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Invariant Hamming Graphs in Infinite Quasi-Median Graphs

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Invariant Hamming Graphs in Infinite Quasi-Median Graphs DISCRETE MATHEMATICS ELSEVIER Discrete Mathematics 160 (1996) 93 - 104 Invariant Hamming graphs in infinite quasi-median graphs Marc Chastand, Norbert Polat* LA.E., Universitb Jean Moulin (Lyon III), 15, quai Claude Bernard, 69239 Lyon Cedex 2, France Received 4 November 1992; revised 13 October 1994 Abstract It is shown that a quasi-median graph G without isometric infinite paths contains a Ham- ming graph (i.e., a cartesian product of complete graphs) which is invariant under any automorphism of G, and moreover if G has no infinite path, then any contraction of G into itself stabilizes a finite Hamming graph. O. Introduction For several classes of graphs, it has been shown that each member of these classes contains a regular subgraph of the same class which is invariant under any automor- phism, or that any contraction of that graph into itself stabilizes a regular subgraph of the same class. One can find various examples of such classes, particularly with finite graphs. See for example: Nowakowski and Rival [8] for trees, Poston [12] for finite contractible graphs, Quillot [13] for finite ball-Helly graphs, Polat F10, 11] for infinite dismantlable graphs and infinite ball-Helly graphs, Bandelt and Mulder [1] for finite pseudo-median graphs, Bandelt and van de Vel [-3] for finite median graphs, and Tardif [16] for infinite median graphs. The graphs that we consider in this paper are the quasi-median graphs. These graphs have been defined independently by several authors and with various ap- proaches. The finite quasi-median graphs were introduced as a generalization of median graphs (see [7]), as connected subgraphs of Hamming graphs (i.e., cartesian products of complete graphs) that are closed under the quasi-median operation (see [-7, 5]), as retracts of Hamming graphs (see [-17, 5]), as graphs in which there exists an optimal strategy for a particular dynamic location problem (see [-5]). Note that median graphs are the bipartite quasi-median graphs and that the regular quasi- median graphs are precisely the Hamming graphs. Bandelt et al. [-2] gave several characterizations of (finite or infinite) quasi-median graphs, by bringing together * Corresponding author 0012-365X/96/$15.00 cO 1996 Elsevier Science B.V. All rights reserved SSDI 0012-365X(95)00151-4 94 M. Chastand, N. Polat /Discrete Mathematics 160 (1996) 93 104 different approaches, and in particular by linking those graphs with some ternary algebras called quasi-median algebras. Some special sets of vertices, called prefibers, related to the structure of metric space which is naturally associated with a graph, are very important for the study of quasi-median graphs. In any graph, the family of all prefibers has the Helly property (i.e., every finite family of pairwise non-disjoint prefibers has a nonempty intersection). Moreover this property also holds for every infinite family of nondisjoint prefibers if the graph has no isometric rays (i.e., no distance-preserving one-way infinite paths), and this enables to prove that: A quasi-median graph without isometric rays contains a Hamming graph which is invariant under any automorphism. This result holds a fortiori if the graph is rayless (i.e., without infinite paths), but in this case we also have the following: Any contraction of a rayless quasi-median graph stabilizes a finite Hamming graph. These results generalize those recently obtained by the first author for finite quasi-median graphs [4] as well as some results of Tardif on median graphs [163. 1. Notation and definitions The graphs we consider are undirected, without loops and multiple edges. We denote by V(G) the vertex set of a graph G, and by E(G) its edge set. Ifx and y are two vertices of a graph G we write x--Gy if X =y or {x,y} eE(G). If xe V(G), the set V(x; G):= {ye V(G): {x, y} e E(G)} is the neighborhood of x. The subgraph of G induced by a subset A of V(G) is denoted by G[A], or simply by A whenever no confusion is likely; and we set G - A := G[V(G) - A]. A path W := <xo .... ,x,> is a graph with V(W)={xo ..... x,}, xi#x~ if i#j, and E(W)={{xi, x~+l}: 0 ~< i < n}; Xo and x, are its endpoints, and W is also called an xox,-path. A ray or one-way infinite path R := <Xo, xl .... > is defined similarly. The (geodesic) distance in G between two vertices x and y, that is the length of an xy-geodesic (i.e. a shortest xy-path) in G, is denoted by do(x, y); and every graph G is endowed with the structure of metric space associated with this distance. A subgraph H of G is isometric ifdn(x, y) = do(x, y) for all vertices x and y ofH. Ifx is a vertex of G and r a nonnegative integer, the set Bo(x, r) := {y e V(G): do(x, y) <~ r} is the ball of center x and radius r in G. If x and y are two vertices of G, then the interval Io(x, y) is the set of vertices of all xy-geodesics. Clearly Io(x,y):= {zs V(G): do(x, z) + do(z, y) = do(x, y)}. A subset C of V(G) is geodesically convex, for short convex, if it contains the interval Io(x, y) for all x, y e C. The convex hull coo(C) of C in G is the smallest convex set of G containing C. Thus coo(C) = 0. ~> o Co where Co = C and C,÷1 = U~,y~c.Io(x, y). M. Chastand, N. Polat/Discrete Mathematics 160 (1996) 93 104 95 Let (ul, u2, u3) be a triple of vertices of a graph G. A quasi-median of (Ul, U2, U3) is a triple of vertices (xl, x2, x3) such that • xl, xj lie on a uiu~-geodesic, i,j ~ {1, 2, 3}; • do(Xl, x2) = do(x2, x3) = d~(x3, xl) = k; • k is minimal with respect to these conditions. If k = 0, then the quasi-median is reduced to a single vertex x, which is called a median of the triple (u~, u2, u3). A median 9raph is a graph in which every triple of vertices has a unique median. If G and H are two graphs, a mapJ: V(G) --, V(H) is a contraction iffpreserves the relation -, i.e., x -oY implies f(x) -nf(Y). Notice that a contraction f: G ~ H is a non-expansive map between the metric spaces (V(G), disto) and (V(H), distil), i.e., distn(f(x), f(y))~< disto(x,y) for all x, y ~ 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[ H to H is the identity. The cartesian product G x H of two graphs G and H is defined by V(G x H) = V(G) x V(H), and (x,y) =-o×n(x',y') if and only if x = x' and y -n)/, or x --- ~ x' and y = y'. Clearly do ×n = dc + du. A contractionJ'of G (into itself) is said to stabilize a set A of vertices (resp. a subgraph H of G) if f (A) = A (resp. f(H) = Hi. A subgraph H of G is said to be invariant if it is stabilized by any automorphism of G. A complete graph is simply called a simplex, and a clique is a simplex which is maximal with respect to inclusion. A Hammin9 9raph (resp. hypercube) is a cartesian product of simplices (resp. K2). As usual K2, 3 (resp. K1,1,2)denotes the complete bipartite (resp. tripartite) graph whose subsets of vertices have 2 and 3 (resp. 1, 1 and 2) elements, respectively. Roughly K1, 1,2 is K~ minus an edge. 2. Prefibers The concept of prefiber generalizes that of fiber of a cartesian product of metric spaces; it has been studied in particular by Dress and Scharlau [6] and by Tardif [14, 15]. 2.1. Definition. Let (X, d) be a metric space. A prefiber (or 9ated set) of ~" is a subset A of Y" such that, for all x s Y', there is y e A with d(x, z) = d(x, y) + d(y, z) for every z s A. The element y is unique, and the map projA :2" --, A defined by y -- projA(x) is the projection onto A. In this paper we will use the following properties: 2.2. Properties (Tardif [14]). (i) IrA and B are two prefibers of a metric space (:~', d), then proja(B) is a prefiber of 9£. Moreover, if Ac~B ~ O, then Ac~B is a prefiber of(?F, d) and projA~,~ = projA o proj~ = projR ° projA. 96 M. Chastand, N. Polat/Discrete Mathematics 160 (1996) 93-104 (ii) The family of prefibers of a metric space has the Helly property (i.e., every finite family of pairwise nondisjoint prefibers has a nonempty intersection). (iii) If (Y(, d) is a complete metric space, and ~ a family of prefibers of 35 such that n ~,~ ~ O, then n ~-~ is a prefiber of 35. When the metric space of a graph is concerned, the projection associated with a prefiber is clearly an idempotent contraction, and a prefiber is a retract of this graph. We will give two properties that will be useful in the following. The first, which is a property on nested prefibers, enables to show the existence of some particular geodesic.
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