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DISTRIBUTIVE LATTICE POLYMORPHISMS ON REFLEXIVE GRAPHS

  • Siggers, Mark (Department of Mathematics Kyungpook National University)
  • Received : 2016.10.24
  • Accepted : 2017.08.17
  • Published : 2018.01.31

Abstract

In this paper we give two characterisations of the class of reflexive graphs admitting distributive lattice polymorphisms and use these characterisations to address the problem of recognition: we find a polynomial time algorithm to decide if a given reflexive graph G, in which no two vertices have the same neighbourhood, admits a distributive lattice polymorphism.

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Figure 1. Poset P and lattice D(P ) in thick light edges. Di-graph A and (the complement of) graph G(P,A) in dark.

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Figure 2. Left: The lattice D(P ) from Figure 1 embedded ina product of three chains, and the graph G(P,A) from Figure1 embedded as an induced subgraph of the product of pathson those chains. Right: The usual labelling on the product ofchains showing D(P ) as P? [1[2], 0[1]]? [2[1], 0[3]].

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Figure 3. Graph (left) with compatible lattice (right) but nocompatible distributive lattice.

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Figure 4. Compatible pair (G,L), poset JL, and the graph red(A c)

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Figure 5. The Game of Conjecture 6.15

Acknowledgement

Supported by : NRF

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