Retract-collapsible graphs and invariant subgraph properties

Retract-Collapsi ble Graphs and Invariant Subgraph Properties

Norbert Polat /A E

UN/V€RS/T€ JEAN MOULlN (LYON 111) LYON, FRANCE

ABSTRACT

A (finite or infinite) graph G is retract-collapsible if it can be dismantled by deleting systematically at each step every vertex that is strictly dominated, in such a way that the remaining subgraph is a retract of G, and so as to get a simplex at the end. A graph is subretract- collapsible if some graph obtained by planting some rayless tree a t each of its vertices is retract-collapsible. It is shown that the subretract-collapsible graphs are cop-win; and that a ball-Helly graph is subretract-collapsible if and only if it has no isometric infinite paths (thus in particular if it has no infinite paths, or if it is bounded). Several fixed subgraph properties are proved. In particular, if G is a subretract- collapsible graph, and f a contraction from G into G, then (i) if G has no infinite simplices, then f ( S ) = Sfor some simplex Sof G; and (ii) if the dismantling of G can be achieved in a finite number of steps and if some family of simplices of G has a compacity property, then there is a simplex S of G such that f ( S ) C S. This last result generalizes a property of bounded ball-Helly graphs. 0 1995 John Wiley & Sons, Inc.

INTRODUCTION

In this paper we continue our investigations of the invariant subgraph properties for finite or infinite graphs. For this purpose we introduced in [3] the concept of retract-collapsible graphs. A graph G is retract-collupsihfe if it can be reduced to a simplex (i.e., a complete subgraph), called the base of G, by deleting systematically every vertex x whose neighborhood is strictly contained in that of an adjacent vertex of x, and in such a way that at each step the subgraph of G that is obtained is a retract of those obtained at the preceding steps. We showed in particular that a rayless (i.e., without infinite paths) connected graph is retract-collapsible if and only if

Journal of Graph Theory, Vol. 19, No. 1, 25-44 (1 995) 0 1995 John Wiley & Sons, Inc. CCC 0364-9024/95/010025-20

26 JOURNAL OF GRAPH THEORY

it is dismantlable, and we studied those retract-collapsible graphs for which any contraction (a function mapping an edge onto an edge or a vertex) stabilizes a finite simplex, and having a nonempty finite simplex invariant under any automorphism.

Different examples of graphs that are not retract-collapsible but close enough to have similar properties, led us in this paper to enlarge this notion by considering subretract-collapsible graphs, that is, graphs that admit some extensions that are retract-collapsible. We show that these graphs are closely related to contractible graphs; that they are cop-win-these are the graphs characterized by Nowakovski and Winkler [2] where, in some pursuit game, a cop can always catch a robber (see Section 6)-and that a ball-Helly graph (i.e., a graph such that any family of pairwise nondisjoint balls has a nonempty intersection) is subretract-collapsible if and only if it contains no isometric (i.e., distance preserving) rays. Thus, in particular bounded or rayless ball-Helly graphs are subretract-collapsible.

In Section 4 we improve substantially the results of [3] on invariant subgraph properties. We get as particular cases of a much more general result (Theorem 4.6) that, if G is a subretract-collapsible graph, and f a contraction from G into G, then (i) there is a vertex x such that f ( x ) is equal or adjacent with x; (ii) if G has no infinite simplices, then f ( S ) = S for some simplex S of G; and (iii) if the dismantling of G can be achieved in a finite number of steps and if some family of simplices of G has a compacity property, then there is a simplex S of G such that f ( S ) S. This last result generalizes an analogous result for bounded ball-Helly graphs by Jawhari et al. [ l ] and Quillot [ 5 ] .

1. NOTATION

The graphs we consider are undirected, without loops and multiple edges. If x and y are two vertices of a graph G, we denote x 5 G y if x = y or { x , y } E E(G). If x E V ( G ) , the set V ( x ; G ) := {y E V(C) : {x,y} E E(G)} is the neighborhood of x. For A V ( G ) we denote by G I A the subgraph of G induced by A, and we set G - A := G I ( V ( G ) - A ) . A path W := (XO ,..., x , ) is a graph with V ( W ) = {XO ,.... x,}, xi # x j if i f j , and E ( W ) = { { x i , x i + , } : 0 5 i C n } . A ray or one-way inJinite path R := (xo,x , , . . .) is defined similarly. A complete graph will be simply called a simplex. The usual distance in G between two vertices x and y, that is, the length of a shortest xy-path in G, will be denoted by disk ( x , y). The diameter of G is diam(G) := sup(disk(x,y): x , y E V ( G ) } . A graph is bounded (respectively r-bounded) if its diameter is finite (respectively 5 r ) ; in particular, a I-bounded graph is a simplex. A subgraph H of G is isometric if distH(x, y) = disk(x, y) for all vertices x and y of H. If x is a vertex of G and r a nonnegative integer, the set &(x, r ) := { y E V ( G ) : distG(x,y) 5 r } is the bull of center x and radius r in G . We will write x 5 G y (respectively

RETRACT-COLLAPSIBLE GRAPHS 27

x < G y ) if BG(x, 1) & ( y , 1) (respectively BG(x , 1) C & ( y , l)), and we will say that x is dominated (respectively strictly dominated) by y in G .

If G and H are two graphs, a map f : V(G) - V(H) is a contraction if f preserves the relation =, i.e., if x = G y then f ( x ) = " f ( y ) . Notice that a contraction f : G - H is a nonexpansive map between the metric spaces (V(G),distc) and (V(H),distH), i.e., distH(f(x),f(y)) 5 disk(x, y ) for all x , y E V(G). A contraction f from G onto an induced subgraph H of G is a retraction, and H is a retract of G, if its restriction f I H to H is the identity. A contraction f of C (Le., from G into G) stabilizes (respectively strictly stabilizes) a subgraph H of G , or H is invariant (respectively strictly invariant) under f, if f (H) C H (respectively f (H) = H ) .

2. COLLAPSIBLE GRAPHS AND RETRACTABLE GRAPHS

2.1. nal a , we define - G'O' := G.

- G(*) := np<, G'p) if a is a limit ordinal.

The ordinal d ( G ) := min{a: G ( * ) = G ( a + i ) } will be called the depth of G, and the subgraph G'"' := G'd'G)) the base of G. Finally, for x E V(G) the depth of x will be d(x ) := max{a: a I d ( G ) and x E V(G'"))}.

Let D ( G ) := { x E V(G): x < G y for some y E V(G)}. For an ordi- inductively as follows:

- ~ ( a + l ) .= G(a) - ~ ( ~ c a ) ) .

2.2. Definition. is a simplex.

A graph G is said to be collapsible if G'") is empty or

2.3. Definition. A graph G is said to be retractable if, for any ordinals a and p with a 5 p 5 d(G) , there exists a retraction fup: G ( @ ) - G ' p ) satisfying the following properties:

(i) f u p = frP o far for any ordinal y with a 5 y I p; (ii) x < G , e , f a a + l ( x ) for any x E D(G'"');

(iii) if p is a limit ordinal > a , then, for any x E V ( G ( " ) ) , there is an ordinal S with a 5 S < p , such that S I y 5 p implies f a & ) = fa+>.

In particular any graph of depth 0 is retractable.

2.4. Proposition. A graph G is retractable if and only if, for every vertex x of G, there is a path (xo,. . . , x , ) with xo = x, X n E V(G'"'), d(x , ) < d(x ;+ l ) and x i < Gld(c,) lx;+l for 0 5 i < n.

ProoJ The necessity is obvious. Let us prove the sufficiency

28 JOURNALOFGRAPHTHEORY

(a) Let a be a well-ordering on V ( G ) . Define a path W ( x ) for every vertex x as follows. If x E V(CCx) ) let W ( x ) := ( x ) . Let x be the least vertex, with respect to a, such that W ( x ) is not yet defined; and let (xo.. . . , x , ) with xo = x and x, E V(G(")) be the path given by the hypothesis of the theorem. Let i be the least integer I n such that W ( x , ) is already defined. Then, for every j < i , put W ( x , ) := (x,, . . . , x I ) U W ( x , ) .

(b) Let a 9 p 9 d ( G ) . Define f a p as follows. Let x E V(G(a' ) , and W ( x ) = (xo,. . . , x n ) with xo = x . Then put f a p ( x ) := x , where i is the smallest integer such that d ( x , ) 2 p. We have only to show that f a p is a contraction to see that it is a retraction, and that the family ( f ap )a5g satisfies the three conditions of Definition 2.3.

Let x , y E V ( G ( a ) ) be such that x = G y . And let (xo, . . . , x , ) and ( y o , . . . , y p ) be the subpaths of W ( x ) and of W ( y ) , respectively, such that xo = x, yo = y , x, = f,p(x), and y p = f , p ( y ) . We first prove by induction that x, = G y I for any i 9 min{n,p}. By hypothesis, xo = G y ~ . Let i < min{n,p}. Suppose that x, = G y r and d ( x , ) 5 d ( y , ) . Then x, < Gldlr,IIxI+I implies X , + I = G y I . Thus y , < Gldl \ , I ly l+ l implies X , + I

- G Y l t l . We are done if n = p . Assume that n < p . Then x, = G y n and d ( y , ) <

p 5 d(x,) . Thus y,, < G i d i \ n l l y f l + l implies x, = G ~ , + ~ . Therefore by induction x,, = I

- -

. .

On account of the property [3, Lemma 3.51 stating that if G contains no injnite simplices, then for any x E D ( G ) there is y 4 D ( G ) such that x < G y . Proposition 2.4 implies immediately the following result:

2.5. Proposition [3, Lemmas 3.6 and 3.71. plices and has a finite depth, or if G is rayless, then G is retractable.

2.6. Proposition. if so has G'"'.

If G contains no infinite sim-

A retractable graph G has an isometric ray if and only

Proof. Only the necessity has to be proved. For that we will show by induction that, for any ordinal a , G'"' has an isometric ray if so has G. Assume that G = G'O' has an isometric ray, and let a 2 0. Suppose that G'p' has an isometric ray for any p < a.

Case 1.

Let R = ( x o , x ~ , . . .) be an isometric ray of G'p). We have fp , , (x i ) # f p u ( x i + k ) for any i 2 1 and k 2 1, where ( f a p ) a 5 p 5 d ( ~ ) is the family of retractions as defined in 2.3. Otherwise, fp"(xi-1) = Gfpa(x i ) and

= G f p n ( x i + k + l ) would imply that distc(xi-l,xi+k+l) 5 2 contrary to distG(xi-l,xi+k+l) = distR(xi-l,xi+k+l) = k + 2 2 3 since R is isometric and k 2 1.

a = p + 1.


Then R' = ( fpa(x1),fpa(x2), . . .) is a ray of G'"). Suppose that it is not isometric. Then there is i 1 1 such that p := distGcU)(fSp(xi),fpu(xi+~)) < k. Thus k + 2 = distc(xi-l,xi+k+l) I p + 2, contrary to p < k. There- fore R' is isometric.

Case 2. ct is a limit ordinal.

Let R = (~0~x1,. . .) be an isometric ray of G. Let W = (yo,. . . ,y,) be

0 5 i < n. Then foa(xo) # foa(x,+l). Otherwise, suppose that foa(xo) = foa(xn+l), and let W' = (z o,..., 2,) be the path of G such that zo = x,+1, z p = f o a ( x n + , ) = foa(xO), and zi < G ( d ( r , ) ) Z ; + l for 0 5 i < p . Then

path of G of length n + p, thus of length less than distR(xo,x,+p+l), contrary to the fact that R is isometric.

Therefore there exists a sequence (i,),?~ of nonnegative integers such that R' = ( fOn (xi,,), foa ( x i l ) , . . .) is a ray of G("). Let xi, and xi,+, be two vertices of R. By condition (iii) of Definition 2.3, ( f o a ( x i n ) , f ~ a ( x ; n + , ) , . . . ,foa (xi ,+,))

is a path of G(p) for some /3 < a. Thus, by the induction hypothesis, diStci~i(fo(l(Xi,),foon(X;.+,)) = diStC(P)(fOa(Xi.).foa(x;.,,)) = k. Hence R' is isometric. I

the path Of G Such that yo = X o , Yn = foa(xo), and y; < Gcdcy,))Yj+l for

w u (fo".(X,+l)rfOd(z,-I)(X,+2), f0d(z,-2)(xn+3)r * * * J , + p + l ) is a xox,+p+1-

2.7. Corollary. A retract-collapsible (i.e.. retractable and collapsible) graph has no isometric ray.

Proof. If G is retract-collapsible, then its base is a simplex, and thus contains no isometric ray. Hence G has no isometric ray by the preceding proposition. I

2.8. Proposition. only if so is ~ ( " 1 .

A retractable graph G of finite depth is bounded if and

We get immediately

2.9. Corollary. A retract-collapsible graph of finite depth is bounded.

The converses of Corollaries 2.7 and 2.9 are false-one can easily find counterexamples; but we will see that they hold for ball-Helly graphs.


3. SUBRETRACT-COLLAPSIBLE GRAPHS

3.1. Let G be a graph, and ( T x ) x E ~ ( ~ ) a family of pairwise disjoint rayless trees such that T, fl G = (x) = T y ) for every x E V(C). Note that any rayless tree is retract-collapsible by Proposition 2.5. Then H := G U UrEV(G) T, will be called a tree-extension o fC . Observe that H'") C G since the base of T, is (x) for every x E V(G).

3.2. Definition. A graph G will be said subretractable (respectively subretract-collapsible) if there is a tree-extension of G, which is retractable (respectively retract-collapsible). The base of a retractable tree-extension of G will be called a subbase of G. The subdepth of G will be the ordinal

sd (G) := min{d(H): H is a retractable tree-extension of G}

Clearly sd(G) = d(G) if C is retractable. We will characterize the subretract-collapsible graphs by using the con-

cepts of contractibility and of dismantlability that we will recall, and the following lemma.

3.3. Lemma. whose depth is a and base is (x,).

For any ordinal a there is a rayless rooted tree (T,,x,)

Proof. By induction on a . For a = 0 let TO := (xo). Let a 2 0. Suppose that the rooted tree (Tp ,xp ) has already been defined for any p < a . Let ( T ; , X ~ ) ~ < ~ be a family of pairwise disjoint rooted trees such that, for p < a and i = 0,1, (T$,x&) is a copy of ( T p , x p ) . Now let (T , ,x , ) be the rooted tree obtained by forming the union of this family, and joining each x i to a new vertex x,. This tree has clearly the desired properties. I

3.4. V(P,) = a + 1 and edge set is E ( P , ) = { {p , p + I}: p < a}.

For an ordinal a we denote by P , the graph whose vertex set is

If G is a graph, a contraction F : G X P , - G will be said to be continuous if F is a continuous function from the product space V(G X Pa) into V ( G ) when the set V(G) is endowed with the discrete topology, and a + 1 with the usual order topology for which { ( y , p ] : y < p I a } U {[O,p]: /3 5 a } is a base. That means that F is continuous if and only if, for any x E V ( G ) and any limit ordinal /3 5 a , there is y ( x ) < p such that y ( x ) 5 y 5 p implies F ( x , y ) = F(x , p).

3.5. Definition. A graph G is said to be contractible if there are an ordinal CT, a vertex a, and a continuous contraction F : G X P , - G such that F ( x , O ) = x and F ( x , a ) = n for every x E V ( G ) .


3.6. Theorem [3, Theorem 4.41. ble if and only if it is contractible.

A retractable connected graph is collapsi-

3.7. Theorem. A graph G is subretract-collapsible if and only if there are an ordinal u, a vertex a, and a continuous contraction F : G X P , - G such that, for any a I u and x E V ( G ) :

( i ) F ( x , O ) = x and F ( x , a) = a; (ii) F ( x , a ) = F ( F ( x , a) , p ) = F ( F ( x , p), a) for every p 5 a.

In particular, a subretract-collapsible graph is contractible, but we do not know if the converse holds.

Proof. (a) Assume that G is subretract-collapsible, and let H := G U UxEV(G) T, be a retract-collapsible tree-extension of G. Denote by ( f a p ) a s p s d ( ~ ) the family of retractions for H as defined in 2.3, and let a be a vertex of H'"), and u := d ( H ) + 1. Define F : G X P , - G as follows: for any a 5 u and x E V ( G ) , let F ( x , a ) = foa(x) or a according as a is or is not less than u. It is straightforward to check that F has the desired properties.

(b) Conversely assume that there is a continuous contraction F : G X P , - G and a vertex a satisfying properties (i) and (ii) of the statement. For every vertex x , let a ( x ) := sup{a 5 u: F ( x , a ) = x}. Since F is continuous and by condition (ii), F ( x , p ) = x for all p I a ( x ) . By Lemma 3.3 there is a rayless rooted tree (T , ,x ) such that T, f l G = ( x ) = T F ) and d(T,) = a(x ) . It is then straightforward to check that H := G U UrEV(G) T, is retract-collapsible with H'"' = (a) , and ( fap)a5p5, such that foa(x) = F ( x , a ) for any x E V ( G ) and a 5 u. I

3.8. Proposition. A subretractable graph is subretract-collapsible if and only if any of its subbases is a simplex.

Proof. The sufficiency is obvious. Let G be a subretractable graph, and let H be a retractable tree-extension of G. By Theorem 3.7, G is contractible, then so is clearly H. Hence H is retract-collapsible by Theorem 3.6, thus H(") is a simplex. I

3.9. Definition. A graph G is said to be dismantlable if there is a well- ordering I on V ( G ) such that, any vertex x that is not the greatest element of ( V ( G ) , 5) if such a greatest element exists, is dominated by some vertex y # x in the subgraph G, of G induced by the set { z E V ( G ) : x I z } .

3.10. Theorem [3, Theorem 4.41. tlable. The converse holds if G is rayless.

A retract-collapsible graph G is disman-


3.11. Theorem. A graph G is subretract-collapsible if and only if there is a well-ordering 5 on V ( G ) with a greatest element a such that, for every vertex x # a, there is a strictly increasing finite sequence x = xo < . + < x, = a where, for 0 5 i < n, the vertex xi is dominated by xi+l in the subgraph of G induced by the set {z E V ( G ) : x i 5 z}.

Proof. (a) Suppose that G is subretract-collapsible, and let H := G U UxEV(G) T, be a retract-collapsible tree-extension of G such that H(") = {a). For a 5 d ( H ) let V , := D(H(")) fl V ( G ) or (a ) according as a # d(H) or not, and let I , be a well ordering on V,. Then the order 5 on V ( G ) such that x 5 y if and only if x,y E V, for some a and x 5 ,y, or (x,y) E V , X V p with a < p, is a well ordering on V ( G ) . Let x E V ( G ) . Then x E V , for some a 5 d ( H ) . Hence x 5 Hl"ly for some vertex y E V ( H ( a ) ) n V ( G ) , thus x < ,y. Then x I G,y where G, is the subgraph of G induced by {z E V ( G ) : x I z}. The result is then an immediate consequence of Proposition 2.4 since H is retractable.

(b) Conversely assume now that there is a well-ordering on V ( G ) that satisfies the property of the statement. Then there is an enumeration ( x f ) f 5 , of the vertices of G with x, = a, and an extensive map g : a + 1 - a + 1 such that, for any [ < a: [ < g(6) , xf is dominated by xg(f) in the subgraph G , of G induced by {x,,: [ 5 7 I a}, and xgncf) = a for some integer n. For any ordinals 6 and y less than or equal to a, there is a nonnegative integer n such that g " ( 6 ) 2 y . Let n ( 6 , y ) := min{n: ~ " ( 6 ) 2 y} , and [ ( y ) := g " ' f . Y ) ( t ) . Then, for any 6 I a and y 5 a, define F ( x g , y ) := ~ g ( ~ ) . F is then a map from V ( G X P a ) onto V ( G ) such that F(xg,O) := xf and F ( x g , P ) := a

F is continuous by the last property of g, which is a consequence of the hypothesis. We will prove that F is also a contraction. Let (xf, y ) and (x,,, 6) be two adjacent vertices of G X P a . Suppose that y P 6, thus y = 6 or y = 6 + 1. If y = 6 then xg and x,, are adjacent. We prove inductively that x ~ ( ~ ) = G x , , ( Y ) . This is obvious if y = 0. Let y 2 0. Suppose that xf(&) = G x O ( E ) for any E < y .

If y = E + 1, then ((7) = g ' ( C ( E ) ) and q(y) = g i ( q ( E ) ) for some i - j E {071>. Hence xt(E) 5 GcI,,xf(y) and x , , ( E ) 5 G , , , e , x , , ( y ) 7 thus x e ( y )

- C X , , ( Y ) . If y is a limit ordinal, then [ ( E ) = ( ( y ) and q ( ~ ) = ~ ( y ) for some

E < y since F is continuous. By the induction hypothesis x f ( E ) = Gx, , (E) , hence x f ( y ) = Gx,, (y) .

Suppose now that y = 6 + 1. By the preceding case, xf(8) = ,+,,(8). We are done if [(a) 2 y . If ((6) < y , thus if [(a) = 6, then xf(&) = Gx,,(e) 5

Therefore F is a continuous contractions that satisfies clearly condition

- -

G,, , ,x6(Y) . hence x ? ( Y ) = Gxfb').

(ii) of Theorem 3.7, hence G is subretract-collapsible by this theorem. I


3.12. Remarks

3.12.1. The concepts of contractibility and of dismantlability as defined in 3.5 and 3.9, respectively, are natural extensions to infinite graphs of the usual finite respective concepts. Since these concepts are strengthened in Theorems 3.7 and 3.1 1, respectively, we could then say that the subretract- collapsible graphs are the “strongly contractible” graphs or the “strongly dismuntlable” graphs. By Proposition 2.5 and Theorems 3.6 and 3.10, for rayless graphs, thus in particular for finite graphs, these strong concepts coincide with the usual ones, as well as the concept of retract-collapsibility with that of subretract-collapsibility.

3.12.2. A ray is clearly the simplest example of a graph that is not subretractable. A graph may be not retractable nor collapsible but subretract- collapsible as is shown by the following example we already gave in [3, Remark 4.51. Let {u; : 0 I i 5 3}, {b;: 0 I i 5 3}, {x,: n E N}, and {y,,: n E N} be four pairwise disjoint sets. Denote by G the graph whose vertex set is the union of these sets, and whose edge set is E ( G ) =

{ ( a o , u l > , { a l , a 2 } , { a 2 , u 3 ) , 0 u {{a;,b;}: 0 5 i 5 3) u {{ai.x,}: 0 5 i 5 3 and n E N} U {{x,,~,,}: n < p} U {{y,,,~,,}: n < p}. This graph is not collapsible since G(’) = G‘”’ is the cycle (ao,al ,a2,a3,ao) of length four, and it is not retractable since no x, is strictly dominated by some vertex of G(’). On the other hand, the map F : G X P2 - G such that F ( z , O ) = z , F ( x , , 1) = x,, F(y,, 1 ) = x,+I, F(a;, 1) = a; = F(b;, 11, and F ( z , 2 ) = XO, for every z E V ( G ) , n E N and 0 5 i 5 3, is a continuous contraction that satisfies conditions (i) and (ii) of Theorem 3.7.

4. INVARIANT SUBGRAPH PROPERTIES

4.1. A graph G will be said to have the invariant r-bounded subgraph property (respectively finite invariant r-bounded subgraph property) if any contraction of G stabilizes a nonempty (respectively nonempty finite) r-bounded subgraph of G. Clearly a finite r-bounded subgraph of G, which is invariant under a contraction f of G, contains a finite r-bounded subgraph that is strictly invariant under fi

Finally, if p is a nonnegative integer, G will be said to have the p-Jixed point property if, for any contraction f of G, there is a vertex x of G such that distc(x,f(x)) 5 p . Such a vertex will be called a p-Jixed point offi

4.2. Let r and s be two positive integers such that r 5 s. The [G,s]- closure of a r-bounded subgraph S of G is the r-bounded subgraph [C,s](S> := G I n { B c ( x , s ) : x E V ( G ) and S C BG(x , s)}. In other words, x E V ( [ G , s ] ( S ) ) if and only if { y E V ( G ) : S B G ( y , s)} C &(x, s). An r-bounded subgraph S is [G, $1-closed if [ G , s] ( S ) = S . For brevity, and if


no confusion arises, we will write G-closure, G-closed, and G ( S ) for [G, r ] - closure, [ G , r]-closed, and [G, r ] ( S ) , respectively, when only r-bounded subgraphs are considered.

A graph G will be said r-bounded-compact (simplex-compact if r = 1) if any countable family of G-closed r-bounded subgraphs of G has a nonempty intersection whenever so has each of its finite subfamily.

4.3. Lemma. If a retractable graph G is r-bounded-compact, then so is G(I).

Proof. Let (S,),,?O be a countable family of G(')-closed r-bounded subgraphs of G(I), whose finite subfamilies have nonempty intersections. For every n L 0, let SL := G(S,) . Every finite subfamily of (SA),2~ has a nonempty intersection, hence nnzO SL # 0 since G is r-bounded-compact. Let x E V(nn2,SL). If x E V(G(')), then x E V(nnzOSn). Suppose that x @ V(G( ' ) ) ; then x < G y for some y E V(G")) since G is retractable. Thus, for every n 2 0, {z E V(G): S, &(z, r ) } C BG(x , r ) C M y , r ) , since BG(x , 1 ) C B G ( y . 1) implies B G ( x , r ) C B G ( y , r ) for any r 2 1; therefore y E G(I)(S,) = S,J. Consequently y E V(nn2,Sn>. I

4.4. Lemma. Let G be an r-bounded-compact graph such that there is a retractionffrom G onto G"' such that x < J(x) for all x E D(G) . Let g be a contraction of C , and g l := f 0 g I Then g stabilizes an r-bounded subgraph if so does g l .

Proof. In the following we will write x = r y for distG(x,y) 5 r . Since gl := f 0 g I G"' is a contraction of G(I), there is a vertex xo of G(' ) such that A := G l { g ! ( x o ) : i 2 0) is an r-bounded subgraph of @ I ) . For simplicity put xi := g;(xo) for any i 2 0. Define the sequences (A,),,o and (S,) , ,-O of r-bounded subgraphs of G so that S , := G(A,) and {xi: i 2 p} V(A,) for any p 2 0. Let A0 := A and SO := G(A0). Let p 2 0, and suppose that A, and S, have already been defined. Let A,+I be the subgraph of G induced by V(g(S,)) U {xi: i 2 p + 1). Let us prove that diam(A,+I) 5 r . The G-closure S , of A,, is an r-bounded subgraph, thus so is g ( S , ) . Hence we have only to show that the distance of any vertex of g ( S , ) with xi for any i 2 p + 1 is less than or equal to r. Let y be a vertex of S, and i I p + 1. Since S , is a r-bounded subgraph and x i - ] one of its vertex by the induction hypothesis, y = r ~ i - l , thus g ( y ) = rg (x i - l ) . Besides, notice thatf(g(xi-,)) = gl(xi-l) = xi, hence < Gxi by the definition off. Therefore g ( y ) = r g ( x i - I ) C Gxi implies that g ( y ) = r x ; . Consequently Ap+l is an r-bounded subgraph. Finally put Sp+l := G(A,+!) .

(S,),,o is then a sequence of G-closed r-bounded subgraphs whose finite subfamilies have nonempty intersections. By the r-bounded-compactness of G, there is a vertex x E n p k o V ( S , ) . Clearly g"(x) E n,,oV(g"(S,N G npZn V(S,). Thus, for every n and m, g"(x> and g m ( x ) are vertices of S, for any p 2 max{n,rn}. Hence g"(x) = ,g'"(x) since S, is an r-bounded


subgraph. Therefore H := G I {g"(x): n 2 0) is an r-bounded subgraph such that g ( H ) C H. I

We get immediately

4.5. Corollary. has the invariant r-bounded subgraph property if so has G(I) .

Let G be an r-bounded-compact retractable graph. Then G

4.6. Theorem. We have the following properties:

Let G be a retractable graph, and r a nonnegative integer.

(i) G has the r-fixed point property, for r 2 1, if and only if so has G'"). (ii) If G contains no infinite r-bounded subgraphs, then G has the finite

invariant r-bounded subgraph property if and only if so has G'"). (iii) If d(G) if finite and if G is r-bounded-compact, then G has the

invariant r-bounded subgraph property if and only if so has G'").

Proof. Let ( fup)aCPCd(G) be the family of retractions as defined in 2.3.

(i) To prove the first statement we will show, what seems to be more general but is actually equivalent, if a is an ordinal, then G has the r-jked pointproperty ifand only ifso has G("). In the following, as in the preceding proof, we will write x = .y for distG(x,y) 5 r.

(La) Suppose that G has the r-fixed point property, and let g be a contraction of G'"). Then h = g 0 fo, is a contraction of G. By the hypothesis, there is a vertex x of G such that h ( x ) = ,x. Then foa(x) = ,foa(h(x)) = h ( x ) since fo, is a contraction and h ( x ) E V(G(*)) . There- fore g(fo,(x)) = h ( x ) = ,foa(x), hence f o , ( x ) is a r-fixed point of g.

(i.b) Conversely, let g be a contraction of G; then g, := foa 0 g I G(") is a contraction of G'"). Notice that, for p < a , by 2.3 (i),

We will prove by induction on a that g has a r-fixed point if so has g,. This is obvious if a = 0 since go = g. Let a 2 0, and suppose that this holds for any p < a , i.e., if gp has a r-fixed point then so has g. Assume that g, has a r-fixed point. Then there is x E V ( G ( * ) ) such that gh(x) = ,gc ' (x) for any i 2 0.

Case I . a is a limit ordinal.

By 2.3 (iii) there is /3 < a such that fop(g (gb(x ) ) ) = fo,(g(gh(x))) for 0 5 i 5 2; thus gp(gh(x)) = g,(gh(x)) = gc ' (x ) for 0 5 i 5 2 . Then


g p ( x ) = g a b ) = ,g:(x) = gp(g,(x)), and this proves that g , ( x ) is a r-fixed point of g p . Therefore, by the induction hypothesis, g has a r-fixed point.

Case 2. a = ,L3 + 1 .

We are done if g p ( x ) E V(G'"'), i.e., g p ( x ) = g,(x). Suppose that g p ( x ) 4 V ( G ' " ' ) . Then fp,(gp(x)) = g,(x) by (1 ) . hence g p ( x ) < G,a,gU(.r). Besides, since g p is a contraction, x = ,g,(x) implies g p ( x ) = ,gp(g,(x)). Consequently these two last results imply gp(g,(x)) = ,g,(x), which proves that g,(x) is a r-fixed point of g p . Therefore, by the induction hypothesis, g has a r-fixed point.

(ii) The second statement is Proposition 3.9 of [ 3 ] if r 5 1. Let us prove it for any r.

(ii.a) Suppose that G has the finite invariant r-bounded subgraph property, and let g be a contraction of G','. Then h = g 0 fo, is a contraction of G. By the hypothesis, there is a finite r-bounded subgraph S of G that is strictly invariant under h. Then S C_ G'"' since S = h ( S ) G("). Therefore g(S) =

g(fo,(S)) = h ( S ) = S. Hence G'"' has the finite invariant r-bounded subgraph property.

(ii.b) Conversely assume that G'"' has the finite invariant r-bounded subgraph property, and let g be a contraction of G; then g, := f~, 0 g I G'"' is a contraction of G'"'. We will prove by induction on (Y that g leaves a finite r-bounded subgraph invariant if so does g,. This is obvious if a = 0 since go = g. Let a 2 0, and suppose that this holds for any /? < a , i.e., if g p stabilizes a finite r-bounded subgraph then SO does g. Assume that g, stabilizes a finite r-bounded subgraph. Then there is x E V(G(" ) ) such that S := C I {g;(x): ti 2 0) is a finite r-bounded subgraph of G'"' strictly invariant under g,. Let n be the least positive integer such that g:(x) = .r and gb(.r) f gL(.r) for any 0 5 i < j < n when n 2 1.

Case I . a is a limit ordinal.

By 2.3(iii) there is p < a such that fop(g(gb(x) ) ) = f o a ( g ( g h ( x ) ) ) for 0 5 i 5 n ; thus g p ( g b ( x ) ) = g,(gb(x)) = gL+'(x) for 0 5 i 5 n . Then, by induction, we get easily that gb(x) = gh(x) for 0 I i 5 n. Hence S is a r-bounded subgraph of G'p' strictly invariant under g p . Therefore, by the induction hypothesis, g leaves a finite r-bounded subgraph strictly invariant.

Case 2. a = p + 1


G'P) is r-bounded-compact since, by hypothesis, G has no infinite r- bounded subgraphs. Thus, by Lemma 4.4 gp stabilizes a finite r-bounded subgraph, and so does g by the induction hypothesis.

Consequently, if G'P) has the finite invariant r-bounded subgraph property, then so has G.

(iii) Assume that G is r-bounded-compact and that its depth is finite.

(iii.a) Suppose that G has the invariant r-bounded subgraph property, and let g be a contraction of G'"' =: G'd' . Then h = g 0 f ~ d is a contraction of G. By the hypothesis, there is an r-bounded subgraph S of G that is invariant under h. Then h(S ) C G'"). Therefore g ( h ( S ) ) = g( fOd(h (S ) ) ) = h2(S) h (S) . Hence h ( S ) is an r-bounded subgraph that is invariant under g.

(iii.b) The converse will be proved by induction on the depth of G. This is obvious if d(G) = 0. Let d 2 1, and suppose that this holds for any r-bounded-compact graph of depth d - 1. Let G be such that d(G) = d, assume that its base dd) has the invariant r-bounded subgraph property. Then G'') is a r-bounded-compact graph by Lemma 4.3, whose depth is d - 1, and whose base is @). Hence G ( ' ) has the invariant r-bounded subgraph property by the induction hypothesis, therefore so has G by Corollary 4.5. And this completes the proof of the theorem. I

The preceding properties- the r-fixed point property, the invariant r-bounded subgraph property-are trivially satisfied by G if its base is itself an r-bounded subgraph. In particular, we will get interesting results by considering retract-collapsible graphs, results that we will extend to subretract-collapsible graphs. For this purpose we will need the following lemma.

4.7. Lemma. Then

Let H = G U U+EV(G) T, be a tree-extension of a graph G.

(i) G is simplex-compact if and only if so is H. (ii) If H has the 1-fixed point property (respectively invariant simplex

property), then so has G.

Proof. We will denote by h the retraction from H onto G such that h( t ) = x for all x E V ( G ) and f E V(T,).

(i.a) Assume that H is simplex-compact, and let (Sn),zO be a sequence of finitewise nondisjoint G-closed simplices of G. Then ( H ( S , ) ) n 2 ~ is a sequence of finitewise nondisjoint simplices of H. For every n 2 0, h(H(S , ) ) = H(S , ) n G = S, since S, is a simplex. Therefore n,,,S, = h ( n n Z O H ( S , ) ) # 0 since h is a retraction and H is simplex-compact.


(i.b) Conversely, assume that G is simplex-compact, and let ( S n ) n 2 ~ be a sequence of finitewise nondisjoint H-closed simplices of H . For every n 2 0, SL := S, n G is a simplex of G, possible empty, which is G-closed if ISLl # 1. If, for some p 2 0 and x E V ( G ) , Sb =: ( x ) is not G-closed, then IT,I > 1 and S, = (1). Thus x E V ( S , ) for all n 2 0, since the S,’S are finitewise nondisjoint; and we are done. Suppose that SL is G-closed for every n 2 0. We have two cases.

If the members of (S;),?O are finitewise nondisjoint, then n,,,S; # 0 since G is simplex-compact; hence n n Z o S n f 0.

If there is a finite subset I of nonnegative integers such that nnE, SL = 0, then n n E , S n C T, - x for some x E V ( G ) . Note that, for any n 2 0, S, fl T, is T,-closed. Indeed, this is clear if S, n T, # ( x ) , or if T, = (x). If S, n T, = (x) # T,, then this follows from the fact that, by the definition of a tree-extension, T, f ( x ) = TIx’ implies that IV(x;T,)I 2 2.

Therefore (S, n T,),20 is a family of finitewise nondisjoint T,-closed simplices of T,. Hence nnZO(Sn n T,) # 0 since any tree is clearly simplex-compact. Consequently nnz0 S, # 0.

(ii) Let g be a contraction of G. Then g 0 h is a contraction of H such that g o h I V ( G ) = g.

(ii.a) Assume that H has the 1-fixed point property. Thus g ( h ( x ) ) = H x for some vertex x of H . If x E V ( G ) than h ( x ) = x and g(x) = G x . If x 4 V ( G ) then x E V(Th(,,), and g ( h ( x ) ) = H x implies that g ( h ( x ) ) =

h ( x ) since g ( h ( x ) ) E V ( G ) . Therefore h ( x ) is a 1-fixed point of g.

(ii.b) Assume now that H has the invariant simplex property. Thus g ( h ( S ) ) c S for some simplex S of H. Then g ( h ( S ) ) c G implies that g ( h ( S ) ) c S n G = h ( S ) . Hence h ( S ) , which is a simplex of G, is invariant under g.

4.8. Theorem. Let G be a subretract-collapsible graph. We have the following properties:

( i j G has the 1-tixed point property. (iij If G contains no infinite simplices, then G has the finite invariant

(iii) If s d ( G ) is finite and if G is simplex-compact, then G has the invariant

(iv) If G is retractable, then G contains a nonempty simplex that is strictly

simplex property.

simplex property.

invariant under any automorphism.

Proof. (i)-(iii): If G is subretract-collapsible then, by Proposition 3.8, any retractable tree-extension H of G is collapsible. Then, by Theorem 4.6 since H‘”’ is a simplex, H has the 1-fixed point (respectively the finite


invariant simplex, respectively invariant simplex) property, and so has G by Lemma 4.7.

Condition (iv) is a consequence of the fact that G'"', as well as G'") for any ordinal a , is strictly invariant under any automorphism of G. I

5. HELLY GRAPHS

In this section, as well as in the following one, we will study the relations between subretract-collapsible graphs and two other well-known classes of graphs.

5.1. A family A of sets has the (Jnite) Helly property if any (finite) subfamily B of pairwise nondisjoint members of A has a nonempty intersection. A connected graph will be said to be a (finitely) ball-Helly graph, or simply a ('nitely) Helfy graph, if the family of its balls has the Helly property.

It was proved recently that finitely Helly graphs without infinite simplices, which we studied in [3, Section 51, are actual Helly graphs. In this section, we will extend the results we got to any Helly graphs.

5.2. Lemma. Let G be Helly graph. Then

(i) for any x E D ( G ) there is y @ D ( G ) such that x < G y ; (ii) if x and y are nonadjacent vertices such that distc(x, u) 5 distG(x, y )

for every neighbor u of y then y < Gz for some vertex z with dist, ( x , z ) < distc(x, y ) ;

(iii) if G is bounded but not complete, then D ( G ) # 0 and diam(G - D ( G ) ) < diam(G).

Proof. (i) This is Lemma 3.5 of [3] if G contains no infinite simplices. Assume that G is a Helly graph, and let x E D(G) . Define an ordinal E and vertices ~ 0 ~ x 1 , . . . , x , such that x , < G ~ p if a 5 p 5 E , and x , E D ( G ) if and only if a # E . Let xo := x. Let a be an ordinal 2 0; and suppose that xq has already been defined for any p < a.

Case 1 . a = f l + 1.

If x p @ D(G) , put E = p. Otherwise x p is strictly dominated in G by a vertex x,. Then, by transitivity, xg < Gx, for any 6 < a .

Case 2. a is a limit ordinal.

Then x p E D ( G ) for any p < a. Let f l 5 y < a. For any y E BG(xp , 1) and z E B G ( x y , l), B G ( y , 1 ) fl B G ( z , 1) f 0 since in particular x y belongs to both these balls. Hence n p < , n y E B G ( x p . I ) B G ( ~ , 1 ) f 0 since


G is a Helly graph. Let x, be any element of this intersection. Clearly, x p < for any < a .

For reasons of cardinality, since G is not complete, there will exist an ordinal E 5 ICl such that x, 4 D ( C ) . Therefore x g < cx, 6? D ( G ) .

(ii) Let s and y be two nonadjacent vertices such that distc(x,u) 5 distc(x,y) =: k ( k 2 2) for any neighbor u of y. The balls B G ( x , k - l) , B G ( ~ , l ) , and BG(u , 1 ) for all neighbors u of y , are pairwise nondisjoint. Thus B G ( x , k - 1) f l n l t E B G , , , , ) B G ( u , 1) # 0 since G is a Helly graph. Then any element of this intersection has the desired properties.

(iii) Assume now that G is bounded but not complete. Then G has no isometric rays since it is bounded; hence D ( G ) # 0 by (ii). Moreover, by a proof similar to that of (ii), we can show that any vertices x and y with distG(x,y) = diam(G) belong to D ( G ) . Consequently, D ( G ) # 0 and diam(G - D ( G ) ) < diam(G). I

5.3. Theorem. no isometric rays.

A Helly graph is subretract-collapsible if and only if it has

Proof. Let G be a Helly graph.

(a) Suppose that G is subretract-collapsible, and let H be a retract- collapsible tree-extension of G. Then, by Proposition 2.6, H has no isometric rays since H'"' is a simplex, thus so is G.

(b) Conversely, assume that G has no isometric rays. Let a E V ( G ) , and u := IV(G)l. We will define by induction a map F : G X P , - G such that if, for any a 5 (T, G , denotes the subgraph of G induced by F ( V ( G ) ) X {a}) , then for every a 5 p 5 u

(1) fap: F ( x , a ) * F ( x , p ) is a retraction from G, onto Gp; (2) distc(a, F ( x , p ) ) < distc(a, F ( x , a ) ) for every x E V ( G ) such that

Put F ( x , O ) := x for every vertex x. Let a 5 u. Suppose that F ( x , p )

F(.x, a ) # F ( x , p).

has already been defined for every x E V ( G ) and p < a.

Case 1. a = p + 1

If V ( G p ) C Bc,(a, l ) , then put F ( x , a ) := a for every vertex x. Otherwise let S, := {.x E V ( G p ) : x f Ga and distc,(u,y) 5 distc,(a,x) for every y E V ( x ; G p ) } . This set is nonempty since Gp has no isometric rays, otherwise we would be able to extend inductively a shortest ax-path into an isometric ray, thus contradicting the hypothesis. Let us now prove that G , is a Helly graph. Let ( B G ~ ( X , , r l ) ) i E , be a family of pairwise nondisjoint balls


of Gp. Then ( B ~ ( x ; , r ; ) ) ~ ~ / is a family of pairwise nondisjoint balls of G, thus n;,, & ( x i , r i ) # 0 since G is a Helly graph. Then niE, B G p ( x i , r i ) =

f o p ( n i , , BG(x; , r ; ) z 0 since f o p is a nonexpansive map from G onto G, that, being a retract of G, is an isometric subgraph of G; and this proves that Gp is a Helly graph. Thus, by condition (ii) of Lemma 5.2, for each x E S, , there is a vertex y, of Gp such that x < Gyx and distG,(a, y x ) < distG,(a, x ) . Define then F ( x , a) as yF(x ,p ) or F ( x , p ) according to whether F ( x , p ) is or is not an element of S , . Conditions (1) and (2) are clearly satisfied.

Case 2. a is a limit ordinal.

By condition (2) of the induction hypothesis, for every vertex x of G, there is an ordinal y, < a such that F ( x , p ) = F ( x , y-) for yx 5 /? < a. Thus put F ( x , a) := F ( x , 7,). Conditions (1) and (2) are, as in Case 1 , clearly satisfied.

Finally, it is straightforward to check that F is a continuous contraction from G X P , onto G that satisfies conditions (i) and (ii) of the statement of Theorem 3.7. Therefore C is subretract-collapsible by this theorem. I

5.4. Remark. A Helly graph without isometric rays is not necessarily retractable, as is shown by the following example. Let H be the complete graph whose vertices are the nonnegative integers, and let ( W n ) n 2 ~ be a family of pairwise disjoint paths such that, for any n 2 0, W,, = (xi,. . . ,XI+,) is a path of length n + 1 having only its end point x: = n in common with H. Then let G be the graph such that V ( G ) = V ( H ) U UnZO V(W,,) and E ( G ) = E ( H ) U U,,20E(Wn) U { { x ; , n + l}: n 2 0). This graph is a Helly graph with G'"' = G'"' = 0 and thus not retractable.

On the other hand, the property of having no isometric rays is afortiori satisfied by rayless or bounded graphs, and in these cases Helly graphs are retractable as is shown hereafter.

5.5. Theorem. following properties:

(i) G is rayless; (ii) G is bounded; and in this case d ( G ) < diam(G).

A Helly graph G is retract-collapsible if it has one of the

Proof. This is Theorem 5.3 and 5.4 of [3] if G has no infinite simplices, but the following proof holds for both cases. Since rayless or bounded graphs have no isometric rays, we have only to prove that G is retractable by Proposition 3.8 and 5.3.

(i) is an immediate consequence of Proposition 2.6.

(ii) Assume that G is bounded. By Proposition 2.6, again we have only to prove that d ( G ) is finite. We will show more precisely that d(G) <


diam(G) =: k. This is clear if k = 1 since G is then a simplex. Let k > 1. Suppose that this holds for any graph of diameter less than k. We first prove that G(') is a Helly (respectively finitely Helly) graph. By 5.2(i), for any x E D ( G ) there is y E V ( G ( ' ) ) such that x < G y . Therefore G(l) is a retract of G; let f o l : C - G(l) be the retraction. The rest of the proof is analogous to that given in the proof ((b) Case 1) of Theorem 5.3. Besides, by 5.2(iii), diam(G"') < diam(G) = k. Hence, by the induction hypothesis, d ( G ( ' ) ) < diam(G"'). Therefore d ( G ) = d ( G ( ' ) ) + 1 < diam(G). I

5.6. Remark. Clearly a Helly graph G is r-bounded-compact for any r, since a G-closed r-bounded subgraph is an intersection of balls. Hence, in particular, a Helly graph is simplex-compact, and, by Theorem 5.5(ii), it is a retract-collapsible graph of finite depth if it is bounded. Thus we get as an immediate consequence of 4.8(iii) the following result of Jawhari et al. [ l , Theorem IV-1.3.21 substantially due to Quillot [ 5 ] : A bounded Helly graph has the invariant simplex property.

6. COP-WIN GRAPHS

Consider the following game that is played on a given graph G. There are two players, the cop and the robber. They move alternatively, the cop beginning. On the first move each player chooses a starting vertex, and on each subsequent move the players stay in their place or move to an adjacent vertex. The object of the game is for the cop to catch the robber, i.e., to occupy the same vertex as him, and for the robber to prevent this from happening. The graphs on which the cop can always win are called cop-win by Nowakowski and Winkler [2] who characterized them. A characterization of finite cop-win graphs was also obtained independently by Quillot [4]:

6.1. Theorem. [Nowakowski and Winkler, Quillot]: win if and only if it is dismantlable.

A finite graph is cop-

We will extend this result to the subretractable graphs.

6.2. Theorem. only if it is subretract-collapsible.

Let C be subretractable graph. Then G is cop-win if and

In the following we will denote by c, and r , the nth positions of the cop and of the robber, respectively. We need the following lemma:

6.3. Lemma. If G is cop-win and D ( G ) = 0, then G is a simplex.

Proof. Assume that D ( G ) = 0 and that G is not complete. We define Gc, for any n. inductively vertices C O , cl , . . . , and r o , r I , . . . such that r,

RETRACT-COLLAPSI BLE GRAPHS 43

Let co be any vertex of G. There is x f Gco. Otherwise if BG(cO, 1) = V ( G ) then, since & ( y , 1) # V ( G ) for some vertex y by the fact that G is not complete, we would have y < G ~ ~ , a contradiction with D ( G ) = 0. Define then ro := x . Let n L 0. Suppose that c, and r, have already been defined such that r, + ,+,. Let c , + ~ be any vertex in BG(c,, 1). Since c, f Gr,, there is x = c, + 1, otherwise r, < Gc, + a contradiction with D ( G ) = 0. Define r,+l := x . Therefore this proves that there is no winning strategy for the cop, contrary to the hypothesis.

r, such that x + I

6.4. Proof of Theorem 6.2. (a) We will first prove that a retractable graph G is cop-win if and only if it is collapsible. Let Cfap)a-=psd(G) be the family of retractions as defined in 2.3.

(a.1) Assume that G is cop-win. By [2, Theorem 11 G'"), which is a retract of G, is cop-win. Hence G'") is a simplex by Lemma 6.3, and thus G is collapsible.

(a.2) Assume now that G is collapsible. Define inductively vertices C O , cl , . . . , and ordinals (YO, a l , . . . such that a0 > a I > . . . , and c, := foan(r,-l) for n 2 1, where r , - l is the (n - 1)th position of the robber.

Let co be any vertex of G(") and a0 := d(G) , and put c1 := foa,(r0). Suppose that co, . . . , c, and (YO,. . . , a, have already been defined such that c, := foan(r,-l). Assume that d(r , ) < a,. Since G is retractable, there is an ordinal a,+l with d(r , ) 5 a ,+~ < a, such that f ~ ~ , + ~ + ~ ( r , - ~ ) = foan(rn-l) = c,. Thus f ~ ~ ~ + ~ ( r , - ~ ) 5 G ( m , , + , l ~ n . (1). Besides r, = Gr,-l im-

- Gc,. Define C,+I := .foan+,(rn). The sequence ((Y~)~=-o being strictly decreasing, there must be an integer p

such that up+] = ap. By what precedes, this must happen if d ( r , ) = ap. But in this case, since by hypothesis, foa,(rp-l) = cp and rp = G r p - l , we must have rp = G c p . We can then choose c , , + ~ := r p . And this proves that co,. .. , c , + ~ is a winning strategy, hence that G is cop-win.

plies foa.+,(rn) G f O a , , + l ( r n - I ) (2). Thus (1) and (2) implies fOa"+l(rn) - -

(b) Assume now that G is subretractable, and let H = G U UxEV(G) T, be a tree-extension of G. Denote by h the retraction from H onto G such that h( t ) = x for all x E V ( G ) and r E V(T,) . Suppose that G is subretract- collapsible, then, by Proposition 3.8, H is retract-collapsible. By (a) H is cop-win; then so is G, since if C O , . . . , c, is a winning strategy for a cop in H when the robber remains in G; then h(co), . . . , h(c,) is a winning strategy for the cop in G.

Conversely suppose that G is a cop-win. Then a winning strategy in G will enable a cop to catch the robber in G, or to block him in some tree T,. In this case, since T, is rayless, thus retract-collapsible by Proposition 2.5, hence cop-win by (a), and since T r ) = ( x ) , the cop will be able to catch the robber in T, by using the strategy introduced in (a.2) and stated precisely hereafter.


Therefore H is cop-win. By (a), H is retract-collapsible, and hence G is subretract-collapsible.

6.5. Part (a.2) of the last proof provides an optimal winning strategy C O , c I , , . . for a cop in a retract-collapsible graph G. Define inductively vertices co, cl , . . . and ordinals aO,aI , . . . as follows.

Let co be any vertex of G‘”’ and aO := min{p 2 d(ro): f o p ( r o ) = Gcg}; and for any n 2 1:

If d ( G ) = n is finite, then the cop catch the robber is at most n + 1 moves. One may need n + 1 moves, for example in a path of length 2n + 1.

References

[ I ] E. Jawhari, D. Misane, and M. Pouzet, Retracts: Graphs and ordered sets

[2] R. Nowakowski and P. Winkler, Vertex-to-vertex pursuit in a graph.

[3] N. Polat, Finite invariant simplices in infinite graphs. Period. Math.

141 A. Quillot, Thkse de 3e cycle, Universitk de Paris VI (1978). [ 5 ] A. Quillot, Thttse de doctorat d’Etat. Univ. Paris VI (1983).

from the metric point of view. Contemp. Math., 57 (1986) 175-226.

Discrete Murh. 43 (1983) 235-239.

Hungar. 27 (1993) 125-136.

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Retract-collapsible graphs and invariant subgraph properties