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You can have a look at the questions tagged graph_generation for more details and explanations about the following, see https://ask.sagemath.org/questions/scope:all/sort:activity-desc/tags:graph_generation/page:1/

• being planar is a property that is stable when we remove some edges
• being $C_5$-free is a property that is stable when we remove some edges
• being connected is not stable when we remove some edges

Hence, we can first generate all planar and $C_5$-free graphs with the graphs generators (using the augment="edges" option), and then filter the connected ones (the cost is not expensive since we already selected few graphs):

sage: C5 = graphs.CycleGraph(5)
sage: good_graphs = [g for g in graphs(6, property=lambda G: G.is_planar() and not C5.is_subgraph(G,induced=False), augment="edges") if g.is_connected()]

sage: len(good_graphs)

sage: for g in good_graphs:
....:     show(g)

You can have a look at the questions tagged graph_generation for more details and explanations about the following, see https://ask.sagemath.org/questions/scope:all/sort:activity-desc/tags:graph_generation/page:1/

• being planar is a property that is stable when we remove some edges
• being $C_5$-free is a property that is stable when we remove some edges
• being connected is not stable when we remove some edges

Hence, we can first generate all planar and $C_5$-free graphs with the graphs generators (using the augment="edges" option), and then filter the connected ones (the cost is not expensive since we already selected few graphs):

sage: C5 = graphs.CycleGraph(5)
sage: good_graphs = [g for g in graphs(6, property=lambda G: G.is_planar() and not C5.is_subgraph(G,induced=False), C5.is_subgraph(G,induced=False, up_to_isomorphism=True ), augment="edges") if g.is_connected()]

sage: len(good_graphs)
43

sage: for g in good_graphs:
....:     show(g)