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An Eulerian-minor of an Eulerian graph is obtained from an Eulerian subgraphof the Eulerian graph by contraction. The Eulerian-minor operation preservesEulerian properties of graphs and moreover Eulerian graphs arewell-quasi-ordered under Eulerian-minor relation. In this paper, wecharacterize Eulerian, planar and outer-planar Eulerian graphs by means ofexcluded Eulerian-minors, and provide a concise recursive characterization to4-regular planar graphs.

Eulerian-minors and a concise recursive characterization of 4-regular planar graphs