Snarks are cyclically 4-edge-connected cubic graphs with girth at least 5 and with no 3-edge-coloring. We construct snarks with a (dominating) circuit C so that no other circuit C′ satisfies V(C)⊆V(C′). These graphs are of interest because two known conjectures about graphs can be reduced on them. The first one is Sabidusi's Compatibility Conjecture which suggests that given an eulerian trail T in an eulerian graph G without 2-valent vertices, there exists a decomposition of G into circuits such that consecutive edges in T belong to different circuits. The second conjecture is the Fixed-Circuit Cycle Double-Cover Conjecture suggesting that every bridgeless graph has a cycle double cover which includes a fixed circuit.
