Congruences and subdirect representations of graphs

• Main Author: Stefan Veldsman
• Format: Article
• Language: English
• Published: Indonesian Combinatorial Society (InaCombS); Graph Theory and Applications (GTA) Research Centre; University of Newcastle, Australia; Institut Teknologi Bandung (ITB), Indonesia 2020-04-01
• Series: Electronic Journal of Graph Theory and Applications
• Subjects: congruence on a graph, quotient graph, subdirect product of graphs, subdirectly irreducible graph, birkhoff's theorem
• Online Access: https://www.ejgta.org/index.php/ejgta/article/view/846

<p class="p1">A basic tool in universal algebra is that of a congruence. It has been shown that congruences can be defined<span class="Apple-converted-space">  </span>for graphs with properties similar to their universal algebraic counterparts. In particular, a subdirect product of graphs and hence also a subdirectly irreducible graph, can be expressed in terms of graph congruences. Here the subdirectly irreducible graphs are determined explicitly. Using congruences, a graph theoretic version of the well-known Birkhoff Theorem from universal algebra is given. This shows that any non-trivial graph is a subdirect product of subdirectly irreducible graphs</p>