, inparticular m ~ M U V ( P ) , and v is non-adjacent to all vertices in M because N ( m ) is maximal. Therefore that v is not 3-vertex, not 4-vertex of P. The vertex v is also not 2-vertex (else there is an induced p a t h P4 with two 3-vertices, or G contains an induced G-~-I). Now we shall show that v is a 1-vertex of P. Suppose the contrary t h a t v is a 0-vertex. T h e n further, we m a y assume t h a t there is some c o m m o n neighbor w of v a n d m (recall G is connected). Since v is a 0-vertex of P, w is not a vertex of P , and since v is non-adjacent to all vertices of M, w does not belong to M . Now, w is not a 3-vertex (else G contains an induced Gs), not a 2-vertex (else G contains an induced G1, or Ps), and not a 1-vertex (else G contains an induced Ps). The vertex w is also a 0-vertex of P , therefore Pl m w vl is an induced p a t h of G. Since the middle edge m w of t h a t p a t h is not a bridge of G, there is some vertex t adjacent to m and w; t does not belong to P because w is a 0-vertex of P (but t is possibly adjacent to v). Now, t is neither a 0-vertex nor a 1-vertex of P (else G contains an induced Ps, or A1). t is also neither a 3-vertex nor a 4-vertex of P (else G contains an induced Gs, or A1). Therefore that t is a 2-vertex of P . But then G contains an induced A1, or Ps, or Gll Thus v is a 1-vertex of P , as claimed. Since G is Ps-free, v is a 1-vertex of type (2) or (3). This yields M = {m}, otherwise G contains an induced G4 (v is non-adjacent to all vertices in M ) . Moreover, v is the only 1-vertex of P (else G contains an induced B2, or Gs, or A1, or G12, or there is an induced p a t h P4 with two 3-vertices). Finally, P has no 2-vertex (else G contains an induced G--~',or Gs, or G3, or A~), no 3-vertex (else G contains an induced G3, or G1, or Gs, or G3), no 0-vertex (else G contains an induced Gs, or Ps). Thus G is the graph X in Figure 5, and we have r s ( G ) = 0, v(G) = 3. Thus we may assume that no induced path P4 has a ~-vertex. (9) There is an induced G4 in G. Let P = vlv2vsv4 be an induced p a t h in a G4 in G. Then P has no 3-vertex (else G contains an induced G10, or Gs, or G3, or Gs), no 2-vertex of type (2) (else G contains an induced G9, or G10, or ~ ) . Thus the middle edge v2 v3 of P is a bridge of G. Thus we may assume that G does not contai~i an induced G4. (10) There is a path P = vlv~vzv4 with (exactly) one 3-vertex in G. We m a y assume that the 3-vertex v is of type (1). Case 10.1. P has a 1-vertex. Let w be a 1-vertex then w is of type (2) (else G contains an induced Ps, or Gs, or there is an induced P4 with a 4-vertex), and w is the only 1-vertex of P (else
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G contains an induced G-'~, or G'-~, or A1; therefore P has no 2-vertex (else there is an induced P4 with a 4-vertex, or G contains an induced G l l , or Ps, or GT, or G1, or At), and no 0-vertex (else G contains an induced G1, or Ps). Thus G is the graph ~: in Figure 5, and so rg(G) = 0, v(G) = 3. Case 10.2. P has a 2-vertex. By 10.1 above, P has no 1-vertex. P has also no 2-vertex of type (3) (else G contains an induced A1, or there is an induced P4 with a 4-vertex). Any 2-vertex of type (1) is adjacent to v (else G contains an induced Ps), and any 2-vertex of type (2) is also adjacent to v (else G contains an induced P4 with a 4-vertex). Let now M, N be the sets of 2-vertices of type (1), respectively, of type (2) of P . Then each of M and N induces a complete subgraph of G (else G contains an induced G'-~, respectively, Gs). Furthermore, M or N must be empty (else G contains an induced A1, or Ps), and P has no 0-vertex (else G contains an induced GI1, or Ps, if M ~ 0; or G contains an induced G1, or Gr, or there is an induced path P4 with a 4-vertex, if N r 0). Thus G is the graph I a _ , or Ia+ in Figure 5 (where a denotes the cardinality of the non-empty set M or N), and so r e ( G ) = $, v(G) = 2 (if M r $), or rS(G) = 0, v(G) = 2 (if g r 0). Case 10.3. P has no 1-vertex, no 2-vertex. If G = I0- = I0+ then r6(G) = 0 v(G) = 2. Thus we may assume that P has a 0-vertex u adjacent to v (recall G is connected). Now, v2 is a 3-vertex and u is a 1-vertex of the induced path vl v v3 v4. As in Case 10.1 above, r9(G) = 0. (11) It remains the case that, no induced P4 of G has a 4-vertez, or a 3-vertez. Let now P = VlV2Vzv4 be an induced path in G. Since vzv3 is not a bridge of G, the set N of 2-vertices of type (2) of P is non-empty, and so P has no 2-vertex of other types (else G contains an induced A1, or Ps). The vertices of N induce a complete subgraph (else G contains an induced Gs). Since G is Ps-free, P has no 1-vertex of type (1) or (4). All other 1-vertices of P are of type (2) or of type (3) (else G contains an induced B2) and none of the 1-vertices is adjacent to a vertex of N (else there is an induced P4 with a 3-vertex), and so P has at most one 1-vertex (else G contains an induced G-'~, or A1, or B~). P has no 0-vertex (else G contains an induced G1, or Ps)- Thus G is the graph Ja, or J a + l , and so rs(G) = 0, v(G) = 2,
or rg(G) = 0, v ( V ) = 2. By (6) - (11), the proof of lemma 4.2 is complete.
9
w 7. P r o o f o f L e m m a 5.1 Let G be a graph satisfying the properties in Lemma 5.1. We use the same notations as in Section 6, and consider several cases again. Set p = p ( G ) > 5. (12) There is an induced path P = v l v 2 . . . v p with a f-vertex. Then p = 5 (else G contains an induced G20, or G21, and the 4-vertex v (say) is the only 4-vertex of P (else G contains an induced G4, or GI9). P has no 3vertex (else G contains an induced G17, or Gas, or A1, or G22), no 2-vertex (else G contains an indeced G22, or A1, or G1, or G15, or G16), no 1-vertex (else G contains
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an induced G22, o r G14, o r G1, or G3, or A1), 0-vertex (else G contains an induced G13). Thus G is the graph Y in Figure 6. We have rg(G) = 0 and v(G) = 2. (13) There is an induced path P = v l v 2 . . . v p with a 3-vertex. Then p = 5 and a 3-vertex v is of type (1) or (3) (else G contains an induced G2r, or G22. v is the only 3-vertex of P (else G contains an induced Gd, or At, or Gls). By (12) P has no 4-vertex. P has also no 1-vertex (else G contains an induced G~5, or Gz, or Gs, or G24, or G23). Without loss of generality, say that the 3-vertex v is of type (1). Then P has no 2-vertex of type (1) (else G contains an induced G20, or G26, no 2-vertex of type (3) (else G contains an induced A1, or G2~). Let M, N be the sets of 2-vertices of type (2), respectively, of type (4) of P. Then M induces a complete subgraph of G (else G contains an induced Gs), and v is adjacent to all vertices in M (else G contains an induced G24). Vertices of N are adjacent to none vertices of M U {v} (recall G is chordal), and N induces also a complete subgraph of G (else G contains an induced G23). Finally, P has no 0-vertex (else G contains an induced G25, or G1, or G27). Thus G is the graph Z(a, 8) in Figure 6 (where a and ~ denote the cardinalities of M, Y respectively), and so rs(G) is empty and v(G) = 2. By (12) and (13) we may assume that no induced P (on p ~_ 5 vertices) has a 4-vertex, or a 3-vertex. Then P has also no 0-vertex (else G contains an induced G1, or G29). Let P = v l v 2 . . , vp be an induced path of G, and let Li be the set of 2-vertices of type (i), 1 < i < p - 1, and let Mj be the set of 1-vertices of type (j), 2 < j < p - 1, of P. Since P has no 0-vertex, no 3-vertex, no 4-vertex, G consists of P, (J Li and (J Mj only. Further, for each 2 < i < p - 2, Li induces a complete subgraph of G (else G contains an induced Gs), and for 3 _< j _< p - 2, Mj is an independent set of G, that is, a set of pairwise non-adjecent vertices (else G contains an induced G2s). Further the subgraphs H and H ' of G induced by L 1U M2 U { v 1}, Lp_ 1U Mp_ t tJ { vp }, respectively, satisfy the Chordality Condition (III). We shall often identify a set of vertices with the subgraph induced by that set. (14) There is an induced path P = v l v 2 . . . v p such that the corresponding set Mj is non-empty for some 3
MORTALITY OF ITERATED GALLAI GRAPHS VAN BANG LE (Berlin)
Abstract For a finite or infinite graph G, the Gallai graph F(G) of G is defined as the graph whose vertex set is the edge set E(G) of G; two distinct edges of G are
adjacent in F(G) if they are incident but do not span a triangle in G. For any positive integer t, the tth iterated Gallai graph F*(G) of G is defined by F(F~-I(G)), where
F~ := G. A graph is said to be Gallai-mortalif some of its iterated Gallal graphs finally equals the empty graph. In this paper we characterize Gallai-mortal graphs in several ways.
w 1. I n t r o d u c t i o n
For a finite or infinite graph G, the Gallai graph I,(G) of G is defined as the graph whose vertex set is the edge set E(G) of G; distinct edges e, e~ of G are adjacent in I,(G) if e = zy, e~ = yz for some vertices z, y and z of G such that z and z are non-adjacent in G. Thus I'(G) is a spanning subgraph of the well-known line graph L(G). Gallai graphs, first considered by T. GALLAI [2], play an important role in the investigation of comparability graphs. Moreover, Gallai graphs are also interesting for the theory of perfect graphs: A (finite) graph is called Gallai-perfect [8] if its Gallai graph does not contain a chordless cycle of odd length at least five; L. SUN [8] showed that Gallai-perfect graphs are perfect in the sense of C. BERGE. Gallai graphs are also used in V. CHV~TAL and N. SBIHI [1]. On the other hand, the strong perfect graph conjecture is open for the class of Gallai graphs. The existence of infinite graphs isomorphic to their Gallai is discussed by E. PRISNER in [5]. In [3] we showed that any graph H is an induced subgraph of some graph G ~ I'(G); moreover, if H is finite, then there exists some locally finite such G. So the I,-operator and the L-operator have quite different behaviour (cf. [4], [6]
also [7l). Mathematics subject classification numbers, 1991. Primary 05C99. Key words and phrases. Gailai graphs, iterated graph-valued functions. Akad~rniai Kiad6, Budapeat Kluwer Academic Publishers, Dordrecht
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For a graph G define r~ := G. Then for any positive integer t, the tth iterated Gallai graph V(G) of G is defined by r(r'-l(G)). A graph G is said to be Gallai-mortalifrt(G) is empty for some t > 0. Gallaimortal graphs play an important role in the investigation of Gallai-periodic graphs [3], and are interesting in their own right; they will be characterized in this paper. I am indebted to Professor H. A. JUNG for his helpful discussions; due to his suggestions I was finally able to characterize the infinite Gallai-mortal graphs. I thank also E. PRISNER and I. SATO for their reading of an earlier version of this paper. The following basic properties of Gallai graphs can be obtained easily from the definition. We use these properties often without reference. PROPOSITION 1.1. (i) If H is an induced subgraph of G, then r ( H ) is an induced subgraph o f r ( G ) . (ii) F(G 9 K1) ~ F(G) (J G for every graph G. Here, the complement of a graph G is denoted by G, and the graph G * K1 is obtained from G by adding a new vertex and all edges between vertices of G and that new vertex. Now we are going to describe a main result. If C is an induced cycle of length at least 4 in a graph G, then clearly r ( c ) is an induced cycle (of the same length) in r(G). Hence no Gallai-mortal graph can contain an induced cycle of length at least 4. Graphs without such cycles are called chordal. Thus, every Gallai-mortal graph G satisfies the CHORDALITY CONDITION (I). For any integer t >_ 0, F~(G) is chordal. (We shall see, as a consequence of the main results, that a finite graph is Gallai-mortal ff and only if it satisfies the Chordality Conditions (I).) Further it is also clear that ff P is an induced path of length 1 in a graph G, then r ( P ) is an induced path of length 1 - 1 in r(G). Thus, every Gallai-mortal graph G satisfies the INDUCED PATH CONDITION. G contains an induced path of maximum length. Notice that any finite graph satisfies the Induced Path Condition. The following example based on Proposition 1.1 (ii) gives a non Gallai-mortal graph which eatisfies the two conditions above. For an integer m >_ 0, a chordless path on m vertices is denoted by Pra, for any cardinal a, Ka denotes a complete graph on a vertices. Consider the graphs Gn, n >_ 1, which are recursively defined by G1 = K1 and Gn+l = Gn * K1. Then for any n > 1, it is easy to check by induction that Gn does not contains P4(~ ~
as an induced subgraph,
and
r"(G.) = r"(~-~.) = ~, but r " - l ( G . ) r $.
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Thus G = Un>I Gn satisfies the Chordality Condition (I) and the Induced Path Condition, whffe G cannot be Gallai-mortal. More precisely,
r(v) = U G.-1 u U r(GT-0 - Gu r(G). n>_2
n_>2
The feature in this situation is, that G contains "neighborhood chains" of any length (here realized in each Gn) in the following sense. For a vertex v of a graph G let Na(v) (or N(v), when the context is clear) be the set of neighbors of v in G, that is the set of all vertices in V(G)\{v) adjacent to v. An induced subgraph H of a graph G with 2 m - 1 vertices vl, x2,. 9 vm, wl, w2, 9.., w,n-1 is called a neighborhood chain of length m in G if
NH(vi) = V(H)\{vi,wl,...,wi_l}, for i = 1 , . . . , m , and
NH(wj) = { v l , . . . , W } , for j = 1 , . . . , m -
v2
1,
w!
a
Ym
w2
Win_ I
Figure 1 Note that in particular Vl, v 2 , . . . , vm are pairwise adjacent, and that w l , . . . , win-1 are pairwise non-adjacent. Thus the graph H is the comparability graph of the poset in Figure 1. Notice also that 1VH(vi) ~ NH(Vi+l), hence our notation. Clearly, i~\{vl, v,~ } is a neighborhood chain of length m - 1 in the complement
(m > 3).
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Now, the edges at vl of H induce a subgraph of F(G) which is isomorphic to H \ { v l } (see Proposition 1.1 (ii).) The latter contains the neighborhood chain [I\{vl, vm}. Consequently, every Gallai-mortal graph G has the following property NEIGHBORHOOD CHAIN CONDITION. G contains a neighborhood chain of mazimnm length. Notice that graphs of bounded degree satisfy the Neighborhood Chain Condition. We are now able to describe one of the main results:
A graph is Gallai-mortal if and only if it satisfies the three conditions above. In the proof we need a description in terms of forbidden induced subgraphs for graphs satisfying the Chorfality Condition (I). This will be done in the next section.
w 2. F o r b i d d e n i n d u c e d s u b g r a p h s for g r a p h s satisfying the C h o r d a l i t y C o n d i t i o n (I). ff G and H are isomorphic graphs, then we also write G = H. For any cardinal a, a G denotes a disjoint copies of the graph G, and for an integer m >_ 3, Cm denotes a chordless cycle on m vertices. A subgraph H of G is called homogeneous in G if every vertex of G outside H is adjacent to all or none vertices in H. Homogeneous cycles play an important role when we investigate the Gallai-periodic graphs of bounded degree. The following observations show that any graph listed in Figures 2, 3 and 4 yields an induced cycle of length at least 4 in some of its iterated Gallai graphs. The proofs of these observations are routine, hence we omitted them. OBSERVATION 2.1. F(G4) : G29 (3 3K1, F(G27 = G29 (3 K1, F(G29) = G1, r ( a l ) = c6. r(G3) = G22 u K~, r(G22) = C6 u K1. r(G30) = r2(G34) = G23, r(G2s) = G2s v K1, r(G~s) = a s u K~, r4(Gb) = G4 U 2K1. F(G2) --- Gs (9 2K1 r(G--~)= c9 and F4(~3) = C4 tJ 2 g l . F 4 ( ~ * g l ) has an induced C6, and F5(~55 9 g l ) an induced C4. OBSERVATION 2.2. There is a non-homogeneous
C4 in
rT(G6), rb(Gn),
r~(G2o), r~(a~6), ,rid r~(G3~). There is a non-homogeneous C6 in r4(GT), rS(Gs), r4(G9), r4(Glo),
r*(al~), r*(al~), r~(a.), rT(a~), r~(alT), rT(a~,), rS(a~5), rS(a~) and rS(a~2). There is a non-homogeneous C, in r(G16), r(G18), r(a21). There is a non-homogeneous C8, respectively, C9 in r(Ga2), respectively, in
r(a19).
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OBSERVATION 2.3. For any k >_ 2, r2(A2k_l) = A2k-3, F 2 ( B 2 k ) = B2k-2 I.J 2K1 and inductively, F2~-l(A2~_1) = F2k(B2~) = C4. Dk-1 is a proper induced subgraph of F(Dk), and inductively, G25 = F(D1) is an induced subgraph of Fk( Dk ). Thus, Fk+8(Dk) contains a non-homogeneous C6. [2(Es) = GsotJK1, so FI~ is a C4. For k >_ 2, E~k-1 is a proper induced subgraph of F2( E2k+l ). Thus the graph E3 is a proper induced subgraph of F2~(E2k+I), and so F2k+l~ contains a non-homogeneous Ca. F(F4) contains an induced Es and so Fll(F4) contains a non-homogeneous C4. For k >__2, F2k is a proper induced subgraph ofr2(F2k+2), hence F2k+ll(F2k+2 ) contains a non-homogeneous C4. Thus by the observations above, a graph G satisfying the Chordality Condition (I) has the following property. CHORDALITY CONDITION (II). G is chordal and does not contain G1, 99 G34, G1, G2, G3, G4 * K1, Gs * K1, and Azk-1, B2k. Dk, E2k+l, F2k+2, k > 1, as induced subgraphs (see Figures $, 3 and $). Notice that B2 is the complement G2 of the graph G2. A consequence of the main results is that the Chordality Conditions (I) and (II) are, in feint, equivalent; see Corollary 3.2.
w 3. C h a r a c t e r i z a t i o n s o f G a l l a i - m o r t a l g r a p h s If a graph G satisfies the Induced Path Condition or the Neighborhood Chain Condition, then we shall denote by p(G), v(G) the number of vertices of an induced path of maximum length in G, the maximum length of neighborhood chains in G, respectively. Graphs that do not contain an induced subgraph isomorphic to a given graph H are called H-free. THEOREM 3.1. For any graph G, the following statements are equivalent. (i) G is Gallai-mortal. (ii) G satisfies the Chordality Condition (I), the Induced Path Condition, and the Neighborhood Chain Condition. (iiii) G satisfies the Chordality Condition (II), the Induced Path Condition, and the Neighborhood Chain Condition. (iv) G satisfies the Induced Path Condition, the Neighborhood Chain Condition, and F'(G) = 0 for some t < p(G) + 2v(G) + 6. Moreover, if G is 194* Kl-free, then t _< p ( a ) + 2 v ( a ) - 1.
In the finite case, the theorem takes a simpler form. THEOREM 3.1'. For any finite graph G, the following statements are equiva. lent.
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G:
G2
G3
G4
05
G6
~
G:
1 and finite induced subgraph F in F*(G), there is some finite induced subgraph F ' in F*-I(G) such that F is an induced subgraph in r(F') (for example, F' is induced by the end vertices of edges in r'-l(G) which appear as vertices of F.) Now, if G does not satisfy the Chordality Condition (I), there is some t > 1 and some induced cycle C of length at last 4 in F*(G). Then, as noted above, there exists some finite induced subgraph H of G such that F*(H) contains C as an induced cycle. But H, as induced subgraph of G, satisfies the Chordality Condition (II) and, as a finite such graph, H satisfies the ChordMity Condition (I) by Theorem 3.1'. Thus we get a contradiction, hence G must satisfy the Chordality Condition (I). 9 The proof of Theorem 3.1 relies on a characterization of strongly Gallai-mortal graphs. Here, a graph G is called strongly Gallai-mortal if both G and G are Gallaimortal. The structure of Gallai-mortal graphs depends on that of strongly Gallaimortal graphs, as the following observation shows. Let H be the subgraph of a graph G induced by the neighborhood of a vertex of G. Since H 9 K1 is an induced subgraph of G, ~r is an induced subgraph of r(G) (see Proposition 1.1). Hence for any Gallai-mortal graph, the "neighborhood graphs" are strongly Gallai-mortal. For this reason we shall first characterize strongly Gallai-mortal graphs in Theorem 3.3 below, and then obtain a proof for Theorem 3.1. Before giving Theorem 3.3 we recall that a graph G satisfies the Neighborhood Chain Condition if and only if G does, as noted in the introduction; the same argument yields a little more: For graphs G satisfying the Neighborhood Chain Condition, v( G ) - 1 < v( G) t. If t is even, then by Claim 1.2, St is edgeless with at least two vertices, or St contains an induced P4. In any case, S / h a s two non-adjacent vertices w,, w~. Then vl, v3, v s , . . . , v , - i together with v2, v4,. 9 v,-2, w tt induce a neighborhood chain of length (t + 1)/2 of G. Again, 2v(G) - 1 >_ t. Let now G be disconnected. I f t = 1 then G is edgeless hence t = 1 < 2v(G) = 2. If t > 2 then, with H2 as in the construction, v(G) = v(H2). Since H2 is connected and has the decomposition {s/li = 2 , . . . , t}, as in the case above, we have t-l 2. (2) W e now show by induction on t that (2.1) if G is connected then the non-trivial components of rt-~(G) are that ofrk(s/) or ofrk(s-,-), fork = 0, I .... ,t- i.
and (2.2) If G is disconnected then the non-tirvial components o f r ' - 2 ( G ) are that ofrk(st)
or
fork = 0,1 .... ,t-
2.
Let namely Hi as in the construction (1). If G is connected, then the nontrivial components of r ( G ) are that of r(H2) and of I&l~. Thus i f t = 2 then we are done; if t > 3 then r(H2) = r ( H s ) and Ha is connected. It follows inductively from (1.4) and (1.5) (applied to Hz and H'~2) that the non-trivial components of and of are that of or for k = 0, 1,..., t - 2. Thus the non-trivial components of
r'-1(a) = rt-l(n2) u ISxlr'-2( ) are that of rk(s/) or rk(~t) for k = 0,1,...,t - i. That is (2.1).
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IF G is disconnected, t h e n / / 2 = G\S1 is connected and r(G) = r(H2); (2.1), the non-trivial components of r t - 2 ( G ) = r t - 2 ( H 2 ) are that of r k ( s t ) rk(s'~t) for k = 0, 1 , . . . , t - 2. That is (2.2). 9 (3) Now, by Claim 1.2, r 2 ( s t ) = r 2 ( ~ ) = 0 if G (hence St) is P4-free. other cases, rg(s,) = rg(s~,) - r (see also the proof of Lemma 4.2). Thus, by we have for connected G rt-l+9(G) = r
117
by or In (2)
and r'-l+2(G) = $ if G is P4-free,
and for disconnected G, rt-2+S(G) = r
and rt-2+2(G) = r ifa is Pa-free.
This and Claim 1.3 yield
r2~(a)+~(G) = O, and r2~(a)(G) = O if G is P4-free. Now Theorem 3.3 follows by symmetry and the fact v(G) l, or i = 2 . An even quasi.millepede (odd quasi-millepede} M* is a graph isomorphic to the Gallai graph of an even millepede (odd millepede) M. More precisely, a graph M* consisting of an induced path P = vlvz . . . vt of maximum length such that * d(Vl) = 1
is an even quasi-millepede of length l, if it has furthermore the following property 9 Any vertex outside P is adjacent tO exactly two vertices vi,vi+l for some even i, and (N(vi) N N ( v I + I ) ) \ V ( P ) induces a complete graph. It is an odd quasi-millepede of length l, if 9 Any vertez outside P is adjacent to exactly two vertices vi,vi+l for some odd i, and (N(vi) N N ( v i + I ) ) \ V ( P ) induces a complete graph. In any case, the vertex vl of M or M* is called the head of M or M*, respectively. Notice that the Gallai graph of a quasi-millepede has at most one non-trivial component which is a millepede. The proof of the following technical lemma is given in Section 7.
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LEMMA 5.1. Let G satisfy the Induced Path Condition and the Chordality Condition (II). If G is connected and p = p(G) > 5 then G is one of the graphs Y, Z, g(a, 8) listed in Figure 6, or G is an even (quasi-) millepede of length p, of G is a substitution of some graph satisfying the Chordality Condition (IlI) for the head of some odd (quasi-) millepede of length p.
u
/A
w
v
Z
w
w
Z(a, ~)
Figure 6 PROOF of Theorem 3.1. The implications "(i) ~ (ii) =# (iii)" are showed in Sections 1 and 2; The implication "(iv) => (i)" is trivial. Let G r 0 satisfy the Property (iii) of Theorem 3.1 we shall prove (iv). We first assume that G is connected and then consider two cases. (4) G has a central vertex. Then G satisfies the Property (iii) of Theorem 3.3, hence F2v(G)+7(G) = and r2~(G)(G) = $ whenever G is P4 * Kl-free. Thus Theorem 3.1 is proved. (5) G has no central vertex. By Lemma 4.1, p = p(G) > 4 and G is one of the graphs described in Lemma 4.2 (ii), or (iii) and in Lemma 5.1. If G is one of the graphs T , X , X , I ~ _ , I ~ + , J ~ , J ~ + (see Figure 5), or Y,Z, Z ( a , ~ ) (see Figure 6) then Fg(G) = 0, and we are done; see also the proofs of Lemmas 4.2 and 5.1. If G is an even (quasi-) millepede of length p then one shows easily by induction that FP+2(G) is empty. Thus we may assume that G is substitution of some graph H satisfying the Chordality Condition (III) for the head of some odd (quasi-) millepede of length p. In this case, the non-trivial components of FP-I(G) are that of rk(H) and of
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rk(_0) for k = 0 , 1 , . . . , p Theorem 3.3 yields
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1. Since v(H) = v ( G ) - 1 and v(~r) < v ( H ) + 1 = v(G),
rp-l+[2~+~l(G) = r p + ~ + 6 ( G ) = r and
r~-l+2~(G) = rp+2~-~(G) = r if H is P4-free, that is, if G is P4 * Kl-free. Thus Theorem 3.1 is proved for connected graphs G. If G is disconnected and Q is a connected component of G then p(Q) < p(G), v(Q) < v(G); Theorem 3.1 follows now by applying the result above for each component Q of G. 9
6. P r o o f o f L e m m a 4.2
An edge e of a connected graph G is a bridge if G\{e} is disconnected. Let P = VlV2... v,~ be an induced path of a graph G. A vertex v of G outside P is called a x-verte~ of Pi if v has x neighbors in P. If G is chordal, Np(v) must be a subpath of P. Then we shall can a x-vertex v of type (i), if Np(v) is the usbpath of P with endvertices vi and vi+~-l, that is, Np(v) = { v i , . . . , ~)i+~-1}. If furthermore G does not contain an induced A1 then P has no x-vertex for k > 5. Note that if P is maximal, P has also no 1-vertex of types (1) and (m). The following observation is used in the proof below. Let G satisfy the Chordality Condition (II). If H is a subgraph of G induced by a neighborhood of a vertex of G, then H satisfies the Chordality Condition (III). Let G be a graph satisfying the properties of Lemma 4.2. Assume that G has no central vertex. We have to show that the Statement (ii) or (iii) of Lemma 4.2 holds. We consider several cases. (6) There is an induced path P = VlV~V3V4 such that the middle edge v2v3 is a bridge of G. Let H be the subgraph of G induced by N(v2)\{v3}; K the subgraph induced by N(va)\{v2}. Then one of H and K must be complete (else G contains an induced B2), say K. Since G has no induced Ps, G is the substitution of H for the vertex Vl, and then of K for the vertex v4 of P. Since H is defined by a neighborhood of a vertex of G, H satisfies the Chordality Condition (III). Thus in this event, the Statement (ii) of the lemma occurs. Thus we may assume that G has no bridge which is a middle edge of'an induced P4. (7) There is a path P = V l V 2 V 3 ? ) 4 with two 3-vertices u, v. Then u and v are of distinct types (else G contains an induced G4; recall G is chordal), therefore u and v are the only 3-vertices. Furthermore, u and v are non-adjacent (else G contains an induced G2). Now, P has no 4-vertex (else G contains an induced A1), no 2-vertex (else G contains an induced Ps, or A1, or G2, or G12), no 1-vertex (else G contains an induced Ps, or G3, or A1), and no 0-vertex (else G contains an induced Ps; recall that G is connected). Thus G is the graph T in Figure 5. We have FS(G) = 0, v(G) = 2.
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Thus we may assume that any induced path P4 has at most one 3-vertex. (8) There is a path P = V l V 2 V 3 V 4 with a 4-vertez. Let M be the set of 4-vertices of P . Since G is chordal, M induc4s a complete subgraph in G. Moreover, for two vertices m l and m2 of M , we have
Na(ml) U {ml} _CNa(r 2) U
or Na(m2) U {m2} C_Na(ml) U {rod
(else G contains an induced G1, Gs, G4, or A1). Let m E M with m a x i m a l neighborhood. Since G has no central vertex, there is some vertex v such that v is non-adjacent to m, inparticular m ~ M U V ( P ) , and v is non-adjacent to all vertices in M because N ( m ) is maximal. Therefore that v is not 3-vertex, not 4-vertex of P. The vertex v is also not 2-vertex (else there is an induced p a t h P4 with two 3-vertices, or G contains an induced G-~-I). Now we shall show that v is a 1-vertex of P. Suppose the contrary t h a t v is a 0-vertex. T h e n further, we m a y assume t h a t there is some c o m m o n neighbor w of v a n d m (recall G is connected). Since v is a 0-vertex of P, w is not a vertex of P , and since v is non-adjacent to all vertices of M, w does not belong to M . Now, w is not a 3-vertex (else G contains an induced Gs), not a 2-vertex (else G contains an induced G1, or Ps), and not a 1-vertex (else G contains an induced Ps). The vertex w is also a 0-vertex of P , therefore Pl m w vl is an induced p a t h of G. Since the middle edge m w of t h a t p a t h is not a bridge of G, there is some vertex t adjacent to m and w; t does not belong to P because w is a 0-vertex of P (but t is possibly adjacent to v). Now, t is neither a 0-vertex nor a 1-vertex of P (else G contains an induced Ps, or A1). t is also neither a 3-vertex nor a 4-vertex of P (else G contains an induced Gs, or A1). Therefore that t is a 2-vertex of P . But then G contains an induced A1, or Ps, or Gll Thus v is a 1-vertex of P , as claimed. Since G is Ps-free, v is a 1-vertex of type (2) or (3). This yields M = {m}, otherwise G contains an induced G4 (v is non-adjacent to all vertices in M ) . Moreover, v is the only 1-vertex of P (else G contains an induced B2, or Gs, or A1, or G12, or there is an induced p a t h P4 with two 3-vertices). Finally, P has no 2-vertex (else G contains an induced G--~',or Gs, or G3, or A~), no 3-vertex (else G contains an induced G3, or G1, or Gs, or G3), no 0-vertex (else G contains an induced Gs, or Ps). Thus G is the graph X in Figure 5, and we have r s ( G ) = 0, v(G) = 3. Thus we may assume that no induced path P4 has a ~-vertex. (9) There is an induced G4 in G. Let P = vlv2vsv4 be an induced p a t h in a G4 in G. Then P has no 3-vertex (else G contains an induced G10, or Gs, or G3, or Gs), no 2-vertex of type (2) (else G contains an induced G9, or G10, or ~ ) . Thus the middle edge v2 v3 of P is a bridge of G. Thus we may assume that G does not contai~i an induced G4. (10) There is a path P = vlv~vzv4 with (exactly) one 3-vertex in G. We m a y assume that the 3-vertex v is of type (1). Case 10.1. P has a 1-vertex. Let w be a 1-vertex then w is of type (2) (else G contains an induced Ps, or Gs, or there is an induced P4 with a 4-vertex), and w is the only 1-vertex of P (else
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G contains an induced G-'~, or G'-~, or A1; therefore P has no 2-vertex (else there is an induced P4 with a 4-vertex, or G contains an induced G l l , or Ps, or GT, or G1, or At), and no 0-vertex (else G contains an induced G1, or Ps). Thus G is the graph ~: in Figure 5, and so rg(G) = 0, v(G) = 3. Case 10.2. P has a 2-vertex. By 10.1 above, P has no 1-vertex. P has also no 2-vertex of type (3) (else G contains an induced A1, or there is an induced P4 with a 4-vertex). Any 2-vertex of type (1) is adjacent to v (else G contains an induced Ps), and any 2-vertex of type (2) is also adjacent to v (else G contains an induced P4 with a 4-vertex). Let now M, N be the sets of 2-vertices of type (1), respectively, of type (2) of P . Then each of M and N induces a complete subgraph of G (else G contains an induced G'-~, respectively, Gs). Furthermore, M or N must be empty (else G contains an induced A1, or Ps), and P has no 0-vertex (else G contains an induced GI1, or Ps, if M ~ 0; or G contains an induced G1, or Gr, or there is an induced path P4 with a 4-vertex, if N r 0). Thus G is the graph I a _ , or Ia+ in Figure 5 (where a denotes the cardinality of the non-empty set M or N), and so r e ( G ) = $, v(G) = 2 (if M r $), or rS(G) = 0, v(G) = 2 (if g r 0). Case 10.3. P has no 1-vertex, no 2-vertex. If G = I0- = I0+ then r6(G) = 0 v(G) = 2. Thus we may assume that P has a 0-vertex u adjacent to v (recall G is connected). Now, v2 is a 3-vertex and u is a 1-vertex of the induced path vl v v3 v4. As in Case 10.1 above, r9(G) = 0. (11) It remains the case that, no induced P4 of G has a 4-vertez, or a 3-vertez. Let now P = VlV2Vzv4 be an induced path in G. Since vzv3 is not a bridge of G, the set N of 2-vertices of type (2) of P is non-empty, and so P has no 2-vertex of other types (else G contains an induced A1, or Ps). The vertices of N induce a complete subgraph (else G contains an induced Gs). Since G is Ps-free, P has no 1-vertex of type (1) or (4). All other 1-vertices of P are of type (2) or of type (3) (else G contains an induced B2) and none of the 1-vertices is adjacent to a vertex of N (else there is an induced P4 with a 3-vertex), and so P has at most one 1-vertex (else G contains an induced G-'~, or A1, or B~). P has no 0-vertex (else G contains an induced G1, or Ps)- Thus G is the graph Ja, or J a + l , and so rs(G) = 0, v(G) = 2,
or rg(G) = 0, v ( V ) = 2. By (6) - (11), the proof of lemma 4.2 is complete.
9
w 7. P r o o f o f L e m m a 5.1 Let G be a graph satisfying the properties in Lemma 5.1. We use the same notations as in Section 6, and consider several cases again. Set p = p ( G ) > 5. (12) There is an induced path P = v l v 2 . . . v p with a f-vertex. Then p = 5 (else G contains an induced G20, or G21, and the 4-vertex v (say) is the only 4-vertex of P (else G contains an induced G4, or GI9). P has no 3vertex (else G contains an induced G17, or Gas, or A1, or G22), no 2-vertex (else G contains an indeced G22, or A1, or G1, or G15, or G16), no 1-vertex (else G contains
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an induced G22, o r G14, o r G1, or G3, or A1), 0-vertex (else G contains an induced G13). Thus G is the graph Y in Figure 6. We have rg(G) = 0 and v(G) = 2. (13) There is an induced path P = v l v 2 . . . v p with a 3-vertex. Then p = 5 and a 3-vertex v is of type (1) or (3) (else G contains an induced G2r, or G22. v is the only 3-vertex of P (else G contains an induced Gd, or At, or Gls). By (12) P has no 4-vertex. P has also no 1-vertex (else G contains an induced G~5, or Gz, or Gs, or G24, or G23). Without loss of generality, say that the 3-vertex v is of type (1). Then P has no 2-vertex of type (1) (else G contains an induced G20, or G26, no 2-vertex of type (3) (else G contains an induced A1, or G2~). Let M, N be the sets of 2-vertices of type (2), respectively, of type (4) of P. Then M induces a complete subgraph of G (else G contains an induced Gs), and v is adjacent to all vertices in M (else G contains an induced G24). Vertices of N are adjacent to none vertices of M U {v} (recall G is chordal), and N induces also a complete subgraph of G (else G contains an induced G23). Finally, P has no 0-vertex (else G contains an induced G25, or G1, or G27). Thus G is the graph Z(a, 8) in Figure 6 (where a and ~ denote the cardinalities of M, Y respectively), and so rs(G) is empty and v(G) = 2. By (12) and (13) we may assume that no induced P (on p ~_ 5 vertices) has a 4-vertex, or a 3-vertex. Then P has also no 0-vertex (else G contains an induced G1, or G29). Let P = v l v 2 . . , vp be an induced path of G, and let Li be the set of 2-vertices of type (i), 1 < i < p - 1, and let Mj be the set of 1-vertices of type (j), 2 < j < p - 1, of P. Since P has no 0-vertex, no 3-vertex, no 4-vertex, G consists of P, (J Li and (J Mj only. Further, for each 2 < i < p - 2, Li induces a complete subgraph of G (else G contains an induced Gs), and for 3 _< j _< p - 2, Mj is an independent set of G, that is, a set of pairwise non-adjecent vertices (else G contains an induced G2s). Further the subgraphs H and H ' of G induced by L 1U M2 U { v 1}, Lp_ 1U Mp_ t tJ { vp }, respectively, satisfy the Chordality Condition (III). We shall often identify a set of vertices with the subgraph induced by that set. (14) There is an induced path P = v l v 2 . . . v p such that the corresponding set Mj is non-empty for some 3
