, Since they are finite and gated, by the Helly property for gated sets, X is nonempty. If X = H, by Claim 10, there exists an edge of H whose carrier does not include all maximal cells, a contradiction. Hence H = X, i.e., H is a finite cell. Consequently, Z ? (G) is a finite cell of G

. Proof, Every automorphism ? of G maps maximal halfspaces to maximal halfspaces, thus ?

. Z-?-(g), By Lemma 21, Z ? (G) is a finite cell, thus every automorphism of G fixes the cell Z ? (G)

, A non-expansive map from a graph G to a graph H is a map f : V (G) ? V (H) such that for any x, y ? V (G) it holds d H (f (x), f (y)) ? d G (x, y)

=. X. , i k ), and w = (j 1 , j 2 , . . . , j k ), where i m , j m ? i m , n m ? j m < n m /2 for all 1 ? m ? k. Since f (u ) = u , f (v ) = v , f (w ) = w and any vertex of X lies on a shortest path between one of the pairs of u , v , w , we conclude that f (X) ? X. It remains to prove that f (X) = X. Without loss of generality assume that among all j m ? i m with 1 ? m ? k, the difference j 1 ? i 1 is minimal. The vertex y = (i 1 , 0, 0, Lemma 22. Let G be a hypercellular graph and f be a non-expansive map from G to itself. Let u, v, w be any three vertices of G and let X = u , v , w be their median-cell. If f (u) = u, f (v) = v, f (w) = w, then f fixes each of the apices u = (uvw), v = (vwu), w = (wuv) and f (X)

.. .. +s, Both must be at distance at most two from each other, at distance at most n 1 /2?1 from y, and a must be adjacent or equal to

.. .. , ,. .. , .. .. , ,. .. , .. ;. et al., In each case y, a, b spans a cycle, since the length of C 1 is at least six and f is a non-expansive map. Thus f (C 1 , x 2 + 1, x 3 , . . . , x k ) is a cycle of the form (f 1 (C) + s, y 2 , y 3 , . . . , y k ) or (f 1 (C), y 2

, An endomorphism r of G with r(G) = H and r(v) = v for all vertices v in H is called a retraction of G and H is called a retract of G. Moreover, if r is just a non-expansive map from G to itself

, = H, where X is the median cell of u, v, w. This proves that H satisfies the median-cell property and by Theorem C, H is hypercellular. We continue with the proof of assertion (ii) of Theorem F. S is finite, also conv(S) is finite, therefore H is finite and nonempty. Notice that f (conv(S)) ? conv(S), thus H ? f (H) ? f (f (H)) ? . . ., but since for every v ? V (H) there exists n v such that f nv (v) = v, the inclusions cannot be strict. Thus f (H) = H and f acts as an automorphism on H, Proof. Let r be a weak retraction of G to H. For arbitrary vertices u, v, w of H it holds r(u) = u, r(v) = v, r(w) = w, thus by Lemma 22 X = r(X) ? r(G)

, If G is a finite regular hypercellular graph, then G is a single cell, i.e., G is isomorphic to a Cartesian product of edges and even cycles

, We will prove that also W (b, a) is minimal. The carrier N (E ab ) is the union of maximal cells of G crossed by E ab . For each such maximal cell X of N (E ab ) there exists a unique automorphism of X that fixes edges of E ab ? X and maps X ? W (a, b) to X ? W (b, a) and vice versa, G such that W (a, b) is an inclusion minimal halfspace

M. Albenque and K. Knauer, Convexity in partial cubes: the hull number, Discr. Math, vol.339, pp.866-876, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01185337

H. Bandelt, Characterizing median graphs, manuscript, 1982.

H. Bandelt, Retracts of hypercubes, J. Graph Th, vol.8, pp.501-510, 1984.

H. Bandelt, Graphs with intrinsic S3 convexities, J. Graph Th, vol.13, pp.215-338, 1989.

H. Bandelt and V. Chepoi, Cellular bipartite graphs, Europ. J. Combin, vol.17, pp.121-134, 1996.

H. Bandelt and V. Chepoi, The algebra of metric betweenness I: subdirect representation and retracts, Europ. J. Combin, vol.28, pp.1640-1661, 2007.

H. Bandelt and V. Chepoi, Metric graph theory and geometry: a survey, Surveys on Discrete and Computational Geometry: Twenty Years Later, Contemp. Math, vol.453, pp.49-86, 2008.

H. Bandelt, V. Chepoi, A. Dress, and J. Koolen, Combinatorics of lopsided sets, Europ. J. Combin, vol.27, pp.669-689, 2006.

H. Bandelt, V. Chepoi, and K. Knauer, COMs: complexes of oriented matroids, J. Comb. Theory, Ser. A, vol.156, pp.195-237, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01457780

H. Bandelt, V. Chepoi, M. Van-de, and . Vel, Pasch-Peano spaces and graphs, 1993.

H. Bandelt and J. Hedlíková, Median algebras, Discr. Math, vol.45, pp.1-30, 1983.

H. Bandelt, M. Van-de, and . Vel, Superextensions and the depth of median graphs, J. Combin. Th. Ser. A, vol.57, pp.187-202, 1991.

A. Berman and A. Kotzig, Cross-cloning and antipodal graphs, Discr. Math, vol.69, pp.107-114, 1988.

A. Björner, P. H. Edelman, and G. M. Ziegler, Hyperplane arrangements with a lattice of regions, Discrete Comput. Geom, vol.5, pp.263-288, 1990.

A. Björner, M. Vergnas, B. Sturmfels, N. White, and G. Ziegler, Oriented Matroids, Encyclopedia of Mathematics and its Applications, vol.46, 1993.

B. Bresar, J. Chalopin, V. Chepoi, T. Gologranc, and D. Osajda, Bucolic complexes, vol.243, pp.127-167, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01198891

M. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, 1999.

J. Chalopin, V. Chepoi, A. Genevois, H. Hirai, and D. Osajda, Helly graphs and Helly groups

J. Chalopin, V. Chepoi, H. Hirai, and D. Osajda, Weakly modular graphs and nonpositive curvature, Memoirs of AMS
URL : https://hal.archives-ouvertes.fr/hal-01787197

V. Chepoi, d-Convex sets in graphs, Dissertation, 1986.

V. Chepoi, Isometric subgraphs of Hamming graphs and d-convexity, Cybernetics, vol.24, pp.6-10, 1988.

V. Chepoi, Classifying graphs by metric triangles, Metody Diskretnogo Analiza, vol.49, pp.75-93, 1989.

V. Chepoi, Separation of two convex sets in convexity structures, J. Geometry, vol.50, pp.30-51, 1994.

V. Chepoi, Graphs of some CAT(0) complexes, Adv. Appl. Math, vol.24, pp.125-179, 2000.

V. Chepoi and O. Topal?, Tverberg numbers for cellular bipartite graphs, Arch. Math, vol.66, pp.258-264, 1996.

M. Deza and M. Laurent, Geometry of Cuts and Metrics, 1997.

D. ?. Djokovi?, Distance-preserving subgraphs of hypercubes, J. Combin. Th. Ser. B, vol.14, pp.263-267, 1973.

J. Doignon, J. R. Reay, and G. Sierksma, A Tverberg-type generalization of the Helly number of a convexity space, J. Geometry, vol.16, pp.117-125, 1981.

A. W. Dress and R. Scharlau, Gated sets in metric spaces, Aequationes Math, vol.34, pp.112-120, 1987.

M. Gromov, Essays in Group theory, vol.8, pp.75-263, 1987.

F. Haglund and F. Paulin, Simplicité de groupes d'automorphismes d'espacesà courbure négative, The Epstein birthday schrift, Geom. Topol. Monogr, vol.1, pp.181-248, 1998.

K. Handa, Topes of oriented matroids and related structures, Publ. Res. Inst. Math. Sci, vol.29, pp.235-266, 1993.

W. Imrich and S. Klav?ar, Product Graphs: Structure and Recognition, 2000.

W. Imrich and S. Klav?ar, Two-ended regular median graphs, Disc. Math, vol.311, pp.1418-1422, 2011.

J. R. Isbell, Median algebra, Trans. Amer. Math. Soc, vol.260, pp.319-362, 1980.

S. Klav?ar and S. V. Shpectorov, Convex excess in partial cubes, J. Graph Theory, vol.69, pp.356-369, 2012.

J. Lawrence, Lopsided sets and orthant-intersection of convex sets, Pacific J. Math, vol.104, pp.155-173, 1983.

N. Polat, Netlike partial cubes I. General properties, Discr. Math, vol.307, pp.2704-2722, 2007.

N. Polat, Netlike partial cubes II. Retracts and netlike subgraphs, Discr. Math, vol.309, pp.1986-1998, 2009.

N. Polat, Netlike partial cubes III. The median cycle property, Discr. Math, vol.309, pp.2119-2133, 2009.

N. Polat, Netlike partial cubes IV. Fixed finite subgraph theorems, Europ. J. Combin, vol.30, pp.1194-1204, 2009.

C. Tardif, On compact median graphs, J. Graph Theory, vol.23, pp.325-336, 1996.

M. Van-de and . Vel, Matching binary convexities, Topology Appl, vol.16, pp.207-235, 1983.

M. Van-de and . Vel, Theory of Convex Structures, 1993.