code: def min_vertex_cover(
      graph: Union[nx.Graph, rx.PyGraph], constrained: bool = True
  ):
      r"""Returns the QAOA cost Hamiltonian and the recommended mixer corresponding to the Minimum Vertex Cover problem,
      for a given graph.
  
      To solve the Minimum Vertex Cover problem, we attempt to find the smallest
      `vertex cover <https://en.wikipedia.org/wiki/Vertex_cover>`__ of a graph --- a collection of vertices such that
      every edge in the graph has one of the vertices as an endpoint.
  
      Args:
          graph (nx.Graph or rx.PyGraph): a graph whose edges define the pairs of vertices on which each term of the Hamiltonian acts
          constrained (bool): specifies the variant of QAOA that is performed (constrained or unconstrained)
  
      Returns:
          (.Hamiltonian, .Hamiltonian): The cost and mixer Hamiltonians
  
      .. details::
          :title: Usage Details
  
          There are two variations of QAOA for this problem, constrained and unconstrained:
  
          **Constrained**
  
          .. note::
  
              This method of constrained QAOA was introduced by Hadfield, Wang, Gorman, Rieffel, Venturelli, and Biswas
              in arXiv:1709.03489.
  
          The Minimum Vertex Cover cost Hamiltonian for constrained QAOA is defined as:
  
          .. math:: H_C \ = \ - \displaystyle\sum_{v \in V(G)} Z_{v},
  
          where :math:`V(G)` is the set of vertices of the input graph, and :math:`Z_i` is the Pauli-Z operator
          applied to the :math:`i`-th vertex.
  
          The returned mixer Hamiltonian is :func:`~qaoa.bit_flip_mixer` applied to :math:`G`.
  
          .. note::
  
              **Recommended initialization circuit:**
                  Each wire in the :math:`|1\rangle` state.
  
          **Unconstrained**
  
          The Minimum Vertex Cover cost Hamiltonian for unconstrained QAOA is defined as:
  
          .. math:: H_C \ = \ 3 \sum_{(i, j) \in E(G)} (Z_i Z_j \ + \ Z_i \ + \ Z_j) \ - \
                    \displaystyle\sum_{i \in V(G)} Z_i
  
          where :math:`E(G)` is the set of edges of :math:`G`, :math:`V(G)` is the set of vertices,
          and :math:`Z_i` is the Pauli-Z operator acting on the :math:`i`-th vertex.
  
          The returned mixer Hamiltonian is :func:`~qaoa.x_mixer` applied to all wires.
  
          .. note::
  
              **Recommended initialization circuit:**
                  Even superposition over all basis states.
  
      """
  
      if not isinstance(graph, (nx.Graph, rx.PyGraph)):
          raise ValueError(
              f"Input graph must be a nx.Graph or rx.PyGraph, got {type(graph).__name__}"
          )
  
      graph_nodes = graph.nodes()
  
      if constrained:
          cost_h = bit_driver(graph_nodes, 0)
          cost_h.grouping_indices = [list(range(len(cost_h.ops)))]
          return (cost_h, qaoa.bit_flip_mixer(graph, 1))
  
      cost_h = 3 * edge_driver(graph, ["11", "10", "01"]) + bit_driver(
          graph_nodes, 0
      )
      mixer_h = qaoa.x_mixer(graph_nodes)
  
      # store the valuable information that all observables are in one commuting group
      cost_h.grouping_indices = [list(range(len(cost_h.ops)))]
  
      return (cost_h, mixer_h)
  
file_path: pennylane/pennylane/qaoa/cost.py
imports: from typing import Iterable, Union
  import networkx
  import rustworkx
  import pennylane
  from pennylane import qaoa
label_desc: Use image node_id 5 for calling a global function with example usage: min_vertex_cover(graph, constrained) and returns: cost_h,  mixer_h, cost_h,  qaoa