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Graphclass: (2K2,C4,P4)-free

Equivalent classes:  Dilworth 1   co-interval cap cograph cap interval   co-trivially perfect cap trivially perfect   cograph cap split   comparability graphs of threshold orders   threshold 
Complement classes: self-complementary
See also: P4 2K2 C4

Inclusions

Minimal superclasses:  (2K2,C4,C5,H,S3,X160,co-(X159),net,rising sun)-free   (2K2,C4,C5,S3,X159,X160,co-(H),co-rising sun,net)-free   (2K2,C4,C5,S3,co-rising sun,net)-free   (2K2,C5,S3,X159,X160,X161,X162,X46,X70,co-(2P3),co-(3K2),co-(H),co-(P2 cup P4),co-(X1),co-rising sun,house,net)-free   (2K2,K3,3,K3,3+e,P4,co-(2P3))-free   2K2-free cap probe trivially perfect   (2K3,2P3,C4,K3 cup P3,P4)-free   (2P3,3K2,C4,C5,H,P2 cup P4,P5,S3,X1,X160,co-(X159),co-(X161),co-(X162),co-(X46),co-(X70),net,rising sun)-free   (3K2,C4 cup P2,C5,P2 cup P4,P5,S3,X1,X46,X70,co-(3K2),co-(C4 cup P2),co-(P2 cup P4),co-(X1),co-(X46),co-(X70),co-fish,co-rising sun,fish,house,net,rising sun)-free   (5,2)-odd-noncrossing-chordal   (A,H,K3,3,K3,3-e,X45,XZ5,domino)-free   (A,H,K3,3,X45,X46,X47,X48,X49,X50,X51,X52,X53,X54,X55,X56,X57,co-(X42))-free   AT-free cap chordal   (C4,co-(A),co-(H))-free   C4-free cap co-comparability   (C5,K2 cup K3,K2,3,P,P2 cup P3,P5,co-(P),co-(P2 cup P3),co-fork,fork,house)-free   (C5,P6,co-(P6))-free   C5-free cap P4-tidy   C5-free cap matrogenic   (Cn+4,H)-free   (Cn+4,P5,bull)-free   (Cn+4,T2,X31,XF2n+1,XF3n)-free   (Cn+4,gem)-free   Dilworth 2   EPT   Gallai   (P,P5,co-(3K2),gem)-free   (P,co-butterfly,co-fork,co-gem)-free   P4-extendible   P4-free   P4-lite   (P5,bull)-free cap interval   (XC1,XC2,XC3,XC4,XC5,XC6,XC7,XC8)-free   (XC7,co-(XC1),co-(XC2),co-(XC3),co-(XC4),co-(XC5),co-(XC6),co-(XC8))-free   bithreshold   boxicity 1   bull-free   chordal cap co-comparability   chordal cap distance-hereditary   chordal cap domination perfect   chordal cap gem-free   cliquewidth 2   co-bithreshold cap split   (co-gem,gem)-free   co-probe threshold   co-threshold tolerance   cograph   comparability cap split   comparability graphs of series-parallel posets   domination perfect   domishold   homogeneously representable   i-triangulated   interval   matroidal   murky   probe co-trivially perfect cap probe trivially perfect   probe threshold   probe threshold cap split   ptolemaic   split cap superperfect   strict 2-threshold   threshold signed   undirected path 

Problems summary

Recognition:Lineardetails
Cliquewidth expression:Lineardetails
Cliquewidth:Boundeddetails
Weighted independent set:Lineardetails
Independent set:Lineardetails
Domination:Lineardetails

Algorithms for Recognition

Linear from threshold  [218] [509]
Polynomial
     Finite forbidden subgraph characterization

Algorithms for Cliquewidth expression

Linear from cliquewidth 2 
Linear from (bull,fork,gem)-free  [1185]
Linear from (P5,bull,house)-free  [1187] [1185]
Linear from (5,1)  [1175]
Linear from (q,q-4), fixed q  [1175]
Linear from (co-(P),fork,gem)-free  [1184] [1185]
Linear from (bull,co-fork,fork)-free  [1124] [1185]
Linear from (P,co-(P),co-fork,fork)-free  [1185]
Linear from partner-limited  [1179]
Linear from semi-P4-sparse  [1186]
Linear from (q, q-3), fixed q>= 7  [1176]
Linear from (bull,fork,house)-free  [1124] [1185]
Linear from (7,3)  [1175]
Linear from P4-tidy  [1175]
Linear from distance-hereditary  [1177]
Linear from (P5,fork,house)-free  [1185] [402]
Linear from (6,2)  [1175]
Linear from (co-(P),fork,house)-free  [1185] [1186]
Linear from (9,6)  [488]
Polynomial from tree-cograph 
     From the decomposition tree.

Polynomial [O(V^2E)] from cliquewidth 3  [1178]
See also : Cliquewidth : Weighted independent set : Domination

Algorithms for Cliquewidth



Bounded from (P,P5,co-fork)-free 
     From the complement .






Bounded from (P,co-fork,co-gem)-free 
     From the complement .


Bounded from (P,co-butterfly,co-fork,co-gem)-free 
     From the complement .


Bounded from cliquewidth 4 
Bounded from (co-gem,gem)-free  [1188] [1185]
Bounded from (P,co-gem,house)-free 
     From the complement .


Bounded from (P5,anti-hole,co-domino,co-gem)-free 
     From the complement .



Bounded from co-probe cograph 
     From the complement  probe cograph  .

Bounded from (co-gem,house)-free 
     From the complement .




Bounded from (bull,co-fork,co-gem)-free 
     From the complement .



Bounded from (C5,co-gem,house)-free 
     From the complement .

Bounded from (3K2,co-(P),co-gem,house)-free 
     From the complement .







Bounded from (P5,gem)-free  [1189] [1171] [1185]

Bounded from (2P3,3K2,C4,C5,H,P2 cup P4,P5,S3,X1,X160,co-(X159),co-(X161),co-(X162),co-(X46),co-(X70),net,rising sun)-free 
     From the complement .




Bounded from (co-(Cn+4),co-gem)-free 
     From the complement .

Bounded from (2K2,C4,C5,H,S3,X160,co-(X159),net,rising sun)-free 
     From the complement .














Bounded from (P5,bull,co-fork)-free 
     From the complement .


Bounded from co-probe threshold 
     From the complement  probe threshold  .


See also : Cliquewidth expression

Algorithms for Weighted independent set

Linear from permutation  [1164]
Linear from (P5,gem)-free  [1170]
Linear from chordal  [1166]
Linear from distance-hereditary 
     Hammer/Maffray's [511] algorithm contained an error that was corrected by Nicolai. [809]

Linear from (2K2,C4)-free 
Polynomial [O(VE)] from (co-(P),fork)-free  [1125]
Polynomial [O(VE)] from (P5,fork)-free  [1125]
Polynomial from nK2-free, fixed n  [1102]
Polynomial from (K2,3,P5)-free  [1110]
Polynomial [O(V^5E^3)] from Berge cap bull-free  [1278]
Polynomial from semi-P4-sparse  [402]
Polynomial from (P,P5)-free  [1353]
Polynomial [O(n logn logn)] from trapezoid 
     Timebound valid only when given the model [1120] ; otherwise O(n^2).

Polynomial [O(n^4)] from (C4,P5)-free 
     Algorithm for (P_5,K_{m,m})-free (fixed m) [1118]

Polynomial from K2 cup claw-free  [1290]
Polynomial [O(VE)] from co-gem-free 
     Because for all v: G[\co{N}(v)] is P_4-free

Polynomial from (C4,C5,T2)-free  [1108]
Polynomial from (P5,house)-free  [1109]
Polynomial from fork-free  [1099]
Polynomial from parity  [170]
Polynomial [O(n^6)] from (K3,3,P5)-free 
     Algorithm for (P_5,K_{m,m})-free (fixed m) [1118]

Polynomial [O(n^4)] from AT-free  [160]
Polynomial from (P5,co-fork)-free  [1161]
Polynomial [O(n^8)] from (K4,4,P5)-free 
     Algorithm for (P_5,K_{m,m})-free (fixed m) [1118]

Polynomial from (K2,3,P,hole)-free  [1107]
Polynomial [O(n^2)] from circle  [1121]
Polynomial from (co-(P),butterfly,fork,gem)-free  [1104]
Polynomial from interval filament  [1159]
Polynomial [O(n^{6p+2})] from (p,q<=2)-colorable  [1116]
Polynomial [O(ln)] from circular arc 
     Where l is the minimum number of arcs passing through a given point on the circle. [995]

Polynomial from perfect  [476]
Polynomial from (P5,X82,X83)-free  [1246]
Polynomial [O(VE)] from (bull,fork)-free  [1124] [307]
Polynomial from (P,P5,co-(3K2),gem)-free  [1114]
Polynomial from 2K2-free  [1160]
Polynomial from (K2,3,P,P5)-free  [1107]
Polynomial [O(V^4)] from weakly chordal  [997]
Polynomial from subtree overlap  [1123]
Polynomial from (C4,P6)-free  [1353]
See also : Cliquewidth expression : Independent set

Algorithms for Independent set

Linear [O(n)] from circular arc  [1105] [1106] [1158]
Linear from partner-limited  [1180]
Linear from co-chordal  [558]
     See also [425] .

Linear from P4-tidy  [440]
Linear from co-Matula perfect  [221]
Linear from co-Welsh-Powell perfect  [221]
Linear from co-comparability  [1100]
Linear from chordal  [425] [931]
Linear from extended P4-laden  [438]
Polynomial from co-biclique separable  [1304]
Polynomial from Gallai  [1081]
Polynomial from (C5,P5,co-(P2 cup P3))-free  [1118]
Polynomial from co-hereditary clique-Helly  [1298]
Polynomial from (C4,P6)-free  [1351] [1352]
Polynomial from comparability  [453]
Polynomial from (K3,3-e,P5)-free  [1246]
Polynomial [O(VE)] from (P,P5)-free  [1117]
Polynomial [O(VE)] from weakly chordal  [530] [1119]
Polynomial from (E,P)-free  [1305]
Polynomial from (K2,3,P,P5)-free  [1346] [1107]
Polynomial [O(V^8)] from (P,P7)-free  [1351]
Polynomial from (C6,K3,3+e,P,P7,X37,X41)-free  [1346]
Polynomial from EPT  [1019]
Polynomial from (P,T2)-free  [1305]
Polynomial from Meyniel  [169]
Polynomial [O(nm)] from (K3,3-e,P5,X98)-free  [1117]
Polynomial from clique separable  [1081]
Polynomial from (P5,co-(P2 cup P3))-free  [1350]
Polynomial from (P,star1,2,5)-free  [1349]
Polynomial [O(V^{5})] from (K3,3-e,P5,X99)-free  [1307]
Polynomial from (P,P8)-free  [1306]
Open from (P,star1,2,3)-free  [1351]
Open from (P,star1,2,4)-free  [1351] [1306]
See also : Weighted independent set

Algorithms for Domination

Linear from distance-hereditary  [1153]
Linear from dually chordal  [143] [332]
Linear from circular arc  [1143] [1158]
Linear [O(V)] from permutation  [1342] [1147] [1148] [1149] [1165]
Linear from interval  [1143]
Polynomial from strongly chordal  [374]
Polynomial from co-interval cup interval 
     From  interval  and  co-interval  .

Polynomial from directed path  [524]
Polynomial from k-polygon  [352]
Polynomial from co-comparability  [1150] [1151]
Polynomial from AT-free  [1152]
Polynomial [O(n^2 log^5 n)] from co-bounded tolerance  [1172]
     Assuming a square embedding of the graph is given; finding this is an open problem.

Polynomial from cograph  [524]
Polynomial from trapezoid  [1155]
Polynomial from probe interval  [1340]
See also : Cliquewidth expression