Minimum Total Dominating Set

Author

Eric W. Weisstein
April 12, 2025

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GraphTheory/MinimumTotalDominatingSet.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/MinimumTotalDominatingSet.html.

©2025 Wolfram Research, Inc. except for portions noted otherwise

Counts table

GraphDataCountTable["MinimumTotalDominatingSetCount",{"Alkane","AlternatingGroupGraph",
"Andrasfai"->016789, (* for n > 1 *)
{"Antelope",2},"Antiprism"->302652,"Apollonian","Barbell"->000012,{"Bishop",2}->321673,{"BlackBishop",2}->303141,"Book","Bruhat"->323517,"Centipede"->000012,"CocktailParty"->046092,"Complete"->000217,{"CompleteBipartite",2}->000290,{"CompleteTripartite",3}->033428,"CrossedPrism"->302946,"Crown"->000537,"CubeConnectedCycle","Cycle"->302653,"CycleComplement"->055998,"DiagonalIntersection"->347641,
"Dipyramid"->005843, (* for n > 4 *)
"Empty"->000007,"FibonacciCube"->323609,{"Fiveleaper",2}->382743,"Flower"->382534,"FoldedCube"->347642,"Gear"->141310,"Goldberg"->382657,{"Grid",2}->303142,{"Grid",3}->303899,"HalvedCube"->323449,"Hanoi"->382695,"Helm"->000012,{"HexagonalGrid",3},"Hypercube"->323515,"Keller"->323489,{"King",2}->303156,{"Knight",2}->323549,"Ladder"->303054,"LadderRung"->000012,"LindgrenSousselier"->382656,"LucasCube"->382533,"MengerSponge","MoebiusLadder"->303046,"Mycielski"->303066,"Odd"->38279,"Pan"->347650,"Pasechnik","Path"->302654,"PathComplement"->000217,"Pell"->382548,"PermutationStar","Prism"->303053,{"Queen",2}->002566,{"Rook",2}->303211,{"RookComplement",2}->303212,"SierpinskiCarpet","SierpinskiGasket"->383022,"SierpinskiTetrahedron"->323516,"Star"->001477,"Sun"->141310,"Sunlet"->000012,"Tetrahedral"->319741,{"TorusGrid",2}->303213,"Transposition"->303143,"Triangular"->304561,"TriangularGrid"->304562,"TriangularHoneycombAcuteKnight"->304563,"TriangularHoneycombBishop"->304564,"TriangularHoneycombObtuseKnight"->304565,"TriangularHoneycombQueen"->304566,"TriangularHoneycombRook"->000007,"TriangularSnake","Web"->000012,
"Wheel"->000027, (* for n > 5 *)
{"WhiteBishop",2}->303144}]

Code

<<MathWorld`Graphs`

transformations[n_,points_]:=Module[{rot90,rot180,rot270,reflH,reflV,reflD1,reflD2},
  rot90={n+1-#2,#} & @@@ points;
  rot180={n+1-#,n+1-#2} & @@@ points;
  rot270={#2,n+1-#} & @@@ points; 
  reflH={#,n+1-#2} & @@@ points;
  reflV={n+1-#,#2} & @@@ points;
  reflD1=Reverse /@ points;
  reflD2={n+1-#2,n+1-#} & @@@ points;
  Union[{points,rot90,rot180,rot270,reflH,reflV,reflD1,reflD2}]
]

SymmetricallyExpandedConfigurations[g_Graph,sets_]:=Module[{emb=GraphEmbedding[g],n=Sqrt[VertexCount[g]],rules},
rules=Thread[emb->Range[n^2]];
Sort[Sort/@Union[Flatten[Union[Sort/@(transformations[n,emb[[#]]])]&/@sets,1]]/.rules]
]

canonicalorientation[n_,points_]:=Union[Sort/@transformations[n,points]][[1]]

DistinctArrangements[g_Graph,sets_]:=Module[{n=Sqrt[VertexCount[g]],emb=GraphEmbedding[g]},
If[Min[Flatten[emb]]==0,emb+=1];
Union[canonicalorientation[n,emb[[#]]]&/@sets]
]

DistinctArrangementsWithTallies[g_Graph,{}]:={}
DistinctArrangementsWithTallies[g_Graph,sets_]:=Module[{n=Sqrt[VertexCount[g]],emb=GraphEmbedding[g],mins},
mins=Min/@Transpose[emb];
emb=#+1-mins&/@emb;
Counts[canonicalorientation[n,emb[[#]]]&/@sets]
]

VerticalDividers[lens_,n_]:=Append[Flatten[Join[{Thick},Table[True,#-1]]&/@Ceiling[lens/n]],Thick]

SymmetricallyDistinctTallyGraphic[class_String,nmax_Integer,rows_:10]:=SymmetricallyDistinctTallyGraphic[class,Range[nmax],rows]

SymmetricallyDistinctTallyGraphic[class_String,prop_,nrange_List,rows_:10,opts___?OptionQ]:=Module[
{
table,res,
label=If[Length[nrange]==1,First@nrange,Style["n",Italic]],
mesh=Lookup[{opts},Mesh,Automatic]
},
table=Table[
ArrayPlot[
SparseArray[Evaluate@Thread[Reverse/@#->1],{n,n}],Mesh->Replace[mesh,Automatic->(n<11)],DataRange->{{0,n},{0,n}},
PlotLabel->#2
]&@@@SortBy[Normal[DistinctArrangementsWithTallies[
GraphData[{class,{n,n}},"Graph","Grid"][[1]],
If[MissingQ[res=GraphData[{class,{n,n}},prop]],Import[
FileNameJoin[{$GraphDataDirectory,"Properties",prop,GraphNameToFileName[{class,{n,n}}]<>".gz"}]],
res]]],
Last
],
{n,nrange}];
Labeled[
GraphicsGrid[
Flatten[Partition[#,UpTo[rows]]&/@table,1],
Dividers->{Thick,VerticalDividers[Length/@table,rows]},
ImageSize->1000
],
Style[Row[{label," × ",label," "<>class}],Directive[18,Bold,FontFamily->"Times"]],
Top
]
]

Parametrized families

<<MathWorld`Graphs`

Andrasfai

seq=Table[GraphData[{"Andrasfai",n},"MinimumTotalDominatingSetCount"],{n,20}]

FindSequenceFunction[Rest@DeleteMissing@seq,n-1]//Factor

gf=x FindGeneratingFunction[DeleteMissing@seq,x]

SeriesCoefficient[gf,{x,0,n}]

Antelope n×n

seq=Table[GraphData[{"Antelope",{n,n}},"MinimumTotalDominatingSetCount"],{n,20}]

Antiprism

A302652

Entity["Graph",{"Antiprism",n}]["MinimumTotalDominatingSetCount"]

seq=Table[GraphData[{"Antiprism",n},"MinimumTotalDominatingSetCount"],{n,3,25}]

(seq=Table[Length@FindTotalDominatingSet[GraphData[{"Antiprism",n}],{GraphData[{"Antiprism",n},"TotalDominationNumber"]}//RootReduce,All],{n,3,14}])//Timing

seq=Import["https://oeis.org/A302652/b302652.txt","Table"][[All,2]]

coef=FindLinearRecurrence[seq]

Length@coef

Take[seq,Length@coef]

LinearRecurrence[{0,0,0,0,0,0,5,0,0,0,0,0,0,-10,0,0,0,0,0,0,10,0,0,0,0,0,0,-5,0,0,0,0,0,0,1},{2,6,12,24,80,48,7,16,237,40,154,1344,208,7,30,1136,68,396,6688,480,7,44,3151,96,750,20800,864,7,58,6730,124,1216,50160,1360,7},200]==seq

s0=seq[[7;;;;7]]

FindSequenceFunction[s0,n/7]

s1=seq[[;;;;7]]

FindSequenceFunction[s1,(n+6)/7]

s2=seq[[2;;;;7]]

FindSequenceFunction[s2,(n+5)/7]//Factor

s3=seq[[3;;;;7]]

FindSequenceFunction[s3,(n+4)/7]//Factor

s4=seq[[4;;;;7]]

FindSequenceFunction[s4,(n+3)/7]//Factor

s5=seq[[5;;;;7]]

FindSequenceFunction[s5,(n+2)/7]//Factor

s6=seq[[6;;;;7]]

FindSequenceFunction[s6,(n+1)/7]//Factor

pieces={{7,Mod[n,7]==0},{2n,Mod[n,7]==1},{n (37+138 n+32 (n^2))/147,Mod[n,7]==2},{4n,Mod[n,7]==3},{2 n (5+4 n)/7,Mod[n,7]==4},{((8 n (2+n) (9+n) (1+4 n))/1029),Mod[n,7]==5},{8 n (1+n)/7,Mod[n,7]==6}};

Table[Piecewise[pieces],{n,200}]==seq

GeneratingFunction[# Boole[#2],n,x]&@@@pieces

gf=Total[{((7(x^7))/(1-(x^7))),((2 x (1+6 (x^7)))/(((-1+(x^7)))^2)),(((x^2) (6+213 (x^7)+224 (x^14)+5 (x^21)))/(((-1+(x^7)))^4)),((4 (x^3) (3+4 (x^7)))/(((-1+(x^7)))^2)),-((2 (x^4) (12+41 (x^7)+3 (x^14)))/(((-1+(x^7)))^3)),-((16 (x^5) (5+59 (x^7)+48 (x^14)))/(((-1+(x^7)))^5)),-((16 (x^6) (3+4 (x^7)))/(((-1+(x^7)))^3))}]

Rest@CoefficientList[Series[gf,{x,0,200}],x]==seq

Refine[SeriesCoefficient[gf,{x,0,n}],n>0]

FullSimplify[%,n∈Integers]

a[n_]:=(470/2401)+((((-1))^(1+((2 n)/7))) (69666-143544 (((-1))^(1/7))+103515 (((-1))^(2/7))-103045 (((-1))^(3/7))+88494 (((-1))^(4/7))-88319 (((-1))^(5/7))+55992 (((-1))^(6/7))+158160 (((-1))^(2 n/7))+231863 (((-1))^((2 (1+n))/7))-13029 (((-1))^((4 (1+n))/7))+159507 (((-1))^((2 (2+n))/7))+172711 (((-1))^((2 (3+n))/7))-232038 (((-1))^((1/7) (1+2 n)))+41551 (((-1))^((2/7) (1+2 n)))-191834 (((-1))^((1/7) (3+2 n)))+470 (((-1))^((2/7) (3+2 n)))-159037 (((-1))^((1/7) (5+2 n)))-73878 (((-1))^((1/7) (1+4 n)))-27000 (((-1))^((1/7) (3+4 n)))-645 (((-1))^((1/7) (5+4 n))))+(143544-69666 (((-1))^(1/7))+55992 (((-1))^(2/7))-88319 (((-1))^(3/7))+88494 (((-1))^(4/7))-103045 (((-1))^(5/7))+103515 (((-1))^(6/7))) (E^(-(2/7) I n π))+(232038-158160 (((-1))^(1/7))+172711 (((-1))^(2/7))-159037 (((-1))^(3/7))+159507 (((-1))^(4/7))-191834 (((-1))^(5/7))+231863 (((-1))^(6/7))) (E^(-(4/7) I n π))+(73878+470 (((-1))^(2/7))-645 (((-1))^(3/7))-13029 (((-1))^(4/7))-27000 (((-1))^(5/7))+41551 (((-1))^(6/7))) (E^(-(6/7) I n π)))/(2401 (3+3 (((-1))^(1/7))-2 (((-1))^(2/7))+(((-1))^(3/7))-(((-1))^(5/7))+2 (((-1))^(6/7))))+((295 (1+n))/2401)+((426 (1+n) (2+n))/2401)+((88 (1+n) (2+n) (3+n))/2401)+((32 (1+n) (2+n) (3+n) (4+n))/7203)+((E^(-(6/7) I n π)) (6333354-5325354 (((-1))^(1/7))+5262942 (((-1))^(2/7))-6152862 (((-1))^(3/7))+7508247 (((-1))^(4/7))-8150709 (((-1))^(5/7))+7545954 (((-1))^(6/7))+(E^((8 I n π)/7)) (76286295-368042 n)+(E^((1/7) I (π+10 n π))) (95682249-265515 n)+(E^((2/7) I (3+4 n) π)) (76936701-261545 n)+(37955+215715 (((-1))^(1/7))-308598 (((-1))^(2/7))+184293 (((-1))^(3/7))+104998 (((-1))^(4/7))-303445 (((-1))^(5/7))+262208 (((-1))^(6/7))) n-3 (E^((1/7) I (5+8 n) π)) (26237041+16527 n)+(E^((1/7) I (5+12 n) π)) (6333354+37955 n)+(E^((2/7) I (π+5 n π))) (-96838857+42505 n)+6 (E^((10 I n π)/7)) (-15871642+46553 n)-6 (E^((3/7) I (π+4 n π))) (-877157+51433 n)+2 (E^((2/7) I (π+4 n π))) (39530094+54941 n)+3 (E^((5/7) I (π+2 n π))) (32257229+56064 n)+3 (E^((2/7) I (π+6 n π))) (-2050954+61431 n)+3 (E^((1/7) I (3+10 n) π)) (32577293+70124 n)+3 (E^((4/7) I (π+3 n π))) (-1775118+71905 n)+(E^((1/7) I (π+12 n π))) (7508247+104998 n)-3 (E^((2/7) I (2+5 n) π)) (32583339+105121 n)+(E^((2/7) I (3+5 n) π)) (-95607459+106589 n)-2 (E^((6/7) I (π+2 n π))) (3772977+131104 n)+(E^((1/7) I (π+8 n π))) (-77192901+215203 n)-(E^((12 I n π)/7)) (8150709+303445 n)+(E^((4/7) I (π+2 n π))) (80254656+322142 n)-(E^((1/7) I (3+8 n) π)) (80492886+363185 n)+(E^((2 I n π)/7)) (96771687-97750017 (((-1))^(1/7))+97731879 (((-1))^(2/7))-96838857 (((-1))^(3/7))+95682249 (((-1))^(4/7))-95229852 (((-1))^(5/7))+95607459 (((-1))^(6/7))+(168192-315363 (((-1))^(1/7))+210372 (((-1))^(2/7))+42505 (((-1))^(3/7))-265515 (((-1))^(4/7))+279318 (((-1))^(5/7))-106589 (((-1))^(6/7))) n)+(E^((4 I n π)/7)) (-78711123+80254656 (((-1))^(1/7))-80492886 (((-1))^(2/7))+79060188 (((-1))^(3/7))-77192901 (((-1))^(4/7))+76286295 (((-1))^(5/7))-76936701 (((-1))^(6/7))+(-49581+322142 (((-1))^(1/7))-363185 (((-1))^(2/7))+109882 (((-1))^(3/7))+215203 (((-1))^(4/7))-368042 (((-1))^(5/7))+261545 (((-1))^(6/7))) n)))/(2401 (17-20 (((-1))^(1/7))+9 (((-1))^(2/7))+9 (((-1))^(3/7))-20 (((-1))^(4/7))+17 (((-1))^(5/7))))+(4/49) (-(((((-1))^(1/7)) (((-(((-1))^(3/7))))^n) (780044-356550 (((-1))^(1/7))+607830 (((-1))^(2/7))-1042214 (((-1))^(3/7))+2508836 (((-1))^(4/7))-3716734 (((-1))^(5/7))+2333714 (((-1))^(6/7))-166462 n+249717 (((-1))^(1/7)) n-308501 (((-1))^(2/7)) n+216460 (((-1))^(3/7)) n+254294 (((-1))^(4/7)) n-543833 (((-1))^(5/7)) n+143013 (((-1))^(6/7)) n-5888 (n^2)+9617 (((-1))^(1/7)) (n^2)-19923 (((-1))^(2/7)) (n^2)+15292 (((-1))^(3/7)) (n^2)+15752 (((-1))^(4/7)) (n^2)-29539 (((-1))^(5/7)) (n^2)+5429 (((-1))^(6/7)) (n^2)))/(2 (((-1+(((-1))^(1/7))))^8) (((1+(((-1))^(1/7))))^17) (((1-(((-1))^(1/7))+(((-1))^(2/7))))^5)))-((((-1))^((5/7)+((4 n)/7))) (-356550+780044 (((-1))^(1/7))-2333714 (((-1))^(2/7))+3716734 (((-1))^(3/7))-2508836 (((-1))^(4/7))+1042214 (((-1))^(5/7))-607830 (((-1))^(6/7))+249717 n-166462 (((-1))^(1/7)) n-143013 (((-1))^(2/7)) n+543833 (((-1))^(3/7)) n-254294 (((-1))^(4/7)) n-216460 (((-1))^(5/7)) n+308501 (((-1))^(6/7)) n+9617 (n^2)-5888 (((-1))^(1/7)) (n^2)-5429 (((-1))^(2/7)) (n^2)+29539 (((-1))^(3/7)) (n^2)-15752 (((-1))^(4/7)) (n^2)-15292 (((-1))^(5/7)) (n^2)+19923 (((-1))^(6/7)) (n^2)))/(2 (((-1+(((-1))^(1/7))))^8) (((1+(((-1))^(1/7))))^17) (((1-(((-1))^(1/7))+(((-1))^(2/7))))^5))+((((-1))^(2 n/7)) (-14074988+15475140 (((-1))^(1/7))-16849758 (((-1))^(2/7))+13472784 (((-1))^(3/7))-11427620 (((-1))^(4/7))+9641806 (((-1))^(5/7))-10681576 (((-1))^(6/7))-200890 n+641152 (((-1))^(1/7)) n-988908 (((-1))^(2/7)) n+50260 (((-1))^(3/7)) n+357357 (((-1))^(4/7)) n-975593 (((-1))^(5/7)) n+548260 (((-1))^(6/7)) n-9126 (n^2)+37616 (((-1))^(1/7)) (n^2)-53478 (((-1))^(2/7)) (n^2)-1422 (((-1))^(3/7)) (n^2)+16917 (((-1))^(4/7)) (n^2)-57987 (((-1))^(5/7)) (n^2)+23894 (((-1))^(6/7)) (n^2)))/(2 (((-1+(((-1))^(1/7))))^8) (((1+(((-1))^(1/7))))^17) (((1+(((-1))^(2/7))))^5) (((1-(((-1))^(1/7))+(((-1))^(2/7))))^5))+((((-1))^(5/7)) (((-(((-1))^(5/7))))^n) (-9641806+11427620 (((-1))^(1/7))-13472784 (((-1))^(2/7))+16849758 (((-1))^(3/7))-15475140 (((-1))^(4/7))+14074988 (((-1))^(5/7))-10681576 (((-1))^(6/7))+975593 n-357357 (((-1))^(1/7)) n-50260 (((-1))^(2/7)) n+988908 (((-1))^(3/7)) n-641152 (((-1))^(4/7)) n+200890 (((-1))^(5/7)) n+548260 (((-1))^(6/7)) n+57987 (n^2)-16917 (((-1))^(1/7)) (n^2)+1422 (((-1))^(2/7)) (n^2)+53478 (((-1))^(3/7)) (n^2)-37616 (((-1))^(4/7)) (n^2)+9126 (((-1))^(5/7)) (n^2)+23894 (((-1))^(6/7)) (n^2)))/(2 (((-1+(((-1))^(1/7))))^8) (((1+(((-1))^(1/7))))^17) (((1+(((-1))^(2/7))))^5) (((1-(((-1))^(1/7))+(((-1))^(2/7))))^5))+((((-1))^((3/7)+((6 n)/7))) (-461531304+472534328 (((-1))^(1/7))-482450718 (((-1))^(2/7))+483519054 (((-1))^(3/7))-474834206 (((-1))^(4/7))+463340538 (((-1))^(5/7))-457245008 (((-1))^(6/7))+1195920 n+2272059 (((-1))^(1/7)) n-4093439 (((-1))^(2/7)) n+2811774 (((-1))^(3/7)) n+622615 (((-1))^(4/7)) n-3530401 (((-1))^(5/7)) n+3828336 (((-1))^(6/7)) n-13638 (n^2)+229025 (((-1))^(1/7)) (n^2)-275665 (((-1))^(2/7)) (n^2)+113356 (((-1))^(3/7)) (n^2)+135949 (((-1))^(4/7)) (n^2)-280045 (((-1))^(5/7)) (n^2)+215690 (((-1))^(6/7)) (n^2)))/(2 (((-1+(((-1))^(1/7))))^7) (((1+(((-1))^(1/7))))^17) (((1+(((-1))^(2/7))))^5) (((1-(((-1))^(1/7))+(((-1))^(2/7))))^5) (((1-(((-1))^(1/7))+(((-1))^(2/7))-(((-1))^(3/7))+(((-1))^(4/7))))^5))-((((-1))^(2/7)) (((-(((-1))^(1/7))))^n) (-958353260+965969772 (((-1))^(1/7))-954985046 (((-1))^(2/7))+934065632 (((-1))^(3/7))-918776312 (((-1))^(4/7))+920585546 (((-1))^(5/7))-938174744 (((-1))^(6/7))-2189159 n+6905213 (((-1))^(1/7)) n-6365498 (((-1))^(2/7)) n+1076139 (((-1))^(3/7)) n+5024256 (((-1))^(4/7)) n-7358737 (((-1))^(5/7)) n+4153016 (((-1))^(6/7)) n+22593 (n^2)+389021 (((-1))^(1/7)) (n^2)-504690 (((-1))^(2/7)) (n^2)+242663 (((-1))^(3/7)) (n^2)+202052 (((-1))^(4/7)) (n^2)-495735 (((-1))^(5/7)) (n^2)+415994 (((-1))^(6/7)) (n^2)))/(2 (((-1+(((-1))^(1/7))))^8) (((1+(((-1))^(1/7))))^17) (((1+(((-1))^(2/7))))^5) (((1-(((-1))^(1/7))+(((-1))^(2/7))))^5) (((1-(((-1))^(1/7))+(((-1))^(2/7))-(((-1))^(3/7))+(((-1))^(4/7))))^5)))-(16/7) (((((-1))^(4/7)) (((-(((-1))^(3/7))))^n) (-3019062+5106174 (((-1))^(1/7))-2915658 (((-1))^(2/7))+1362 (((-1))^(3/7))+1255230 (((-1))^(4/7))-757884 (((-1))^(5/7))-412458 (((-1))^(6/7))-506856 n+1113404 (((-1))^(1/7)) n-424536 (((-1))^(2/7)) n-274774 (((-1))^(3/7)) n+624839 (((-1))^(4/7)) n-721741 (((-1))^(5/7)) n+423442 (((-1))^(6/7)) n-42903 (n^2)+88758 (((-1))^(1/7)) (n^2)-30945 (((-1))^(2/7)) (n^2)-9351 (((-1))^(3/7)) (n^2)+38163 (((-1))^(4/7)) (n^2)-68475 (((-1))^(5/7)) (n^2)+46617 (((-1))^(6/7)) (n^2)-1035 (n^3)+1900 (((-1))^(1/7)) (n^3)-585 (((-1))^(2/7)) (n^3)+247 (((-1))^(3/7)) (n^3)+472 (((-1))^(4/7)) (n^3)-1880 (((-1))^(5/7)) (n^3)+1415 (((-1))^(6/7)) (n^3)))/(6 (((-1+(((-1))^(1/7))))^10) (((1+(((-1))^(1/7))))^23) (((1-(((-1))^(1/7))+(((-1))^(2/7))))^7))-((((-1))^((4/7)+((4 n)/7))) (1255230+1362 (((-1))^(1/7))-2915658 (((-1))^(2/7))+5106174 (((-1))^(3/7))-3019062 (((-1))^(4/7))+412458 (((-1))^(5/7))+757884 (((-1))^(6/7))+624839 n-274774 (((-1))^(1/7)) n-424536 (((-1))^(2/7)) n+1113404 (((-1))^(3/7)) n-506856 (((-1))^(4/7)) n-423442 (((-1))^(5/7)) n+721741 (((-1))^(6/7)) n+38163 (n^2)-9351 (((-1))^(1/7)) (n^2)-30945 (((-1))^(2/7)) (n^2)+88758 (((-1))^(3/7)) (n^2)-42903 (((-1))^(4/7)) (n^2)-46617 (((-1))^(5/7)) (n^2)+68475 (((-1))^(6/7)) (n^2)+472 (n^3)+247 (((-1))^(1/7)) (n^3)-585 (((-1))^(2/7)) (n^3)+1900 (((-1))^(3/7)) (n^3)-1035 (((-1))^(4/7)) (n^3)-1415 (((-1))^(5/7)) (n^3)+1880 (((-1))^(6/7)) (n^3)))/(6 (((-1+(((-1))^(1/7))))^10) (((1+(((-1))^(1/7))))^23) (((1-(((-1))^(1/7))+(((-1))^(2/7))))^7))+((((-1))^(2/7)) (((-(((-1))^(1/7))))^n) (56796269154-56558746008 (((-1))^(1/7))+56318725068 (((-1))^(2/7))-56256767172 (((-1))^(3/7))+56419601322 (((-1))^(4/7))-56684588994 (((-1))^(5/7))+56852298402 (((-1))^(6/7))+99964654 n-51149425 (((-1))^(1/7)) n-36149043 (((-1))^(2/7)) n+96247720 (((-1))^(3/7)) n-83869793 (((-1))^(4/7)) n+8341213 (((-1))^(5/7)) n+73493690 (((-1))^(6/7)) n+10107705 (n^2)-8681619 (((-1))^(1/7)) (n^2)+720912 (((-1))^(2/7)) (n^2)+7784448 (((-1))^(3/7)) (n^2)-10427958 (((-1))^(4/7)) (n^2)+5219289 (((-1))^(5/7)) (n^2)+3921459 (((-1))^(6/7)) (n^2)+276023 (n^3)-339572 (((-1))^(1/7)) (n^3)+147483 (((-1))^(2/7)) (n^3)+155708 (((-1))^(3/7)) (n^3)-341647 (((-1))^(4/7)) (n^3)+270320 (((-1))^(5/7)) (n^3)+4597 (((-1))^(6/7)) (n^3)))/(6 (((-1+(((-1))^(1/7))))^11) (((1+(((-1))^(1/7))))^23) (((1+(((-1))^(2/7))))^7) (((1-(((-1))^(1/7))+(((-1))^(2/7))))^7) (((1-(((-1))^(1/7))+(((-1))^(2/7))-(((-1))^(3/7))+(((-1))^(4/7))))^7))+((((-1))^(1/7)) (((-(((-1))^(5/7))))^n) (105516972-90438078 (((-1))^(1/7))+85206942 (((-1))^(2/7))-92222970 (((-1))^(3/7))+104466930 (((-1))^(4/7))-117338706 (((-1))^(5/7))+114581682 (((-1))^(6/7))+948415 n+3139110 (((-1))^(1/7)) n-4960514 (((-1))^(2/7)) n+2416947 (((-1))^(3/7)) n+915922 (((-1))^(4/7)) n-4778516 (((-1))^(5/7)) n+3873718 (((-1))^(6/7)) n+66987 (n^2)+229665 (((-1))^(1/7)) (n^2)-416859 (((-1))^(2/7)) (n^2)+183657 (((-1))^(3/7)) (n^2)+59991 (((-1))^(4/7)) (n^2)-381738 (((-1))^(5/7)) (n^2)+321465 (((-1))^(6/7)) (n^2)+1526 (n^3)+3975 (((-1))^(1/7)) (n^3)-9781 (((-1))^(2/7)) (n^3)+4062 (((-1))^(3/7)) (n^3)+533 (((-1))^(4/7)) (n^3)-8092 (((-1))^(5/7)) (n^3)+7667 (((-1))^(6/7)) (n^3)))/(6 (((-1+(((-1))^(1/7))))^11) (((1+(((-1))^(1/7))))^23) (((1+(((-1))^(2/7))))^7) (((1-(((-1))^(1/7))+(((-1))^(2/7))))^7))-((((-1))^((1/7)+((2 n)/7))) (104466930-92222970 (((-1))^(1/7))+85206942 (((-1))^(2/7))-90438078 (((-1))^(3/7))+105516972 (((-1))^(4/7))-114581682 (((-1))^(5/7))+117338706 (((-1))^(6/7))+915922 n+2416947 (((-1))^(1/7)) n-4960514 (((-1))^(2/7)) n+3139110 (((-1))^(3/7)) n+948415 (((-1))^(4/7)) n-3873718 (((-1))^(5/7)) n+4778516 (((-1))^(6/7)) n+59991 (n^2)+183657 (((-1))^(1/7)) (n^2)-416859 (((-1))^(2/7)) (n^2)+229665 (((-1))^(3/7)) (n^2)+66987 (((-1))^(4/7)) (n^2)-321465 (((-1))^(5/7)) (n^2)+381738 (((-1))^(6/7)) (n^2)+533 (n^3)+4062 (((-1))^(1/7)) (n^3)-9781 (((-1))^(2/7)) (n^3)+3975 (((-1))^(3/7)) (n^3)+1526 (((-1))^(4/7)) (n^3)-7667 (((-1))^(5/7)) (n^3)+8092 (((-1))^(6/7)) (n^3)))/(6 (((-1+(((-1))^(1/7))))^11) (((1+(((-1))^(1/7))))^23) (((1+(((-1))^(2/7))))^7) (((1-(((-1))^(1/7))+(((-1))^(2/7))))^7))+((((-1))^((5/7)+((6 n)/7))) (56852298402-56684588994 (((-1))^(1/7))+56419601322 (((-1))^(2/7))-56256767172 (((-1))^(3/7))+56318725068 (((-1))^(4/7))-56558746008 (((-1))^(5/7))+56796269154 (((-1))^(6/7))+73493690 n+8341213 (((-1))^(1/7)) n-83869793 (((-1))^(2/7)) n+96247720 (((-1))^(3/7)) n-36149043 (((-1))^(4/7)) n-51149425 (((-1))^(5/7)) n+99964654 (((-1))^(6/7)) n+3921459 (n^2)+5219289 (((-1))^(1/7)) (n^2)-10427958 (((-1))^(2/7)) (n^2)+7784448 (((-1))^(3/7)) (n^2)+720912 (((-1))^(4/7)) (n^2)-8681619 (((-1))^(5/7)) (n^2)+10107705 (((-1))^(6/7)) (n^2)+4597 (n^3)+270320 (((-1))^(1/7)) (n^3)-341647 (((-1))^(2/7)) (n^3)+155708 (((-1))^(3/7)) (n^3)+147483 (((-1))^(4/7)) (n^3)-339572 (((-1))^(5/7)) (n^3)+276023 (((-1))^(6/7)) (n^3)))/(6 (((-1+(((-1))^(1/7))))^11) (((1+(((-1))^(1/7))))^23) (((1+(((-1))^(2/7))))^7) (((1-(((-1))^(1/7))+(((-1))^(2/7))))^7) (((1-(((-1))^(1/7))+(((-1))^(2/7))-(((-1))^(3/7))+(((-1))^(4/7))))^7)))+256 (-(((((-(((-1))^(3/7))))^n) (-3239784+1852488 (((-1))^(1/7))-573384 (((-1))^(2/7))+188400 (((-1))^(3/7))-1167912 (((-1))^(4/7))+2876712 (((-1))^(5/7))-3789600 (((-1))^(6/7))-426706 n+38800 (((-1))^(1/7)) n+394106 (((-1))^(2/7)) n-573248 (((-1))^(3/7)) n+286102 (((-1))^(4/7)) n+245602 (((-1))^(5/7)) n-542348 (((-1))^(6/7)) n-41477 (n^2)+11552 (((-1))^(1/7)) (n^2)+31423 (((-1))^(2/7)) (n^2)-54018 (((-1))^(3/7)) (n^2)+30135 (((-1))^(4/7)) (n^2)+17265 (((-1))^(5/7)) (n^2)-45282 (((-1))^(6/7)) (n^2)-1322 (n^3)+800 (((-1))^(1/7)) (n^3)+622 (((-1))^(2/7)) (n^3)-1624 (((-1))^(3/7)) (n^3)+1082 (((-1))^(4/7)) (n^3)+182 (((-1))^(5/7)) (n^3)-1024 (((-1))^(6/7)) (n^3)-7 (n^4)+16 (((-1))^(1/7)) (n^4)-7 (((-1))^(2/7)) (n^4)-6 (((-1))^(3/7)) (n^4)+9 (((-1))^(4/7)) (n^4)-9 (((-1))^(5/7)) (n^4)+6 (((-1))^(6/7)) (n^4)))/(24 (((-1+(((-1))^(1/7))))^14) (((1+(((-1))^(1/7))))^28) (((1-(((-1))^(1/7))+(((-1))^(2/7))))^9)))+((((-1))^((3/7)+((4 n)/7))) (-248256-365040 (((-1))^(1/7))+301560 (((-1))^(2/7))+547848 (((-1))^(3/7))-1407240 (((-1))^(4/7))+1527360 (((-1))^(5/7))-1022256 (((-1))^(6/7))+137860 n-288846 (((-1))^(1/7)) n+253502 (((-1))^(2/7)) n+7900 (((-1))^(3/7)) n-278202 (((-1))^(4/7)) n+295046 (((-1))^(5/7)) n-99060 (((-1))^(6/7)) n+16276 (n^2)-25201 (((-1))^(1/7)) (n^2)+20081 (((-1))^(2/7)) (n^2)+2816 (((-1))^(3/7)) (n^2)-27319 (((-1))^(4/7)) (n^2)+26699 (((-1))^(5/7)) (n^2)-4724 (((-1))^(6/7)) (n^2)+680 (n^3)-642 (((-1))^(1/7)) (n^3)+382 (((-1))^(2/7)) (n^3)+200 (((-1))^(3/7)) (n^3)-882 (((-1))^(4/7)) (n^3)+742 (((-1))^(5/7)) (n^3)+120 (((-1))^(6/7)) (n^3)+8 (n^4)+(((-1))^(1/7)) (n^4)-5 (((-1))^(2/7)) (n^4)+4 (((-1))^(3/7)) (n^4)-5 (((-1))^(4/7)) (n^4)+(((-1))^(5/7)) (n^4)+8 (((-1))^(6/7)) (n^4)))/(24 (((-1+(((-1))^(1/7))))^13) (((1+(((-1))^(1/7))))^28) (((1-(((-1))^(1/7))+(((-1))^(2/7))))^9))-((((-1))^(4/7)) (((-(((-1))^(5/7))))^n) (-3555936+9256176 (((-1))^(1/7))-10428264 (((-1))^(2/7))+5620608 (((-1))^(3/7))+1244136 (((-1))^(4/7))-4973424 (((-1))^(5/7))+3104064 (((-1))^(6/7))-294450 n+2055234 (((-1))^(1/7)) n-2339496 (((-1))^(2/7)) n+846600 (((-1))^(3/7)) n+1326796 (((-1))^(4/7)) n-2408134 (((-1))^(5/7)) n+1800850 (((-1))^(6/7)) n-11747 (n^2)+181817 (((-1))^(1/7)) (n^2)-216704 (((-1))^(2/7)) (n^2)+93864 (((-1))^(3/7)) (n^2)+111196 (((-1))^(4/7)) (n^2)-217183 (((-1))^(5/7)) (n^2)+175753 (((-1))^(6/7)) (n^2)+1650 (n^3)+4470 (((-1))^(1/7)) (n^3)-6972 (((-1))^(2/7)) (n^3)+4800 (((-1))^(3/7)) (n^3)+1772 (((-1))^(4/7)) (n^3)-6170 (((-1))^(5/7)) (n^3)+6650 (((-1))^(6/7)) (n^3)+83 (n^4)-17 (((-1))^(1/7)) (n^4)-52 (((-1))^(2/7)) (n^4)+96 (((-1))^(3/7)) (n^4)-52 (((-1))^(4/7)) (n^4)-17 (((-1))^(5/7)) (n^4)+83 (((-1))^(6/7)) (n^4)))/(24 (((-1+(((-1))^(1/7))))^14) (((1+(((-1))^(1/7))))^28) (((1+(((-1))^(2/7))))^9) (((1-(((-1))^(1/7))+(((-1))^(2/7))))^9))+((((-1))^((2/7)+((2 n)/7))) (-209510136+210682224 (((-1))^(1/7))-216382464 (((-1))^(2/7))+223042464 (((-1))^(3/7))-224911824 (((-1))^(4/7))+221182536 (((-1))^(5/7))-214317792 (((-1))^(6/7))+2339496 n-2055234 (((-1))^(1/7)) n+294450 (((-1))^(2/7)) n+1800850 (((-1))^(3/7)) n-2408134 (((-1))^(4/7)) n+1326796 (((-1))^(5/7)) n+846600 (((-1))^(6/7)) n+216704 (n^2)-181817 (((-1))^(1/7)) (n^2)+11747 (((-1))^(2/7)) (n^2)+175753 (((-1))^(3/7)) (n^2)-217183 (((-1))^(4/7)) (n^2)+111196 (((-1))^(5/7)) (n^2)+93864 (((-1))^(6/7)) (n^2)+6972 (n^3)-4470 (((-1))^(1/7)) (n^3)-1650 (((-1))^(2/7)) (n^3)+6650 (((-1))^(3/7)) (n^3)-6170 (((-1))^(4/7)) (n^3)+1772 (((-1))^(5/7)) (n^3)+4800 (((-1))^(6/7)) (n^3)+52 (n^4)+17 (((-1))^(1/7)) (n^4)-83 (((-1))^(2/7)) (n^4)+83 (((-1))^(3/7)) (n^4)-17 (((-1))^(4/7)) (n^4)-52 (((-1))^(5/7)) (n^4)+96 (((-1))^(6/7)) (n^4)))/(24 (((-1+(((-1))^(1/7))))^14) (((1+(((-1))^(1/7))))^28) (((1+(((-1))^(2/7))))^9) (((1-(((-1))^(1/7))+(((-1))^(2/7))))^9))-((((-1))^(2/7)) (((-(((-1))^(1/7))))^n) (-1635102975504+1634730006240 (((-1))^(1/7))-1634421070704 (((-1))^(2/7))+1634408802960 (((-1))^(3/7))-1634702443176 (((-1))^(4/7))+1635080876376 (((-1))^(5/7))-1635259127760 (((-1))^(6/7))-144537342 n+49794400 (((-1))^(1/7)) n+82445042 (((-1))^(2/7)) n-152602142 (((-1))^(3/7)) n+107846528 (((-1))^(4/7)) n+18120628 (((-1))^(5/7)) n-130441342 (((-1))^(6/7)) n-18678271 (n^2)+13364276 (((-1))^(1/7)) (n^2)+2013329 (((-1))^(2/7)) (n^2)-15874891 (((-1))^(3/7)) (n^2)+17782318 (((-1))^(4/7)) (n^2)-6299218 (((-1))^(5/7)) (n^2)-9927209 (((-1))^(6/7)) (n^2)-843138 (n^3)+909800 (((-1))^(1/7)) (n^3)-291362 (((-1))^(2/7)) (n^3)-546478 (((-1))^(3/7)) (n^3)+972808 (((-1))^(4/7)) (n^3)-666592 (((-1))^(5/7)) (n^3)-141578 (((-1))^(6/7)) (n^3)-11345 (n^4)+18196 (((-1))^(1/7)) (n^4)-11345 (((-1))^(2/7)) (n^4)-4049 (((-1))^(3/7)) (n^4)+16394 (((-1))^(4/7)) (n^4)-16394 (((-1))^(5/7)) (n^4)+4049 (((-1))^(6/7)) (n^4)))/(24 (((-1+(((-1))^(1/7))))^14) (((1+(((-1))^(1/7))))^28) (((1+(((-1))^(2/7))))^9) (((1-(((-1))^(1/7))+(((-1))^(2/7))))^9) (((1-(((-1))^(1/7))+(((-1))^(2/7))-(((-1))^(3/7))+(((-1))^(4/7))))^9))+((((-1))^(6 n/7)) (723517166064-723673318320 (((-1))^(1/7))+723495066936 (((-1))^(2/7))-723116633736 (((-1))^(3/7))+722822993520 (((-1))^(4/7))-722835261264 (((-1))^(5/7))+723144196800 (((-1))^(6/7))+144537342 n-130441342 (((-1))^(1/7)) n+18120628 (((-1))^(2/7)) n+107846528 (((-1))^(3/7)) n-152602142 (((-1))^(4/7)) n+82445042 (((-1))^(5/7)) n+49794400 (((-1))^(6/7)) n+18678271 (n^2)-9927209 (((-1))^(1/7)) (n^2)-6299218 (((-1))^(2/7)) (n^2)+17782318 (((-1))^(3/7)) (n^2)-15874891 (((-1))^(4/7)) (n^2)+2013329 (((-1))^(5/7)) (n^2)+13364276 (((-1))^(6/7)) (n^2)+843138 (n^3)-141578 (((-1))^(1/7)) (n^3)-666592 (((-1))^(2/7)) (n^3)+972808 (((-1))^(3/7)) (n^3)-546478 (((-1))^(4/7)) (n^3)-291362 (((-1))^(5/7)) (n^3)+909800 (((-1))^(6/7)) (n^3)+11345 (n^4)+4049 (((-1))^(1/7)) (n^4)-16394 (((-1))^(2/7)) (n^4)+16394 (((-1))^(3/7)) (n^4)-4049 (((-1))^(4/7)) (n^4)-11345 (((-1))^(5/7)) (n^4)+18196 (((-1))^(6/7)) (n^4)))/(24 (((-1+(((-1))^(1/7))))^14) (((1+(((-1))^(1/7))))^28) (((1+(((-1))^(2/7))))^9) (((1-(((-1))^(1/7))+(((-1))^(2/7))))^9) (((1-(((-1))^(1/7))+(((-1))^(2/7))-(((-1))^(3/7))+(((-1))^(4/7))))^9)))

Table[a[n]-a[n-7],{n,8,15}]//Round

Apollonian

seq=Table[GraphData[{"Apollonian",n},"MinimumTotalDominatingSetCount"],{n,6}]

Barbell

seq=Table[GraphData[{"Barbell",n},"MinimumTotalDominatingSetCount"],{n,3,20}]

FindSequenceFunction[DeleteMissing@seq,n-2]

Bishop n×n

A321673

seq=Table[GraphData[{"Bishop",{n,n}},"MinimumTotalDominatingSetCount"],{n,20}]

Times@@@Transpose[Table[GraphData[{#,{n,n}},"MinimumTotalDominatingSetCount"],{n,2,12}]&/@{"BlackBishop","WhiteBishop"}]

BlackBishop n×n

A303141

seq=Table[GraphData[{"BlackBishop",{n,n}},"MinimumTotalDominatingSetCount"],{n,20}]

FindSequenceFunction[DeleteMissing@seq,n]//Head//Timing

FindGeneratingFunction[DeleteMissing@seq,x]//Head//Timing

Book

seq=Table[GraphData[{"Book",n},"MinimumTotalDominatingSetCount"],{n,20}]

FindSequenceFunction[DeleteMissing@Rest@seq,n-1]//FullSimplify

Bruhat

A323517

seq=Table[GraphData[{"Bruhat",n},"MinimumTotalDominatingSetCount"],{n,1,6}]

Camel n×n

seq=Table[GraphData[{"Camel",{n,n}},"MinimumTotalDominatingSetCount"],{n,20}]

Centipede

seq=Table[GraphData[{"Centipede",n},"MinimumTotalDominatingSetCount"],{n,10}]

FindSequenceFunction[DeleteMissing@seq,n]

CocktailParty

seq=Table[GraphData[{"CocktailParty",n},"MinimumTotalDominatingSetCount"],{n,20}]

FindSequenceFunction[DeleteMissing@Rest@seq,n-1]//FullSimplify

Complete

Entity["Graph",{"Complete",n}]["MinimumTotalDominatingSetCount"]

seq=Table[GraphData[{"Complete",n},"MinimumTotalDominatingSetCount"],{n,20}]

FindSequenceFunction[DeleteMissing@Rest@seq,n-1]

CompleteBipartite m×n

Entity["Graph",{"CompleteBipartite",{m,n}}]["MinimumTotalDominatingSetCount"]

Table[GraphData[{"CompleteBipartite",{m,n}},"MinimumTotalDominatingSetCount"],{n,10},{m,n}]//Column

Table[m n,{n,10},{m,n}]//Column

CompleteBipartite n×n

Entity["Graph",{"CompleteBipartite",{n,n}}]["MinimumTotalDominatingSetCount"]

seq=Table[GraphData[{"CompleteBipartite",{n,n}},"MinimumTotalDominatingSetCount"],{n,20}]

FindSequenceFunction[DeleteMissing@seq,n]

CompleteTripartite k×m×n

Entity["Graph",{"CompleteTripartite",{k,m,n}}]["MinimumTotalDominatingSetCount"]

TotalDominationPolynomial:

2-(((1+x))^k)-(((1+x))^m)-(((1+x))^n)+(((1+x))^(k+m+n))

Coefficient of (x^2):

Binomial[k+m+n,2]-Binomial[k,2]-Binomial[m,2]-Binomial[n,2]//Factor

DeleteCases[Flatten[Table[{
{k,m,n},
GraphData[{"CompleteTripartite",{k,m,n}},"MinimumTotalDominatingSetCount"],
k m+k n+m n
},
{n,10},{m,n},{k,m}],2],{_,x_,x_}]

CompleteTripartite n×n×n

Entity["Graph",{"CompleteTripartite",{n,n,n}}]["MinimumTotalDominatingSetCount"]

seq=Table[GraphData[{"CompleteTripartite",{n,n,n}},"MinimumTotalDominatingSetCount"],{n,10}]

FindSequenceFunction[DeleteMissing@seq,n]

FindGeneratingFunction[DeleteMissing@seq,x]

CrossedPrism

A302946

=== MinimalTotalDominatingSetCount

seq=Table[GraphData[{"CrossedPrism",2n},"MinimumTotalDominatingSetCount"],{n,2,10}]

seq={4,36,196,1156,6724,39204,228484,1331716,7761796,45239076,263672644,1536796804,8957108164,52205852196,304278004996,1773462177796,10336495061764,60245508192804,351136554095044,2046573816377476,11928306344169796,69523264248641316};

FindLinearRecurrence[DeleteMissing[seq]]

LinearRecurrence[{5,5,-1},{4,36,196},20]

FindGeneratingFunction[DeleteMissing[seq],x]

CoefficientList[Series[-((4 (-1-4 x+(x^2)))/(1-5 x-5 (x^2)+(x^3))),{x,0,20}],x]

FullSimplify[SeriesCoefficient[-((4 (-1-4 x+(x^2)))/(1-5 x-5 (x^2)+(x^3))),{x,0,n}],n>0]

Table[4(((-1))^n)((ChebyshevT[n,I])^2),{n,10}]

Table[2(ChebyshevT[n,3]+(-1)^n),{n,20}]

2ChebyshevT[Range[0,20],3]

Table[2ChebyshevT[n,3],{n,0,20}]

Table[(((3-2 (Sqrt[2])))^n)+(((3+2 (Sqrt[2])))^n),{n,0,20}]//Expand

Join[{2},Table[((Fibonacci[4n,2])/(Fibonacci[2n,2])),{n,20}]]

FunctionExpand[((Fibonacci[4n,2])/(Fibonacci[2n,2]))]

FullSimplify[%,n∈Integers]

Table[(1+Sqrt[2])^(2n)+(1+Sqrt[2])^(-2n),{n,0,20}]//FullSimplify

Crown

A000537

seq=Table[GraphData[{"Crown",n},"MinimumTotalDominatingSetCount"],{n,3,20}]

FindSequenceFunction[DeleteMissing@seq,n-2]//Factor

CubeConnectedCycle

Entity["Graph",{"CubeConnectedCycle",n}]["MinimumTotalDominatingSetCount"]

seq=Table[GraphData[{"CubeConnectedCycle",n},"MinimumTotalDominatingSetCount"],{n,3,8}]

Cycle

A302653

Entity["Graph",{"Cycle",n}]["MinimumTotalDominatingSetCount"]

seq=Table[GraphData[{"Cycle",n},"MinimumTotalDominatingSetCount"],{n,3,20}]

seq[[;;;;2]]

FindSequenceFunction[DeleteMissing@seq[[;;;;2]],(n-1)/2]//Factor

seq[[2;;;;4]]

seq[[4;;;;4]]

seq=Table[Piecewise[{{n,Mod[n,2]==1},{4,Mod[n,4]==0},{(((n/2))^2),Mod[n,4]==2}}],{n,3,100}]

coef=FindLinearRecurrence[seq]

Length[coef]

Take[seq,Length[coef]]

seq=LinearRecurrence[{0,0,0,3,0,0,0,-3,0,0,0,1},{3,4,5,9,7,4,9,25,11,4,13,49},{-1,30}]

LinearRecurrence[{0,0,0,3,0,0,0,-3,0,0,0,1},{1,1,3,4,5,9,7,4,9,25,11,4},20]

RSolve[Append[Table[a[k]==seq[[k]],{k,12}],a[n]==3a[n-4]-3a[n-8]+a[n-12]],a[n],n]

FullSimplify[%,n∈Integers]

Table[((((-1))^n) (((n-4))^2)+(((4+n))^2)-2 ((n^2)-16) Cos[((n π)/2)])/16,{n,30}]

FindGeneratingFunction[seq,x]

seq=Table[Piecewise[{{(((n/2))^2),Mod[n,4]==2}}],{n,3,20}]

FindGeneratingFunction[seq,x]

Series[%,{x,0,20}]

seq=Table[Piecewise[{{(((n/2))^2),Mod[n,4]==2}}],{n,3,25}]

FindGeneratingFunction[seq,x]

Series[%,{x,0,30}]

seq=Table[Piecewise[{{(((n/2))^2),Mod[n,4]==2}}],{n,3,30}]

FindGeneratingFunction[seq,x]

seq=Table[Piecewise[{{(((n/2))^2),Mod[n,4]==2}}],{n,3,100}]

FindGeneratingFunction[seq,x]

FindGeneratingFunction[Table[Piecewise[{#}],{n,3,20}],x]&/@{{n,Mod[n,2]==1},{4,Mod[n,4]==0},{(((n/2))^2),Mod[n,4]==2}}

Series[((3-(x^2))/(((-1+(x^2)))^2))-((4 x)/(-1+(x^4)))+(x^3) (9-(2450/243) Log[1-((243 (x^4))/98)]+((3529 (98 I π-243 (x^4)+98 Log[98]-98 Log[-98+243 (x^4)]))/59049)),{x,0,20}]

DiagonalIntersection

A347641

seq=Table[GraphData[{"DiagonalIntersection",n},"MinimumTotalDominatingSetCount"],{n,3,14}]

DorogovtsevGoltsevMendes

seq=Table[GraphData[{"DorogovtsevGoltsevMendes",n},"MinimumTotalDominatingSetCount"],{n,7}]

Empty

Entity["Graph",{"Empty",n}]["MinimumTotalDominatingSetCount"]

seq=Table[GraphData[{"Empty",n},"MinimumTotalDominatingSetCount"],{n,30}]

FibonacciCube

A323609

seq=Table[GraphData[{"FibonacciCube",n},"MinimumTotalDominatingSetCount"],{n,15}]

FindSequenceFunction[DeleteMissing@seq,n]//Head//Timing

FindGeneratingFunction[DeleteMissing@seq,x]//Head//Timing

FindLinearRecurrence[Rest@DeleteMissing@seq]//Head//Timing

Fiveleaper n×n

A382743

seq=Table[GraphData[{"Fiveleaper",{n,n}},"MinimumTotalDominatingSetCount"],{n,20}]

Table[GraphData[{"Fiveleaper",{n,n}},"Connected"],{n,20}]

Flower

A382534

seq=Table[GraphData[{"Flower",n},"MinimumTotalDominatingSetCount"],{n,5,20}]

seq[[;;;;4]]

Table[12n+3,{n,5}]

seq[[2;;;;4]]

Table[(6n+3)^2,{n,5}]

Solve[n==4k+2,k]

(6n+3)^2/.n->(1/4) (-2+k)//Simplify

seq[[3;;;;4]]

Table[12n+9,{n,5}]

seq[[4;;;;4]]

seq==Table[Piecewise[{{36,Mod[n,4]==0},{3n,Mod[n,2]==1},{9n^2/4,Mod[n,4]==2}}],{n,5,20}]

seq=Table[Piecewise[{{36,Mod[n,4]==0},{3n,Mod[n,2]==1},{9n^2/4,Mod[n,4]==2}}],{n,40}]

coef=FindLinearRecurrence[seq]

OEISRecurrenceIndexString[coef]

OEISRecurrenceString[coef]

Take[seq,Length@coef]

seq==LinearRecurrence[{0,0,0,3,0,0,0,-3,0,0,0,1},{3,9,9,36,15,81,21,36,27,225,33,36},40]

FindGeneratingFunction[seq,x]//Head//Timing

seq=Table[Piecewise[{{36,Mod[n,4]==0},{3n,Mod[n,2]==1},{9n^2/4,Mod[n,4]==2}}],{n,60}]

FindGeneratingFunction[seq,x]//Head//Timing

SeriesCoefficient[gf,{x,0,n}]

FoldedCube

A347642

seq=Table[GraphData[{"FoldedCube",n},"MinimumTotalDominatingSetCount"],{n,2,10}]

Gear

A141310

seq=Table[GraphData[{"Gear",n},"MinimumTotalDominatingSetCount"],{n,3,20}]

seq=Table[Piecewise[{{n,Mod[n,2]==1},{2,Mod[n,2]==0}}],{n,3,20}]

Table[Floor[(n+1)/2]/(n+1),{n,0,20}]//Denominator

FindLinearRecurrence[seq]

LinearRecurrence[{0,2,0,-1},{3,2,5,2},{-1,20}]

seq=LinearRecurrence[{0,2,0,-1},{2,3,2,5},{0,20}]

FullSimplify[FindSequenceFunction[seq,n+1],n∈Integers]

seq=Table[((((-1))^n)(n-1)+n+3)/2,{n,0,20}]

FindGeneratingFunction[seq,x]

CoefficientList[Series[((1+2 x+(x^2)-2 (x^3))/(((-1+(x^2)))^2)),{x,0,20}],x]

Giraffe n×n

seq=Table[GraphData[{"Giraffe",{n,n}},"MinimumTotalDominatingSetCount"],{n,20}]

Goldberg

A382657

seq=Table[GraphData[{"Goldberg",n},"MinimumTotalDominatingSetCount"],{n,3,20}]

seq[[;;;;5]]

seq[[2;;;;5]]

seq[[3;;;;5]]

seq[[4;;;;5]]

Grid m×n

A303293

Entity["Graph",{"Grid",{m,n}}]["MinimumTotalDominatingSetCount"]

seq=Table[GraphData[{"Grid",{m,n}},"MinimumTotalDominatingSetCount"],{n,10},{m,n}]//Column//TraditionalForm

data=Import["https://oeis.org/A303293/b303293.txt","Table"][[All,2]]

Quiet@Table[data[[Binomial[i,2]+(1/2) (j-1) (2 i+j)+1]],{i,10},{j,i}]/._Part:>Missing["NotAvailable"]//Column//TraditionalForm

Grid n×n

A303142

Entity["Graph",{"Grid",{n,n}}]["MinimumTotalDominatingSetCount"]

seq=Table[GraphData[{"Grid",{n,n}},"MinimumTotalDominatingSetCount"],{n,20}]

seq={0,4,2,16,160,144,4,256,1364,484,6,784,5032,1444,8,2116,12972,3364,10,4624,27376,6724,12,8836,50820,12100,14,15376,86264,20164};

Length[seq]

FindLinearRecurrence[seq]//Head//Timing

FindGeneratingFunction[seq,x]//Head//Timing

s0=seq[[4;;;;4]]

Sqrt[s0]

FindSequenceFunction[Sqrt@s0,n,ValidationLength->3]

s1=seq[[;;;;4]]

FindSequenceFunction[Rest@s1,n]//Factor

s2=seq[[2;;;;4]]

Sqrt[s2]

FindSequenceFunction[Sqrt@Rest@s2,n]

s3=seq[[3;;;;4]]

Grid k×m×n

Entity["Graph",{"Grid",{k,m,n}}]["MinimumTotalDominatingSetCount"]

seq=Table[GraphData[{"Grid",{k,m,n}},"MinimumTotalDominatingSetCount"],{n,10}]

Grid n×n×n

A303899

Entity["Graph",{"Grid",{n,n,n}}]["MinimumTotalDominatingSetCount"]

seq=Table[GraphData[{"Grid",{n,n,n}},"MinimumTotalDominatingSetCount"],{n,10}]

HalvedCube

A323449

seq=Table[GraphData[{"HalvedCube",n},"MinimumTotalDominatingSetCount"],{n,10}]

FindSequenceFunction[DeleteMissing@seq,n]

FindGeneratingFunction[DeleteMissing@seq,x]

Hanoi

A382695

seq=Table[GraphData[{"Hanoi",n},"MinimumTotalDominatingSetCount"],{n,5}]

Helm

seq=Table[GraphData[{"Helm",n},"MinimumTotalDominatingSetCount"],{n,3,20}]

FindSequenceFunction[DeleteMissing@seq,n-2]

Hypercube

A323515

Entity["Graph",{"Hypercube",n}]["MinimumTotalDominatingSetCount"]

seq=Table[GraphData[{"Hypercube",n},"MinimumTotalDominatingSetCount"],{n,0,10}]

FindSequenceFunction[DeleteMissing@seq,n]

Keller

A323489

Entity["Graph",{"Keller",n}]["MinimumTotalDominatingSetCount"]

seq=Table[GraphData[{"Keller",n},"MinimumTotalDominatingSetCount"],{n,5}]

FindSequenceFunction[DeleteMissing@seq,n]

FindSequenceFunction[Rest@DeleteMissing@seq,n]

FindGeneratingFunction[DeleteMissing@seq,x]

King m×n

A303335

Table[GraphData[{"King",{m,n}},"MinimumTotalDominatingSetCount"],{n,10},{m,n}]//Column//TraditionalForm

data=Import["https://oeis.org/A303335/b303335.txt","Table"][[2;;,2]]

Quiet@Table[data[[Binomial[i,2]+(1/2) (j-1) (2 i+j)+1]],{i,10},{j,i}]/._Part:>Missing["NotAvailable"]//Column//TraditionalForm

King n×n

A303156

seq=Table[GraphData[{"King",{n,n}},"MinimumTotalDominatingSetCount"],{n,20}]

FindGeneratingFunction[seq,x]//Head//Timing

FindSequenceFunction[Rest@seq,n-1]//Head//Timing

Knight m×n

Entity["Graph",{"Knight",{m,n}}]["MinimumTotalDominatingSetCount"]

Knight n×n

A323549

Entity["Graph",{"Knight",{n,n}}]["MinimumTotalDominatingSetCount"]

seq=Table[GraphData[{"Knight",{n,n}},"MinimumTotalDominatingSetCount"],{n,20}]

Ladder

A303054

Entity["Graph",{"Ladder",n}]["MinimumTotalDominatingSetCount"]

seq=Table[GraphData[{"Ladder",n},"MinimumTotalDominatingSetCount"],{n,20}]

(seq=Table[Length@FindTotalDominatingSet[GraphData[{"Ladder",n}],{2 Floor[((2+n)/3)]},All],{n,13}])//Timing

seq[[3;;;;3]]

seq[[;;;;3]]

Sqrt[%]

Table[n(n+3)/2-1,{n,5}]

(((n(n+3)/2-1))^2)/.n->(n+2)/3//Simplify

Table[(1/324) (((4+13 n+(n^2)))^2),{n,1,13,3}]

seq[[2;;;;3]]

Table[((((n+4)/3))^2),{n,2,10,3}]

Table[Piecewise[{{1,Mod[n,3]==0},{(1/324) (((4+13 n+(n^2)))^2),Mod[n,3]==1},{((((n+4)/3))^2),Mod[n,3]==2}}],{n,20}]

seq=Table[Piecewise[{{1,Mod[n,3]==0},{((((((n^2)+13n+4)/18)))^2),Mod[n,3]==1},{((((n+4)/3))^2),Mod[n,3]==2}}],{n,100}]

seq=Table[Piecewise[{{1,Mod[n,3]==0},{((n^2+13 n+4)/18)^2,Mod[n,3]==1},{((n+4)/3)^2,Mod[n,3]==2}}],{n,100}]

coef=FindLinearRecurrence[seq]

Length@coef

Take[seq,Length@coef]

LinearRecurrence[{0,0,5,0,0,-10,0,0,10,0,0,-5,0,0,1},{1,4,1,16,9,1,64,16,1,169,25,1,361,36,1},20]

gf=x FindGeneratingFunction[seq,x,ValidationLength->5]

InputForm[gf]

CoefficientList[Series[((-1-4 x-(x^2)-11 (x^3)+11 (x^4)+4 (x^5)+6 (x^6)-11 (x^7)-6 (x^8)+(x^9)+5 (x^10)+4 (x^11)-(x^12)-(x^13)-(x^14))/(((-1+(x^3)))^5)),{x,0,20}],x]

coef=Refine[SeriesCoefficient[gf,{x,0,n}],n>1]

FullSimplify[ComplexExpand@coef,n∈Integers]

Factor[916+n (392+n (213+n (26+n)))]

Factor[-56+n (392+n (213+n (26+n)))]

Factor[(-20+n (7+n)) (28+n (19+n))]

Table[(916+392 n+213 (n^2)+26 (n^3)+(n^4)-(-56+392 n+213 (n^2)+26 (n^3)+(n^4)) Cos[((2 n π)/3)]+(Sqrt[3]) (-20+7 n+(n^2)) (28+19 n+(n^2)) Sin[((2 n π)/3)])/972,{n,20}]

LadderRung

seq=Table[GraphData[{"LadderRung",n},"MinimumTotalDominatingSetCount"],{n,20}]

FindSequenceFunction[seq,n]

LindgrenSousselier

A382656

seq=Table[GraphData[{"LindgrenSousselier",n},"MinimumTotalDominatingSetCount"],{n,11}]

seq==Table[LucasL[2n+1]+4n+2,{n,11}]

FindSequenceFunction[DeleteMissing@seq,n]//FullSimplify

Table[(1/2) (-((((1/2) (3-(Sqrt[5]))))^n) (-1+(Sqrt[5]))+(1+(Sqrt[5])) ((((1/2) (3+(Sqrt[5]))))^n)),{n,20}]//Expand

Table[LucasL[2n+1],{n,11}]

seq=Table[LucasL[2n+1]+4n+2,{n,0,29}]

StringLength@ToString@seq

gf=FindGeneratingFunction[seq,x]//Factor

OEISGeneratingFunctionString[gf,x]

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n≥0&&Element[n,Integers]]

Table[(2^(-1-n)) ((2^(2+n))-(((3-(Sqrt[5])))^n) (-1+(Sqrt[5]))+(1+(Sqrt[5])) (((3+(Sqrt[5])))^n))+4 n,{n,0,20}]//Expand

CoefficientList[Series[(((1+x) (3-8 x+3 (x^2)))/((((-1+x))^2) (1-3 x+(x^2)))),{x,0,20}],x]

coef=FindLinearRecurrence[seq]

LinearRecurrence[{5,-8,5,-1},{10,21,43,94},{0,20}]

OEISRecurrenceString[coef]

OEISRecurrenceIndexString[coef]

LucasCube

A382533

seq=Table[GraphData[{"LucasCube",n},"MinimumTotalDominatingSetCount"],{n,15}]

MengerSponge

seq=Table[GraphData[{"MengerSponge",n},"MinimumTotalDominatingSetCount"],{n,3}]

MoebiusLadder

A303046

Entity["Graph",{"MoebiusLadder",n}]["MinimumTotalDominatingSetCount"]

seq=Table[GraphData[{"MoebiusLadder",n},"MinimumTotalDominatingSetCount"],{n,3,25}]

(seq=Table[Length@FindTotalDominatingSet[GraphData[{"MoebiusLadder",n}],{(1/6) (3+(((-1))^(1+n))+4 n+Cos[((n π)/3)]-3 Cos[((2 n π)/3)]+(Sqrt[3]) Sin[((n π)/3)]+((Sin[((2 n π)/3)])/(Sqrt[3])))},All],{n,3,12}])//Timing

(seq=Table[Length@FindTotalDominatingSet[GraphData[{"MoebiusLadder",n}],{(1/6) (3+(((-1))^(1+n))+4 n+Cos[((n π)/3)]-3 Cos[((2 n π)/3)]+(Sqrt[3]) Sin[((n π)/3)]+((Sin[((2 n π)/3)])/(Sqrt[3])))},All],{n,3,13}])//Timing

seq=Import["https://oeis.org/A303046/b303046.txt","Table"][[All,2]]

FindSequenceFunction[seq,n]

coef=FindLinearRecurrence[seq]

Length[coef]

Take[seq,Length@coef]

LinearRecurrence[{0,0,0,0,0,5,0,0,0,0,0,-10,0,0,0,0,0,10,0,0,0,0,0,-5,0,0,0,0,0,1},{1,6,9,8,25,3,196,56,9,20,121,3,1521,154,9,32,289,3,5776,300,9,44,529,3,15625,494,9,56,841,3},200]==seq

s1=seq[[;;;;6]]

FindSequenceFunction[s1,(n+5)/6]//Factor

s2=seq[[2;;;;6]]

FindSequenceFunction[s2,(n+4)/6]//Factor

s3=seq[[3;;;;6]]

FindSequenceFunction[s3,(n+3)/6]//Factor

s4=seq[[4;;;;6]]

FindSequenceFunction[s4,(n+2)/6]//Factor

s5=seq[[5;;;;6]]

FindSequenceFunction[s5,(n+1)/6]//Factor

s0=seq[[6;;;;6]]

FindSequenceFunction[s0,(n+5)/6]//Factor

Table[Piecewise[{{3,Mod[n,6]==0},{(n(n+5)/6)^2,Mod[n,6]==1},{n(2n+5)/3,Mod[n,6]==2},{9,Mod[n,6]==3},{2n,Mod[n,6]==4},{n^2,Mod[n,6]==5}}],{n,200}]==seq

pieces={{3,Mod[n,6]==0},{(n(n+5)/6)^2,Mod[n,6]==1},{n(2n+5)/3,Mod[n,6]==2},{9,Mod[n,6]==3},{2n,Mod[n,6]==4},{n^2,Mod[n,6]==5}};

GeneratingFunction[#1Boole[#2],n,x]&@@@pieces

gf=Total[{((3(x^6))/(1-(x^6))),-((x (1+191 (x^6)+551 (x^12)+121 (x^18)))/(((-1+(x^6)))^5)),-((2 (x^2) (3+19 (x^6)+2 (x^12)))/(((-1+(x^6)))^3)),-((9 (x^3))/(-1+(x^6))),((4 (x^4) (2+(x^6)))/(((-1+(x^6)))^2)),-(((x^5) (25+46 (x^6)+(x^12)))/(((-1+(x^6)))^3))}]

Rest@CoefficientList[Series[((3 (x^6))/(1-(x^6)))-((9 (x^3))/(-1+(x^6)))+((4 (x^4) (2+(x^6)))/(((-1+(x^6)))^2))-(((x^5) (25+46 (x^6)+(x^12)))/(((-1+(x^6)))^3))-((2 (x^2) (3+19 (x^6)+2 (x^12)))/(((-1+(x^6)))^3))-((x (1+191 (x^6)+551 (x^12)+121 (x^18)))/(((-1+(x^6)))^5)),{x,0,200}],x]==seq

FullSimplify[SeriesCoefficient[gf,{x,0,n}],n∈Integers&&n>0]

a[n_]:=(1/432) (E^(-(2/3) I n π)) (12 (72-36 (((-1))^n)+I (1+(((-1))^n)) (11 I+(Sqrt[3])) n)+(n^2) (-85+37 (((-1))^n)-35 I (Sqrt[3])-13 I (((-1))^n) (Sqrt[3])+(-1+(((-1))^n)) (1-I (Sqrt[3])) n (10+n))-2 (E^((5 I n π)/3)) (216+n (-132+n (37+n (10+n))))+2 (E^((2 I n π)/3)) (432+n (132+n (85+n (10+n))))+(E^((4 I n π)/3)) (864+n (-132-12 I (Sqrt[3])+n (-85+35 I (Sqrt[3])+(-1-I (Sqrt[3])) n (10+n))))+(E^((I n π)/3)) (-432+n (-132-12 I (Sqrt[3])+n (37+13 I (Sqrt[3])+(1+I (Sqrt[3])) n (10+n)))))

FullSimplify[a[n],Mod[n,3]==0]//ExpToTrig

FullSimplify[%,n∈Integers]

Table[6+(((-1))^(1+n)) (1+2 Cos[((2 n π)/3)]),{n,3,20,3}]

Table[6+(((-1))^(1+n)) (1+2 Cos[((2 n π)/3)]),{n,3,20,3}]

FullSimplify[6+(((-1))^(1+n)) (1+2 Cos[((2 n π)/3)])/.n->3n,n∈Integers]

Table[6-3 (((-1))^n),{n,66}]==seq[[3;;;;3]]

Table[6+(((-1))^(1+n))+(((-1))^((3+n)/3))+(((-1))^(1+((5 n)/3))),{n,20}]//FullSimplify

Table[6-(((-1))^n) (1+2 Cos[((2 n π)/3)]),{n,20}]

FullSimplify/@Collect[Refine[ExpToTrig[%],n∈Integers],{Cos[((n π)/3)],Sin[((n π)/3)],Cos[((2 n π)/3)],Sin[((2 n π)/3)],Cos[((4 n π)/3)],Sin[((4 n π)/3)],Cos[((5n π)/3)],Sin[((5 n π)/3)]}]

Mycielski

A303066

=== MinimalTotalDominatingSetCount

Entity["Graph",{"Mycielski",n}]["MinimumTotalDominatingSetCount"]

seq=Table[GraphData[{"Mycielski",n},"MinimumTotalDominatingSetCount"],{n,11}]

Odd

A382795

Entity["Graph",{"Odd",n}]["MinimumTotalDominatingSetCount"]

seq=Table[GraphData[{"Odd",n},"MinimumTotalDominatingSetCount"],{n,7}]

Pan

A347650

seq=Table[GraphData[{"Pan",n},"MinimumTotalDominatingSetCount"],{n,3,20}]

FindGeneratingFunction[DeleteMissing@seq,x]//Head//Timing

Table[Piecewise[{{2,Mod[n,4]==0},{(n+1)/2,Mod[n,4]==1},{2((((n+2)/4))^2),Mod[n,4]==2},{(n+3)/2,Mod[n,4]==3}}],{n,3,20}]

seq[[;;;;4]]

seq[[2;;;;4]]

seq[[3;;;;4]]

seq[[4;;;;4]]

seq=Table[Piecewise[{{(n+3)/2,Mod[n,4]==3},{2,Mod[n,4]==0},{(n+1)/2,Mod[n,4]==1},{(n+2)^2/8,Mod[n,4]==2}}],{n,100}]

StringTake[ToString[seq],{2,262}]

coef=FindLinearRecurrence[seq]

Length[coef]

Take[seq,12]

LinearRecurrence[{0,0,0,3,0,0,0,-3,0,0,0,1},{1,2,3,2,3,8,5,2,5,18,7,2},20]

gf=x FindGeneratingFunction[seq,x,ValidationLength->2]

StringReplace[ToString[gf,InputForm]," "->""]

CoefficientList[Series[((-1-2 x-3 (x^2)-2 (x^3)-2 (x^5)+4 (x^6)+4 (x^7)+(x^8)-(x^10)-2 (x^11))/(((-1+(x^4)))^3)),{x,0,20}],x]

a=Simplify[SeriesCoefficient[gf,{x,0,n}],n>0]

FullSimplify[a,#]&/@{Mod[n,4]==3,Mod[n,4]==0,Mod[n,4]==1,Mod[n,4]==2}

Path

A302654

Entity["Graph",{"Path",n}]["MinimumTotalDominatingSetCount"]

seq=Table[GraphData[{"Path",n},"MinimumTotalDominatingSetCount"],{n,20}]

seq=Table[Piecewise[{{1,Mod[n,4]==0},{((((n+2)/4))^2),Mod[n,4]==2},{(n-1)/4,Mod[n,4]==1},{(n+5)/4,Mod[n,4]==3}}],{n,30}]

FindGeneratingFunction[seq,x]

coef=FindLinearRecurrence[seq]

Length[coef]

Take[seq,Length@coef]

LinearRecurrence[{0,0,0,3,0,0,0,-3,0,0,0,1},{0,1,2,1,1,4,3,1,2,9,4,1},100]

RSolve[Append[Table[a[k]==seq[[k]],{k,12}],a[n]==3a[n-4]-3a[n-8]+a[n-12]],a[n],n]

FullSimplify[(1/64) (36+4 (((-1))^n)+(12-24 I) (((-I))^n)+(12+24 I) (I^n)+12 n-4 (((-1))^n) n-4 (((-I))^n) n-4 (I^n) n+(n^2)+(((-1))^n) (n^2)-(((-I))^n) (n^2)-(I^n) (n^2)),n∈Integers]

Table[((((-1))^n) (((n-2))^2)+(((6+n))^2)-2 (n-2) (n+6) Cos[((n π)/2)]-48 Sin[((n π)/2)])/64,{n,20}]

PathComplement

=== MinimalTotalDominatingSetCount

seq=Table[GraphData[{"PathComplement",n},"MinimumTotalDominatingSetCount"],{n,20}]

FindSequenceFunction[Rest@seq,n-1]//Factor

Pell

A382548

seq=Table[GraphData[{"Pell",n},"MinimumTotalDominatingSetCount"],{n,9}]

Prism

A303053

Entity["Graph",{"Prism",n}]["MinimumTotalDominatingSetCount"]

seq=Table[GraphData[{"Prism",n},"MinimumTotalDominatingSetCount"],{n,3,25}]

(seq=Table[Length@FindTotalDominatingSet[GraphData[{"Prism",n}],{(1/6) (3+(((-1))^n)+4 n-Cos[((n π)/3)]-3 Cos[((2 n π)/3)]-(Sqrt[3]) Sin[((n π)/3)]+((Sin[((2 n π)/3)])/(Sqrt[3])))},All],{n,3,13}])//Timing

(seq=Table[Length@FindTotalDominatingSet[GraphData[{"Prism",n}],{(1/6) (3+(((-1))^n)+4 n-Cos[((n π)/3)]-3 Cos[((2 n π)/3)]-(Sqrt[3]) Sin[((n π)/3)]+((Sin[((2 n π)/3)])/(Sqrt[3])))},All],{n,3,14}])//Timing

seq[[;;;;6]]

seq[[2;;;;6]]

seq[[3;;;;6]]

seq[[4;;;;6]]

seq[[5;;;;6]]

seq[[6;;;;6]]

FindGeneratingFunction[DeleteMissing@seq,x]

Queen n×n

A002566

seq=Table[GraphData[{"Queen",{n,n}},"MinimumTotalDominatingSetCount"],{n,20}]

FindSequenceFunction[Rest@DeleteMissing@seq,n-1]//Head//Timing

FindGeneratingFunction[seq,x]//Head//Timing

Rook m×n

Table[GraphData[{"Rook",{m,n}},"MinimumTotalDominatingSetCount"],{n,10},{m,n}]//Column//TraditionalForm

Rook n×n

A303211

seq=Table[GraphData[{"Rook",{n,n}},"MinimumTotalDominatingSetCount"],{n,20}]

seq=Table[Sum[(((-1))^k)((Binomial[n,k])^2)k!(2If[k==n,1,(((n-k))^(n-k))]-(n-k)!),{k,0,n}],{n,20}]

(((-1))^k)((Binomial[n,k])^2)k!(2If[k==n,1,(((n-k))^(n-k))]-(n-k)!)/.k->n

Sum[(((-1))^k)((Binomial[n,k])^2)k!(2(((n-k))^(n-k))),{k,0,n-1}]

Sum[(((-1))^k)((Binomial[n,k])^2)k!(n-k)!,{k,0,n-1}]

FullSimplify[-(((-1))^(1+n)) n!+(((-1))^n) n!]

seq=2Table[(((-1))^n) n!+Sum[(((-1))^k)((Binomial[n,k])^2)k!(((n-k))^(n-k)),{k,0,n-1}],{n,20}]

FunctionExpand[(((-1))^k)((Binomial[n,k])^2)k!(((n-k))^(n-k))]//FullSimplify

Sum[(((((-1))^k) (((-k+n))^(-k+n)) ((Gamma[1+n])^2))/(Gamma[1+k] ((Gamma[1-k+n])^2))),{k,0,n-1}]

Sum[(((-1))^k)((Binomial[n,k])^2)k!(((n-k))^(n-k)),{k,0,n-1}]

seq2=Table[Sum[(((-1))^k)((Binomial[n,k])^2)k!(((n-k))^(n-k)),{k,0,n-1}],{n,20}]

FindLinearRecurrence[Drop[seq2,2]]//Head

FindSequenceFunction[Drop[seq2,2],n]//Head//Timing

FindGeneratingFunction[seq2,x]//Head//Timing

RookComplement n×n

A303212

seq=Table[GraphData[{"RookComplement",{n,n}},"MinimumTotalDominatingSetCount"],{n,20}]

FindSequenceFunction[Drop[seq,2],n-2]//Factor

Factor[n^2*(2-3*n+n^2)^2/6]

6((Binomial[n,3])^2)

seq=Table[If[n==2,1,6((Binomial[n,3])^2)],{n,20}]

seq=Table[n^2(2-3n+n^2)^2/6,{n,20}]

coef=FindLinearRecurrence[seq]

Length@coef

Take[seq,Length@coef]

Join[{0,1},LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,0,6,96,600,2400,7350},{3,20}]]

gf=x FindGeneratingFunction[seq,x]

CoefficientList[Series[((x (-1+x-75 (x^2)-19 (x^3)-41 (x^4)+21 (x^5)-7 (x^6)+(x^7)))/(((-1+x))^7)),{x,0,20}],x]

SierpinskiCarpet

seq=Table[GraphData[{"SierpinskiCarpet",n},"MinimumTotalDominatingSetCount"],{n,4}]

SierpinskiGasket

A383022

seq=Table[GraphData[{"SierpinskiGasket",n},"MinimumTotalDominatingSetCount"],{n,5}]

SierpinskiTetrahedron

A323516

seq=Table[GraphData[{"SierpinskiTetrahedron",n},"MinimumTotalDominatingSetCount"],{n,6}]

StackedBook

DeleteCases[{#,GraphData[#,"MinimumTotalDominatingSetCount"]}&/@GraphData["StackedBook"],{_,1}|{{"Grid"|"Ladder"|"Path"|"Star",_}|"ClawGraph"|"SquareGraph",_}]

{BookGraph[9,6,VertexLabels->"Name"],BookGraph[3,9,VertexLabels->"Name"]}

Star

Entity["Graph",{"Star",n}]["MinimumTotalDominatingSetCount"]

seq=Table[GraphData[{"Star",n},"MinimumTotalDominatingSetCount"],{n,3,25}]

FindSequenceFunction[DeleteMissing@seq,n-2]

Sun

seq=Table[GraphData[{"Sun",n},"MinimumTotalDominatingSetCount"],{n,3,20}]

Table[Piecewise[{{n,Mod[n,2]==1},{2,Mod[n,2]==0}}],{n,3,20}]

Sunlet

seq=Table[GraphData[{"Sunlet",n},"MinimumTotalDominatingSetCount"],{n,3,20}]

FindSequenceFunction[DeleteMissing@seq,n-2]

Tetrahedral

A319741

seq=Table[GraphData[{"Tetrahedral",n},"MinimumTotalDominatingSetCount"],{n,6,10}]

FindSequenceFunction[DeleteMissing@seq,n-5]//Head//Timing

FindGeneratingFunction[DeleteMissing@seq,x]//Head//Timing

TorusGrid n×n

A303213

seq=Table[GraphData[{"TorusGrid",{n,n}},"MinimumTotalDominatingSetCount"],{n,3,20}]

Transposition

A303143

Matches A060545^2 for a(2)-a(4)

seq=Table[GraphData[{"Transposition",n},"MinimumTotalDominatingSetCount"],{n,6}]

Sqrt[DeleteMissing@seq]

FindSequenceFunction[DeleteMissing@seq,n]

FindGeneratingFunction[DeleteMissing@seq,x]

Triangular

A304561

seq=Table[GraphData[{"Triangular",n},"MinimumTotalDominatingSetCount"],{n,2,20}]

Solve[n==3k+1,k]

{
(18-11*k-21*k^2+32*k^3)*(3*k)!/(18*2^k*k!)/.k->(n/3),
(3*k+1)!/(2^k*k!)/.k->(n-1)/3,
4*k*(3*k+2)!/(3*2^k*k!)/.k->(n-2)/3
}//FullSimplify[#,n∈Integers]&

Expand[486+(n-3) n (33+32 n)]//Factor

seq=Table[Piecewise[{
{(((2^(-(n/3+1))) (486-99 n-63 (n^2)+32 (n^3)) n!)/(243 (n/3)!)),Mod[n,3]==0},
{(((2^((1-n)/3)) n!)/(Gamma[(n+2)/3])),Mod[n,3]==1},
{(((2^( (8-n)/3)) n!)/(3 Gamma[(n-2)/3])),Mod[n,3]==2}
}],{n,2,30}]

FindSequenceFunction[DeleteMissing@Drop[seq,1],n-2]//Head//Timing

FindLinearRecurrence[DeleteMissing@Drop[seq,1]]//Head//Timing

FindGeneratingFunction[DeleteMissing@seq,x]//Head//Timing

FindGeneratingFunction[Table[(((2^( (8-n)/3)) n!)/(3 Gamma[(n-2)/3])),{n,20}],x]

TriangularGrid

A304562

seq=Table[GraphData[{"TriangularGrid",n},"MinimumTotalDominatingSetCount"],{n,0,20}]

FindSequenceFunction[DeleteMissing@Drop[seq,1],n-1]//Head//Timing

FindLinearRecurrence[DeleteMissing@Drop[seq,1]]//Head//Timing

FindGeneratingFunction[DeleteMissing@seq,x]//Head//Timing

TriangularHoneycombAcuteKnight

A304563

seq=Table[GraphData[{"TriangularHoneycombAcuteKnight",n},"MinimumTotalDominatingSetCount"],{n,20}]

FindSequenceFunction[DeleteMissing@Drop[seq,8],n-8]//Head//Timing

FindLinearRecurrence[DeleteMissing@Drop[seq,8]]//Head//Timing

FindGeneratingFunction[DeleteMissing@seq,x]//Head//Timing

TriangularHoneycombBishop

A304564

seq=Table[GraphData[{"TriangularHoneycombBishop",n},"MinimumTotalDominatingSetCount"],{n,20}]

FindSequenceFunction[Drop[seq,1],n-1]//Head//Timing

FindLinearRecurrence[Drop[seq,1]]//Head//Timing

FindGeneratingFunction[seq,x]//Head//Timing

TriangularHoneycombObtuseKnight

A304565

seq=Table[GraphData[{"TriangularHoneycombObtuseKnight",n},"MinimumTotalDominatingSetCount"],{n,20}]

FindSequenceFunction[Drop[seq,4],n-4]//Head//Timing

FindLinearRecurrence[Drop[seq,4]]//Head//Timing

FindGeneratingFunction[seq,x]//Head//Timing

TriangularHoneycombQueen

A304566

seq=Table[GraphData[{"TriangularHoneycombQueen",n},"MinimumTotalDominatingSetCount"],{n,20}]

FindSequenceFunction[DeleteMissing@Drop[seq,1],n-1]//Head//Timing

FindLinearRecurrence[DeleteMissing@Drop[seq,1]]//Head//Timing

FindGeneratingFunction[DeleteMissing@seq,x]//Head//Timing

TriangularHoneycombRook

seq=Table[GraphData[{"TriangularHoneycombRook",n},"MinimumTotalDominatingSetCount"],{n,20}]

TriangularSnake

seq=Table[GraphData[{"TriangularSnake",2n-1},"MinimumTotalDominatingSetCount"],{n,10}]

FindSequenceFunction[seq,n]

FindGeneratingFunction[seq,x]

Table[GraphData[{"TriangularSnake",2n-1},"TotalDominationNumber"],{n,10}]

Table[Length@Flatten@FindTotalDominatingSet[TriangularSnakeGraph[2n-1],Method->"Gurobi"],{n,30}]

Table[Floor[2n/3],{n,30}]

(seq=Table[Length@FindTotalDominatingSet[TriangularSnakeGraph[2n-1],{Floor[2n/3]},All],{n,14}])//Timing

seq[[;;;;3]]

seq[[2;;;;3]]

seq[[3;;;;3]]

Table[n(n+1)/2+1,{n,10}]

Table[Piecewise[{{1,Mod[n,3]==1},{(n-2)/3,Mod[n,3]==2},{(1+n/3)(1+n/3+1)/2+1,Mod[n,3]==0}}],{n,14}]

Web

seq=Table[GraphData[{"Web",n},"MinimumTotalDominatingSetCount"],{n,3,20}]

Wheel

Entity["Graph",{"Wheel",n}]["MinimumTotalDominatingSetCount"]

seq=Table[GraphData[{"Wheel",n},"MinimumTotalDominatingSetCount"],{n,4,20}]

WhiteBishop n×n

A303144

seq=Table[GraphData[{"WhiteBishop",{n,n}},"MinimumTotalDominatingSetCount"],{n,2,20}]

FindSequenceFunction[DeleteMissing@seq,n]//Head//Timing

FindGeneratingFunction[DeleteMissing@seq,x]//Head//Timing

Zebra n×n

seq=Table[GraphData[{"Zebra",{n,n}},"MinimumTotalDominatingSetCount"],{n,20}]

Algorithm

TotalDominatingSetFunctions[g_Graph,x_,opts___?OptionQ]:=Module[
{n=VertexCount[g],v=VertexList[g],obj,con1,adj,con2,var,j},If[Min[VertexDegree[g]]==0,Return[If[n==1,{1},$Failed]]];(*contains isolated point(s)*)
var=Array[x,n];
obj=Total[var];
con1=Thread[0<=var<=1];
adj=AdjacencyList[g]/. Thread[v->Range[n]];
con2=Table[Total[x/@adj[[j]]]>=1,{j,n}];
{obj,Flatten[{con1,con2,Element[var,Integers]}]}
]

Reduce

{obj,cons}=TotalDominatingSetFunctions[GraphData["Q3"],x]

Sort[Flatten[Position[#,1]]&/@(Array[x,8]/.{ToRules[Reduce[obj==4&&And@@Most[cons],Array[x,8],Integers]]})]

GraphData["Q3","MinimumTotalDominatingSets"]

ILP

For an description of total dominating set, see https://mathworld.wolfram.com/TotalDominatingSet.html

Basically, it a total dominating set S is a subset of vertices in a connected graph such that every vertex in the graph is adjacent to at least one member of S.  A smallest such set is called a minimum total dominating set, and the size of such a set is called the total domination number.

Consider the Petersen graph:

g=GraphData[gn="PetersenGraph","Graph","Labeled"]

n=VertexCount[g]

Create an array of variables with each set to 1 if it is a member of a total dominating set and 0 if not:

var=Array[x,n]

To find a minimum total dominating set, find a set of values for the x[i] that has the least number of 1's possible.  This is equivalent to minimizing an objective function consisting of the sum of the x[i]'s (which is just the number of elements in the set):

obj=Total[var]

As discussed, each x[i] must be 1 (a member of the total dominating set) or 0 (not a member), corresponding to the constraints:

con1=Thread[0<=var<=1]

The second set of constraints encode the adjacencies of the graph:

adj=AdjacencyList[g]

In a total dominating set, each vertex must be adjacent to (dominated by) at least one other, meaning the sum of x[j]'s adjacent to each x[i] must be at least 1:

con2=Table[Total[x/@adj[[j]]]>=1,{j,n}]

The final constraint is that each x[i] not only satisfy 0 <= x[i] <=1 but be an integer, meaning x[i] can only be 0 or 1:

cons3=Element[var,Integers]

The full set of constraints is then:

cons=Flatten[{con1,con2,cons3}]

The problem can than be solved my minimizing the objective function subject to the constraints:

sol=NArgMin[{obj,cons},var]

The positions of 1's gives the elements of a minimum total dominating set:

tds=Flatten[Position[sol,1]]

As can be seen, every vertex in the original graph is indeed adjacent to at least one member of the set:

HighlightGraph[GraphData[gn,"Graph","Labeled"],tds,VertexSize->Medium]

The total dominating number of the Petersen graph is:

Length[tds]