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Graphclass: (2K2,C4,C5,S3,X159,X160,co-(H),co-rising sun,net)-free

Equivalent classes:  probe threshold cap split 
Complement classes:  (2K2,C4,C5,H,S3,X160,co-(X159),net,rising sun)-free 
See also: co-(H) X159 net X160 co-rising sun S3 2K2 C5 C4

Inclusions

Minimal superclasses:  (2K2,C4,C5,S3,co-rising sun,net,rising sun)-free   (2K2,C4,C5,S3,co-rising sun,net)-free   (2K2,C4,C5,S3,net)-free   (2K2,C4)-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-free cap probe trivially perfect   (2K3,2K3 + e,co-(A),co-(H),co-(X45),co-(XZ5),co-domino)-free   (C4,co-(A),co-(H))-free   HHDS-free   (S3,net)-free cap split   chordal cap co-chordal cap co-comparability cap comparability   co-bithreshold cap split   co-interval cap interval   comparability cap split   hereditary homogeneously orderable   (house,hole,domino,sun)-free   permutation cap split   probe co-trivially perfect cap probe trivially perfect   probe threshold   pseudo-split   split cap superperfect   split cap threshold signed 
Maximal subclasses:  (2K2,C4,P4)-free   Dilworth 1   co-interval cap cograph cap interval   co-trivially perfect cap trivially perfect   cograph cap split   comparability graphs of threshold orders   threshold 

Problems summary

Recognition:Lineardetails
Cliquewidth expression: Unknown to ISGCI details
Cliquewidth:Boundeddetails
Weighted independent set:Lineardetails
Independent set:Lineardetails
Domination:Lineardetails

Algorithms for Recognition

Linear from probe threshold cap split 
     From the constituent classes.

Polynomial
     Finite forbidden subgraph characterization


Algorithms for Cliquewidth expression

See also : Cliquewidth : Weighted independent set : Domination

Algorithms for Cliquewidth

Bounded from cliquewidth 4 
See also : Cliquewidth expression

Algorithms for Weighted independent set

Linear from permutation  [1164]
Linear from chordal  [1166]
Linear from (2K2,C4)-free 
Polynomial from nK2-free, fixed n  [1102]
Polynomial from (K2,3,P5)-free  [1110]
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 from (C4,C5,T2)-free  [1108]
Polynomial from (P5,house)-free  [1109]
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 [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 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 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 co-chordal  [558]
     See also [425] .

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