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**Quasi-median graphs and algebras.**
*(English)*
Zbl 0810.05057

The generalization of median structures yields certain resemblances between nonbipartite graphs and nonsymmetric ternary algebras, and this leads to quasi-median graphs and their algebras. A finite quasi-median graph is a retract of a Cartesian product of complete graphs, where a retraction is an idempotent mapping that preserves or collapses edges, and a quasi-median algebra is a subdirect product of ternary algebras for which the term \((xyz)\) equals \(y\) if \(y= z\) and \(x\) otherwise. Already known results are presented, and the results of the authors are summarized in four theorems which are proved applying numerous notions (they are given in the preliminaries) and operations from graph theory and finitary algebras.

Let \(G= G(V,E)\) be a finite connected graph. In Theorem 1 a list of five conditions is given which are equivalent to the quasi-median property of \(G\). Theorem 2 proves that \(G\) is a quasi-median graph iff every set of vertices that minimizes the total distance to families of vertices (such a set is called median set) is connected and contains no induced \(K_{1,1,2}\). Analogous to Theorem 1 in Theorem 3 statements are given which are equivalent for a finite ternary algebra \(V\). These statements are special axioms for \(V\), and it is shown that \(V\) satisfying several equivalent sets of axioms is associated with a quasi-median graph, for a quasi-median graph with the vertex set \(V\) can be turned into a ternary algebra by a certain given ternary operation. In Theorem 4 it is shown that an intrinsic ternary algebra of \(G\) satisfies at least one of the algebraic identities mentioned in Theorem 3 iff \(G\) is a quasi-median graph.

Let \(G= G(V,E)\) be a finite connected graph. In Theorem 1 a list of five conditions is given which are equivalent to the quasi-median property of \(G\). Theorem 2 proves that \(G\) is a quasi-median graph iff every set of vertices that minimizes the total distance to families of vertices (such a set is called median set) is connected and contains no induced \(K_{1,1,2}\). Analogous to Theorem 1 in Theorem 3 statements are given which are equivalent for a finite ternary algebra \(V\). These statements are special axioms for \(V\), and it is shown that \(V\) satisfying several equivalent sets of axioms is associated with a quasi-median graph, for a quasi-median graph with the vertex set \(V\) can be turned into a ternary algebra by a certain given ternary operation. In Theorem 4 it is shown that an intrinsic ternary algebra of \(G\) satisfies at least one of the algebraic identities mentioned in Theorem 3 iff \(G\) is a quasi-median graph.

Reviewer: H.-J.Presia (Ilmenau)

### MSC:

05C75 | Structural characterization of families of graphs |

08A62 | Finitary algebras |

05C12 | Distance in graphs |

05C99 | Graph theory |

08A40 | Operations and polynomials in algebraic structures, primal algebras |

### Keywords:

median structures; ternary algebras; quasi-median graph; quasi-median algebra; finitary algebras; connected graph
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\textit{H.-J. Bandelt} et al., J. Graph Theory 18, No. 7, 681--703 (1994; Zbl 0810.05057)

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