(*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 0, 0] NotebookDataLength[ 49948, 1156] NotebookOptionsPosition[ 46613, 1092] NotebookOutlinePosition[ 47014, 1108] CellTagsIndexPosition[ 46971, 1105] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Get", "[", "\"\\"", "]"}]], "Input", CellChangeTimes->{{3.832151022612398*^9, 3.832151030465872*^9}, { 3.8321511379667025`*^9, 3.8321511785398226`*^9}, 3.832151627377406*^9}, CellLabel->"In[1]:=",ExpressionUUID->"db347c93-28af-46ae-97e4-0f025565b28b"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"The currently installed versions of IGraph/M are: \"\>", "\[InvisibleSpace]", RowBox[{"{", "\<\"0.5.1\"\>", "}"}]}], SequenceForm[ "The currently installed versions of IGraph/M are: ", {"0.5.1"}], Editable->False]], "Print", CellChangeTimes->{3.832337217807197*^9, 3.832358883859474*^9}, CellLabel-> "During evaluation of \ In[1]:=",ExpressionUUID->"91a871f7-7ad4-4d9b-a238-89534a13d73b"], Cell[BoxData[ TemplateBox[{ "System`PacletInstall", "samevers", "\"A paclet named \\!\\(\\*RowBox[{\\\"\\\\\\\"IGraphM\\\\\\\"\\\"}]\\) \ with the same version number \ (\\!\\(\\*RowBox[{\\\"\\\\\\\"0.5.1\\\\\\\"\\\"}]\\)) is already installed. \ Use PacletUninstall to remove the existing version first, or call \ PacletInstall with ForceVersionInstall -> True.\"", 2, 1, 1, 26206974750047846679, "Local"}, "MessageTemplate"]], "Message", "MSG", CellChangeTimes->{3.832337222565331*^9, 3.832358887306924*^9}, CellLabel-> "During evaluation of \ In[1]:=",ExpressionUUID->"1bd7e6c6-08bf-40e0-ada5-1b9c296f7ca2"], Cell[BoxData[ InterpretationBox[ RowBox[{"\<\"Installation failed. 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About O(2^n) and not likely able \ to improve. There are about O(2^n) cycles (it\[CloseCurlyQuote]s been \ rigorously proven to be P-hard to count the number of cycles), and I don\ \[CloseCurlyQuote]t see a way of getting around that.\ \>", "Text", CellChangeTimes->{{3.83235854587254*^9, 3.832358618254396*^9}},ExpressionUUID->"83ebdaf9-e7b9-4912-83aa-\ 66018e3b4210"], Cell[BoxData[ RowBox[{ RowBox[{"primePolyhedronQ", "[", "graph_", "]"}], ":=", RowBox[{"Block", "[", RowBox[{ RowBox[{"{", RowBox[{ "cycles", ",", "valid", ",", "faceDecomposable", ",", "vertexDecomposable", ",", "dualGraph"}], "}"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"cycles", "=", RowBox[{ RowBox[{"FindCycle", "[", RowBox[{"graph", ",", "Infinity", ",", "All"}], "]"}], "/.", RowBox[{ RowBox[{"(", RowBox[{"x_", "\[UndirectedEdge]", "y_"}], ")"}], "\[Rule]", "x"}]}]}], ";", "\[IndentingNewLine]", RowBox[{"valid", "=", RowBox[{"Select", "[", RowBox[{"cycles", ",", RowBox[{ RowBox[{"Not", "[", RowBox[{"IGConnectedQ", "[", RowBox[{"VertexDelete", "[", RowBox[{"graph", ",", "#"}], "]"}], "]"}], "]"}], "&"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"faceDecomposable", "=", RowBox[{"AnyTrue", "[", RowBox[{"valid", ",", RowBox[{ RowBox[{"With", "[", RowBox[{ RowBox[{"{", RowBox[{"components", "=", RowBox[{"ConnectedComponents", "[", RowBox[{"VertexDelete", "[", RowBox[{"graph", ",", "#"}], "]"}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"KVertexConnectedGraphQ", "[", RowBox[{ RowBox[{"Subgraph", "[", RowBox[{"graph", ",", RowBox[{"Join", "[", RowBox[{ RowBox[{ "components", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}], ",", "#"}], "]"}]}], "]"}], ",", "3"}], "]"}], "&&", RowBox[{"KVertexConnectedGraphQ", "[", RowBox[{ RowBox[{"Subgraph", "[", RowBox[{"graph", ",", RowBox[{"Join", "[", RowBox[{ RowBox[{ "components", "\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}], ",", "#"}], "]"}]}], "]"}], ",", "3"}], "]"}]}]}], "]"}], "&"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"If", "[", RowBox[{"faceDecomposable", ",", RowBox[{"Return", "[", "False", "]"}]}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"dualGraph", "=", RowBox[{"IGDualGraph", "[", "graph", "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"cycles", "=", RowBox[{ RowBox[{"FindCycle", "[", RowBox[{"dualGraph", ",", "Infinity", ",", "All"}], "]"}], "/.", RowBox[{ RowBox[{"(", RowBox[{"x_", "\[UndirectedEdge]", "y_"}], ")"}], "\[Rule]", "x"}]}]}], ";", "\[IndentingNewLine]", RowBox[{"valid", "=", RowBox[{"Select", "[", RowBox[{"cycles", ",", RowBox[{ RowBox[{"Not", "[", RowBox[{"IGConnectedQ", "[", RowBox[{"VertexDelete", "[", RowBox[{"dualGraph", ",", "#"}], "]"}], "]"}], "]"}], "&"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"Not", "[", RowBox[{"AnyTrue", "[", RowBox[{"valid", ",", RowBox[{ RowBox[{"With", "[", RowBox[{ RowBox[{"{", RowBox[{"components", "=", RowBox[{"ConnectedComponents", "[", RowBox[{"VertexDelete", "[", RowBox[{"dualGraph", ",", "#"}], "]"}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"KVertexConnectedGraphQ", "[", RowBox[{ RowBox[{"Subgraph", "[", RowBox[{"dualGraph", ",", RowBox[{"Join", "[", RowBox[{ RowBox[{ "components", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}], ",", "#"}], "]"}]}], "]"}], ",", "3"}], "]"}], "&&", RowBox[{"KVertexConnectedGraphQ", "[", RowBox[{ RowBox[{"Subgraph", "[", RowBox[{"dualGraph", ",", RowBox[{"Join", "[", RowBox[{ RowBox[{ "components", "\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}], ",", "#"}], "]"}]}], "]"}], ",", "3"}], "]"}]}]}], "]"}], "&"}]}], "]"}], "]"}]}]}], "\[IndentingNewLine]", "]"}]}]], "Input", CellChangeTimes->{{3.8323514934530973`*^9, 3.832351646852213*^9}, { 3.8323516804671373`*^9, 3.83235168336065*^9}}, CellLabel->"In[6]:=",ExpressionUUID->"1445a032-9806-4ee7-8079-9849d6ac9896"], Cell["\<\ Function to find the possible decompositions of a graph. Returns a list where \ the first element of the list is the number of possible face decompositions \ and the second is the number of possible vertex decompositions. Should be \ easy to modify to actually return the decompositions. I wonder if we could \ extend this recursively to \[OpenCurlyDoubleQuote]prime factorize\ \[CloseCurlyDoubleQuote] polyhedra?\ \>", "Text", CellChangeTimes->{{3.832358623451818*^9, 3.832358720650798*^9}},ExpressionUUID->"a601bc96-4656-4520-9aeb-\ 158895d9368a"], Cell[BoxData[ RowBox[{ RowBox[{"decomposable", "[", "graph_", "]"}], ":=", RowBox[{"Block", "[", RowBox[{ RowBox[{"{", RowBox[{ "cycles", ",", "valid", ",", "faceDecomposable", ",", "vertexDecomposable", ",", "dualGraph"}], "}"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"cycles", "=", RowBox[{ RowBox[{"FindCycle", "[", RowBox[{"graph", ",", "Infinity", ",", "All"}], "]"}], "/.", RowBox[{ RowBox[{"(", RowBox[{"x_", "\[UndirectedEdge]", "y_"}], ")"}], "\[Rule]", "x"}]}]}], ";", "\[IndentingNewLine]", RowBox[{"valid", "=", RowBox[{"Select", "[", RowBox[{"cycles", ",", RowBox[{ RowBox[{"Not", "[", RowBox[{"IGConnectedQ", "[", RowBox[{"VertexDelete", "[", RowBox[{"graph", ",", "#"}], "]"}], "]"}], "]"}], "&"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"faceDecomposable", "=", RowBox[{"Length", "[", RowBox[{"Select", "[", RowBox[{"valid", ",", RowBox[{ RowBox[{"With", "[", RowBox[{ RowBox[{"{", RowBox[{"components", "=", RowBox[{"ConnectedComponents", "[", RowBox[{"VertexDelete", "[", RowBox[{"graph", ",", "#"}], "]"}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"KVertexConnectedGraphQ", "[", RowBox[{ RowBox[{"Subgraph", "[", RowBox[{"graph", ",", RowBox[{"Join", "[", RowBox[{ RowBox[{ "components", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}], ",", "#"}], "]"}]}], "]"}], ",", "3"}], "]"}], "&&", RowBox[{"KVertexConnectedGraphQ", "[", RowBox[{ RowBox[{"Subgraph", "[", RowBox[{"graph", ",", RowBox[{"Join", "[", RowBox[{ RowBox[{ "components", "\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}], ",", "#"}], "]"}]}], "]"}], ",", "3"}], "]"}]}]}], "]"}], "&"}]}], "]"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"dualGraph", "=", RowBox[{"IGDualGraph", "[", "graph", "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"cycles", "=", RowBox[{ RowBox[{"FindCycle", "[", RowBox[{"dualGraph", ",", "Infinity", ",", "All"}], "]"}], "/.", RowBox[{ RowBox[{"(", RowBox[{"x_", "\[UndirectedEdge]", "y_"}], ")"}], "\[Rule]", "x"}]}]}], ";", "\[IndentingNewLine]", RowBox[{"valid", "=", RowBox[{"Select", "[", RowBox[{"cycles", ",", RowBox[{ RowBox[{"Not", "[", RowBox[{"IGConnectedQ", "[", RowBox[{"VertexDelete", "[", RowBox[{"dualGraph", ",", "#"}], "]"}], "]"}], "]"}], "&"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"vertexDecomposable", "=", RowBox[{"Length", "[", RowBox[{"Select", "[", RowBox[{"valid", ",", RowBox[{ RowBox[{"With", "[", RowBox[{ RowBox[{"{", RowBox[{"components", "=", RowBox[{"ConnectedComponents", "[", RowBox[{"VertexDelete", "[", RowBox[{"dualGraph", ",", "#"}], "]"}], "]"}]}], "}"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"KVertexConnectedGraphQ", "[", RowBox[{ RowBox[{"Subgraph", "[", RowBox[{"dualGraph", ",", RowBox[{"Join", "[", RowBox[{ RowBox[{ "components", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}], ",", "#"}], "]"}]}], "]"}], ",", "3"}], "]"}], "&&", RowBox[{"KVertexConnectedGraphQ", "[", RowBox[{ RowBox[{"Subgraph", "[", RowBox[{"dualGraph", ",", RowBox[{"Join", "[", RowBox[{ RowBox[{ "components", "\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}], ",", "#"}], "]"}]}], "]"}], ",", "3"}], "]"}]}]}], "]"}], "&"}]}], "]"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"faceDecomposable", ",", "vertexDecomposable"}], "}"}]}]}], "\[IndentingNewLine]", "]"}]}]], "Input", CellChangeTimes->{{3.832340233432548*^9, 3.832340427782093*^9}, { 3.8323404633434258`*^9, 3.832340465470111*^9}, {3.832341064134386*^9, 3.832341105249135*^9}, {3.832341215107959*^9, 3.832341309459516*^9}, { 3.83234151306717*^9, 3.8323415312540283`*^9}}, CellLabel->"In[7]:=",ExpressionUUID->"58712807-a428-4b89-8b30-be4ce33480f8"], Cell["\<\ Function to read data from a file and convert it into a list of graphs. The \ easiest way to use this is to download a massive list from plantri (in graph6 \ format if it is very large, to keep the file small), and give that filename \ here. I highly recommend you put a semicolon at the end of that line to \ suppress the output; it tends to be pretty massive.\ \>", "Text", CellChangeTimes->{{3.832358728187701*^9, 3.832358787855624*^9}, { 3.832358819591075*^9, 3.8323588520926237`*^9}},ExpressionUUID->"ab257888-085b-4fde-8617-\ 8abd299bbd66"], Cell[BoxData[ RowBox[{ RowBox[{"readData", "[", "filename_", "]"}], ":=", RowBox[{ RowBox[{ RowBox[{"graphFromPlantri", "[", RowBox[{"#", ",", RowBox[{"format", "\[Rule]", "graph6"}]}], "]"}], "&"}], "/@", RowBox[{ RowBox[{"Flatten", "[", RowBox[{"StringSplit", "[", RowBox[{ RowBox[{"ReadList", "[", RowBox[{ RowBox[{"FileNameJoin", "[", RowBox[{"{", RowBox[{ RowBox[{"NotebookDirectory", "[", "]"}], ",", "filename"}], "}"}], "]"}], ",", "String"}], "]"}], ",", "\"\<: \>\""}], "]"}], "]"}], "\[LeftDoubleBracket]", RowBox[{"2", ";;", RowBox[{"-", "1"}], ";;", "2"}], "\[RightDoubleBracket]"}]}]}]], "Input",\ CellChangeTimes->{{3.8323510610419703`*^9, 3.832351096305499*^9}, { 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