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PathGraphQ

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Tests whether a graph is a simple connected graph with all degrees at most 2 (a digraph: at most one arc in and one out per vertex)

PathGraphQ[g]
{TreeGraphQ[g], PathGraphQ[g]} → {False, False}{ConnectedGraphQ[g], PathGraphQ[g]} → {False, False}{PathGraphQ[g] && AcyclicGraphQ[g], SimpleGraphQ[g]} → {False, False}
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Related

GraphDirectedGraphTreeGraphPathGraphConnectedGraphQLoopFreeGraphQTreeGraphQVertexDegreeAcyclicGraphQBipartiteGraphQEulerianGraphQHamiltonianGraphQ

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