Directed and undirected Hamiltonian path and cycle decision problems

Let $D=(V,A)$ be a finite directed graph in the sense of Multigraphs, loops and directed graphs as variants distinct from the default finite simple graph. A directed Hamiltonian path in $D$ is an ordering $v_1,\dots ,v_n$ of all vertices of $V$ such that $(v_i,v_{i+1})\in A$ for each $1\le i<n$. A directed Hamiltonian cycle in $D$ is a cyclic ordering $v_1,\dots ,v_n,v_1$ of all vertices of $V$ such that $(v_i,v_{i+1})\in A$ for $1\le i<n$ and $(v_n,v_1)\in A$.

Let $G$ be a finite simple graph. An undirected Hamiltonian path in $G$ is a path in the sense of Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges that visits every vertex exactly once, and an undirected Hamiltonian cycle in $G$ is a cycle there that visits every vertex exactly once.

The four decision problems are:

• directed Hamiltonian path: given $D$, decide whether $D$ has a directed Hamiltonian path;
• directed Hamiltonian cycle: given $D$, decide whether $D$ has a directed Hamiltonian cycle;
• undirected Hamiltonian path: given $G$, decide whether $G$ has an undirected Hamiltonian path;
• undirected Hamiltonian cycle: given $G$, decide whether $G$ has an undirected Hamiltonian cycle.