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A Hamiltonian circuit is a cycle that visits every vertex of a graph exactly once and returns to the start. Determining its existence for an arbitrary graph is a classic NP-complete problem, so there is no known efficient algorithm that always works quickly in the worst case. However, there are powerful structural tests that can quickly certify the presence or absence of a Hamiltonian circuit in many practical instances. Since the specific graph is not provided here, the most rigorous way to proceed is to describe a systematic checklist that settles the question whenever possible and then, if the checklist does not decide it, to use a constructive search method that either yields a Hamiltonian circuit or proves that none exists by exhaustion. The first pass uses easy necessary conditions to rule out Hamiltonicity. If the graph has any vertex of degree 0 or 1, it cannot have a Hamiltonian circuit because every vertex on a Hamiltonian cycle must have two incident edges of the cycle. More generally, a Hamiltonian graph is 2-connected: it has no cut vertex whose removal disconnects the graph. If the given graph has an articulation point or a bridge (a cut edge), then a Hamiltonian circuit cannot exist. The reason is that any Hamiltonian cycle is a 2-edge-connected spanning subgraph; if a bridge existed, the cycle would have to cross that bridge twice to return, violating the “exactly once” rule. Therefore, in a preliminary scan, compute degrees, identify bridges (via a depth-first search with lowlink values), and check for articulation points. Discovering either a vertex of degree 1, a bridge, or a cut _________ ________ _______ _________ ____ _____ _____ _________ ________ ___.
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