code: def is_tree_decomp(graph, decomp):
      """Check if the given tree decomposition is valid."""
      for x in graph.nodes():
          appear_once = False
          for bag in decomp.nodes():
              if x in bag:
                  appear_once = True
                  break
          assert appear_once
  
      # Check if each connected pair of nodes are at least once together in a bag
      for x, y in graph.edges():
          appear_together = False
          for bag in decomp.nodes():
              if x in bag and y in bag:
                  appear_together = True
                  break
          assert appear_together
  
      # Check if the nodes associated with vertex v form a connected subset of T
      for v in graph.nodes():
          subset = []
          for bag in decomp.nodes():
              if v in bag:
                  subset.append(bag)
          sub_graph = decomp.subgraph(subset)
          assert nx.is_connected(sub_graph)
  
imports: import itertools
  import networkx
  from networkx.algorithms.approximation import treewidth_min_degree, treewidth_min_fill_in
  from networkx.algorithms.approximation.treewidth import MinDegreeHeuristic, min_fill_in_heuristic
label_desc: Use image node_id 1 for calling a global function with example usage: is_tree_decomp(graph, decomp) without return types