Plot relationships of graph classes

Subset[a_,b_]:=a⋂b==a

classes=GraphData["Classes"];
graphs={};
AppendTo[graphs,Sort[GraphData[#]]]&/@classes;
adjmat=Outer[Boole[#1=!=#2∧#1⊂#2]&,graphs,graphs,1,1];

(* show counts of classes *)
counts=Total[adjmat];
decorated=Transpose[{Total[adjmat],classes}];
Sort[decorated]//Reverse//TableForm

(* select interesting classes *)
intr=Select[Range[Length[adjmat]],0<Total[adjmat][[#]]≤44∧classes[[#]]≠"Class1"∧classes[[#]]≠"Class2"&];
intrCl=classes[[intr]];
adjmat2=adjmat[[intr,intr]];
irrel[k_]:=Total[adjmat2[[k]]]==Total[adjmat2[[All,k]]]==0

(* remove classes with no connections *)
intr2=Select[Range[Length[adjmat2]],Not[irrel[#]]&];
intrCl2=intrCl[[intr2]];
adjmat3=adjmat2[[intr2,intr2]];

AdjacencyGraph[adjmat3,VertexLabels->(#->intrCl2[[#]]&/@Range[Length[adjmat3]]),EdgeShapeFunction->GraphElementData[{"HalfFilledArrow","ArrowSize"->.01}] ]

Test graph class membership

(* returns list of graph classes to which g belongs *)
graphClasses[g_]:=(
propPairs={{AcyclicGraphQ,"Acyclic"},{BipartiteGraphQ,"Bipartite"},{ConnectedGraphQ,"Connected"},{EulerianGraphQ,"Eulerian"},{HamiltonianGraphQ,"Hamiltonian"},{PathGraphQ,"Path"},{TreeGraphQ,"Tree"}};
Select[propPairs,#[[1]][g]&][[All,2]]
);

(* gets graphs with same class and number of vertices *)
getEquivalentGraphs[g_]:=(
vertGraphs=GraphData[VertexCount[g]];
typeGraphs=GraphData[graphClasses[g]];
vertGraphs⋂typeGraphs
);

findGraph[g_]:=(
Select[getEquivalentGraphs[g],IsomorphicGraphQ[g,GraphData[#]]&]
);

findGraph[LineGraph[CompleteGraph[{3,3}]]]

Testing Combinatorica graph classes (very slow, not fully tested)

(* give alias to all Combinatorica functions and set $Post to remove Combinatorica from ContextPath automatically *)
Needs["Combinatorica`"];
combNames=Names["Combinatorica`*"];
{Set@@{ToExpression["c"<>Last[StringSplit[#,"`"]]],ToExpression[#]}}&/@Names["Combinatorica`*"];
Needs["GraphUtilities`"];
s2c=GraphUtilities`ToCombinatoricaGraph;
(* remove Combinatorica for CompleteGraph and GraphUtilities for VertexList *)
$Post=($ContextPath=DeleteCases[$ContextPath,"Combinatorica`"|"GraphUtilities`"];#)&;

cPropPairs={{cBiconnectedQ,"Biconnected"},{cPerfectQ,"Perfect"},{cPlanarQ,"Planar"},{cRegularQ,"Regular"},{cSelfComplementaryQ,"SelfComplementary"},{cSymmetricQ,"Symmetric"}};

cGraphClasses[g_]:=(
cPropPairs={{cBiconnectedQ,"Biconnected"},{cPerfectQ,"Perfect"},{cPlanarQ,"Planar"},{cRegularQ,"Regular"},{cSelfComplementaryQ,"SelfComplementary"},{cSymmetricQ,"Symmetric"}};
Select[cPropPairs,#[[1]][s2c[g]]&][[All,2]]
);