## The algebra of metric betweenness. I: Subdirect representation and retraction.(English)Zbl 1122.05083

The authors bring together concepts from graph theory and algebra. They aim for developing a structure theory for graphs from the algebraic point of view. For that they use equational classes, subdirect products, retractions and gated amalgamations.
They investigate classes of graphs which posses distinctive features of the geometry of their shortest paths. So the starting point are median graphs which can be characterized by the property that for each triple $$(u,v,w)$$ of vertices there is a unique vertice $$x$$, the median of $$(u,v,w)$$, lying simultaneously on shortest paths between the three pais of the triplet. Thus there is a ternary operation and this operation can be used to define equational classes of the corresponding algebras. Generalizations lead to the new classes of quasi-median and weakly median graphs. These classes allow a decomposition into simple building blocks, which have a geometric interpretation.
As the main result the authors show that successive fiber amalgamations from Cartesian products lead to the subdirect representation of the resulting associated algera by subdirectly irreducibles whenever they begin with a class of (“prime”) graphs that posses only trivial gated subgraphs. The authors also consider infinite weakly median graphs.

### MSC:

 05C75 Structural characterization of families of graphs 05C12 Distance in graphs

### Keywords:

median graph; retraction; gated amalgamation; fiber amalgamation
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### References:

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