A note on the dominating circuit conjecture and subgraphs of essentially 4-edge-connected cubic graphs

The well-known dominating circuit conjecture has several interesting reformulations, for example conjectures of Fleischner, Matthews and Sumner, and Thomassen. We present another equivalent version of the dominating circuit conjecture considering subgraphs of essentially 4-edge-connected cubic graphs. Let image be a set of four distinct vertices of a graph image and image be a set of all vertices of degree 2 of a graph image. We say that image is image-strongly dominating if the graph arising from image after adding two new edges image and image such that image has a dominating closed trail containing image and image. We show that the dominating circuit conjecture is equivalent to the statement that any subgraph image of an essentially 4-edge-connected cubic graph with image and minimum degree image is strongly image-dominating.