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Peano Partial Cubes, I.E., the Partial Cubes That Satisfy the Pash and the Peano Properties

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Peano Partial Cubes, I.E., the Partial Cubes That Satisfy the Pash and the Peano Properties Peano partial cubes Norbert Polat I.A.E., Université Jean Moulin (Lyon 3) 6 cours Albert Thomas 69355 Lyon Cedex 08, France [email protected] Abstract Peano partial cubes are the partial cubes that have the Pash and Peano properties, and thus they are the bipartite graphs whose geodesic interval spaces are (closed) join spaces. These graphs are the partial cubes all of whose finite convex subgraphs have a pre-hull number which is at most 1. Special Peano partial cubes are median graphs, cellular bipartite graphs and netlike partial cubes. Analogous properties of these graphs are satis- fied by Peano partial cubes. In particular the convex hull of any isometric cycle of such a graph is a gated quasi-hypertori (i.e., the Cartesian product of copies of K2 and even cycles). The finite quasi-hypertori are the finite regular Peano partial cubes, and they turn out to be the Peano partial cubes that are antipodal. Moreover, for any Peano partial cubes G that contains no isometric rays, there exists a finite qasi-hypertorus which is fixed by all automorphisms of G, and any self-contraction of G fixes some finite quasi-hypertorus. A Peano partial cube G is called a hyper-median partial cube if any triple of vertices of G has either a median or a hyper- median, that is, a quasi-median whose convex-hull induces a hypertorus (i.e., the Cartesian product of even cycles such that at least one of them has length greater than 4). These graphs have several properties similar to that of median graphs. In particular a graph is a hyper-median partial cube if and only if all its finite convex subgraphs are obtained by a se- quence of gated amalgamations from finite quasi-hypertori. Also a finite graph is a hyper-median partial cube if and only if it can be obtained arXiv:1901.09075v3 [math.CO] 1 Sep 2020 from K1 by a sequence of special expansions. The class of Peano partial cubes and that of hyper-median partial cubes are closed under convex subgraphs, retracts, Cartesian products and gated amalgamations. We study two convex invariants: the Helly number of a Peano partial cube, and the depth of a hyper-median partial cube that contains no isometric rays. Finally, for a finite Peano partial cube G, we prove an Euler-type formula, and a similar formula giving the isometric dimension of G. Keywords: Bipartite graph; Partial cube; Peano partial cube; Ph- homogeneous partial cube; Hyper-median partial cube; Netlike partial cube; Median graph; Hypercube; Hypertorus; Prism; Cartesian product; Gated amalgam; Retract; Geodesic convexity; Weak geodesic topology; 1 Helly number; Depth; Euler-type property; Fixed subgraph property. 2010 Mathematics Subject Classification: Primary(05C75, 05C63, 05C12, 05C30, 05C76);Secondary(52A37) 1 Introduction Median graphs are certainly the most extensively studied and characterized partial cubes (i.e., isometric subgraphs of hypercubes). Several classes of partial cubes containing median graphs as special instances have already been defined and studied, often with emphasis on properties generalizing some well-known properties of median graphs. In particular the class of netlike partial cubes was introduced and studied in a series of papers [43, 44, 45, 46, 47] as a class of partial cubes containing, in addition to median graphs, even cycles, cellular bipartite graphs [4] and benzenoid graphs. Median graphs and netlike partial cubes have the common property that any of their finite convex subgraphs has a pre-hull number [53] which is at most 1. A graph that has this property is said to be ph-homogeneous. These graphs were introduced in [51] as the bipartite graphs whose geodesic interval spaces are (closed) join spaces, i.e., which share a number of geometrical properties with Euclidean spaces. It was proved, that ph-homogeneous partial cubes are the Pash-Peano partial cubes, i.e., the partial cubes that satisfy the Pash and the Peano Properties. In fact, for partial cubes, the Peano Property is stronger than the Pash Property. This is why we call them Peano partial cubes. The class of Peano partial cubes is closed under convex subgraphs, Cartesian products, gated amalgams and retracts, as is the class of netlike partial cubes. It follows that the class of Peano partial cubes is a variety. We recall that, in graph theory, varieties are classes of graphs closed under retracts and products; the choice of products is not unique: it could be either the strong products or the Cartesian products. This paper is organized as follows. After the introduction of the basic con- cepts of Peano partial cubes and characterizations by convexity properties, sev- eral different structural characterizations of Peano partial cubes are given in Section 4. Particular properties follows from these characterizations. A Peano partial cube G is a median graph (resp. a netlike partial cube) if the convex hull of any isometric cycle of G is a hypercube (resp. this cycle itself or a hypercube). Any Peano partial cube G also satisfies an analogous but more general property, namely the convex hull of any isometric cycle of G is a gated quasi-hypertorus, that is, the Cartesian product of copies of K2 and even cycles. The finite quasi- hypertori, which turn out to be the finite regular Peano partial cubes, are also the Peano partial cubes that are antipodal (a graph G is antipodal if for any vertex x of G there is a vertexx ¯ such that the interval between x andx ¯ is equal to the vertex set of G). In Section 5, by analogy with the fact that median graphs are the particular Peano partial cubes for which any triple of vertices has a unique median, we 2 define a hyper-median partial cube as a Peano partial cube G such that any triple of vertices of G has either a median or a hyper-median, that is, a quasi- median whose convex-hull induces a hypertorus. Only the existence of a median or a hyper-median for each triple of vertices is necessary in the definition, the uniqueness comes as a consequence. We show in particular that a partial cube is a hyper-median partial cube if and only if all its finite convex subgraphs are obtained by a sequence of gated amalgamations from finite quasi-hypertori. This generalizes an analogous property of finite median graph [28, 61] stating that any finite median graph can be obtained by a sequence of gated amalgamations from finite hypercubes. Fairly recently we found out the existence of the paper of Chepoi, Knauer − and Marc [19] entitled Partial cubes without Q3 minors. This paper and the first version of ours were independently and almost simultaneously written, and it turns out that the graphs studied in [19], also called hypercellular partial cubes, coincide with our hyper-median partial cubes (Subsection 5.5). It follows that some of our results or parts thereof are also proved in [19]. Several other properties of these graphs are given in [19]. Since Mulder [36] and Chepoi [17] introduced the expansion procedure for median graphs and partial cubes, different kinds of finite partial cubes have already been constructed from K1 by sequences of special expansions (see [26]). In [45, Subsection 6] we showed that, if one can obtain all partial cubes by Chepoi’s theorem, not all graphs in the middle of expansion are netlike, and even ph-homogeneous. More precisely, there exist infinitely many finite netlike partial cubes, in particular some benzenoid graphs, which are not the expansion of any Peano partial cubes. In Subsection 5.7 we proved that there exists a particular kind of expansion that enables to construct all finite hyper-median partial cubes from K1. In Section 7 we focus on fixed subgraph properties, generalizing some re- sults on netlike partial cubes [46], which themselves are generalizations of three results of Tardif [59] on median graphs. For infinite partial cubes we use the topological concepts and results of [48, 49]. We prove three main results for any compact Peano partial cube G (i.e., containing no isometric rays). The first deals with the existence of a finite gated quasi-hypertorus that is fixed by all automorphisms of G. The second deals with the fact that any self-contraction (i.e., non-expansive self-map) of G fixes a finite quasi-hypertorus. The third extends the second result to any commuting family of self-contractions of G, and is of the same kind as two well-known theorems stating that commuting families of endomorphisms of certain structures have a common fixed point: the Markov-Kakutani Theorem [35, 29] for compact convex sets of locally convex linear topological spaces, and the Tarski’s theorem [60] for complete lattices. The proofs of these properties require some preliminary results, the most important of which also has another interesting consequence: the closure of the class of Peano partial cubes under retracts. This property is one of the main results of Section 6 which is devoted to the study of retracts and hom- retracts and their relation with convex subgraphs and more generally with the so-called “strongly faithful” subgraphs of Peano partial cubes, that is isometric 3 subgraphs that are stable under median and hyper-median. We show that any convex subgraph is a retract, and that any retract is strongly faithful. However the characterization of the strongly faithful subgraphs of a Peano partial cube that are retracts (or hom-retracts) of this graph is still an open problem. Section 8 deals with two convex invariants. First, a classical invariant: the Helly number, that is, the smallest integer h, if it exists, such that any finite family of h-wise non-disjoint convex sets has a non-empty intersection.
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