Graphclass: (3K₁,3K₂)-free

The speed of a class $X$ is the function $n \mapsto |X_n|,ドル where $X_n$ is the set of $n$-vertex labeled graphs in $X$.

Depending on the rate of growths of the speed of the class, ISGCI distinguishes the following values of the parameter:
Constant
Polynomial
Exponential
Factorial
Superfactorial (2ドル^{o(n^2)}$ )
Superfactorial (2ドル^{\Theta(n^2)}$ )

V.E. Alekseev
On lower layers of a lattice of hereditary classes of graphs
Diskretn. Anal. Issled. Oper. Ser. 1 4:1 3-12 (1997)

E.R. Scheinerman, J. Zito
On the size of hereditary classes of graphs
J. Combin. Th. B Vol. 61 No.1 16-39 (1994)

J. Balogh, B. Bollobas, D. Weinreich
The speed of hereditary properties of graphs
J. Combin. Th. B Vol. 79 No.2 131-156 (2000)

The acyclic chromatic number of a graph $G$ is the smallest size of a vertex partition $\{V_1,\dots,V_l\}$ such that each $V_i$ is an independent set and for all $i,j$ that graph $G[V_i\cup V_j]$ does not contain a cycle.

The bandwidth of a graph $G$ is the shortest maximum "length" of an edge over all one dimensional layouts of $G$. Formally, bandwidth is defined as $\min_{i \colon V \rightarrow \mathbb{N}\;}\{\max_{\{u,v\}\in E\;} \{|i(u)-i(v)|\}\mid i\text{ is injective}\}$.

A book embedding of a graph $G$ is an embedding of $G$ on a collection of half-planes (called pages) having the same line (called spine) as their boundary, such that the vertices all lie on the spine and there are no crossing edges. The book thickness of a graph $G$ is the smallest number of pages over all book embeddings of $G$.

F. Bernhart, P.C. Kainen
The book thickness of a graph
J. of Combin. Th. (B) 27 320-331 (1979)

Consider the following decomposition of a graph $G$ which is defined as a pair $(T,L)$ where $T$ is a binary tree and $L$ is a bijection from $V(G)$ to the leaves of the tree $T$. The function $\text{cut-bool} \colon 2^{V(G)} \rightarrow R$ is defined as $\text{cut-bool}(A)$ := $\log_2|\{S \subseteq V(G) \backslash A \mid \exists X \subseteq A \colon S = (V(G) \backslash A) \cap \bigcup_{x \in X} N(x)\}|$. Every edge $e$ in $T$ partitions the vertices $V(G)$ into $\{A_e,\overline{A_e}\}$ according to the leaves of the two connected components of $T - e$. The booleanwidth of the above decomposition $(T,L)$ is $\max_{e \in E(T)\;} \{ \text{cut-bool}(A_e)\}$. The booleanwidth of a graph $G$ is the minimum booleanwidth of a decomposition of $G$ as above.

A branch decomposition of a graph $G$ is a pair $(T,\chi),ドル where $T$ is a binary tree and $\chi$ is a bijection, mapping leaves of $T$ to edges of $G$. Any edge $\{u, v\}$ of the tree divides the tree into two components and divides the set of edges of $G$ into two parts $X, E \backslash X,ドル consisting of edges mapped to the leaves of each component. The width of the edge $\{u,v\}$ is the number of vertices of $G$ that is incident both with an edge in $X$ and with an edge in $E \backslash X$. The width of the decomposition $(T,\chi)$ is the maximum width of its edges. The branchwidth of the graph $G$ is the minimum width over all branch-decompositions of $G$.

Consider a decomposition $(T,\chi)$ of a graph $G$ where $T$ is a binary tree with $|V(G)|$ leaves and $\chi$ is a bijection mapping the leaves of $T$ to the vertices of $G$. Every edge $e \in E(T)$ of the tree $T$ partitions the vertices of the graph $G$ into two parts $V_e$ and $V \backslash V_e$ according to the leaves of the two connected components in $T - e$. The width of an edge $e$ of the tree is the number of edges of a graph $G$ that have exactly one endpoint in $V_e$ and another endpoint in $V \backslash V_e$. The width of the decomposition $(T,\chi)$ is the largest width over all edges of the tree $T$. The carvingwidth of a graph is the minimum width over all decompositions as above.

The chromatic number of a graph is the minimum number of colours needed to label all its vertices in such a way that no two vertices with the same color are adjacent.

The cliquewidth of a graph is the number of different labels that is needed to construct the graph using the following operations:

• creation of a vertex with label $i,ドル
• disjoint union,
• renaming labels $i$ to label $j,ドル and
• connecting all vertices with label $i$ to all vertices with label $j$.

The cochromatic number of a graph $G$ is the minimum number of colours needed to label all its vertices in such a way that that every set of vertices with the same colour is either independent in G, or independent in $\overline{G}$.

The cutwidth of a graph $G$ is the smallest integer $k$ such that the vertices of $G$ can be arranged in a linear layout $v_1, \ldots, v_n$ in such a way that for every $i = 1, \ldots,n - 1,ドル there are at most $k$ edges with one endpoint in $\{v_1, \ldots, v_i\}$ and the other in $\{v_{i+1}, \ldots, v_n\}$.

Let $G$ be a graph and consider the following algorithm:

• Find a vertex $v$ with smallest degree.
• Delete vertex $v$ and its incident edges.
• Repeat as long as the graph is not empty.

The degeneracy of a graph $G$ is the maximum degree of a vertex when it is deleted in the above algorithm.

The diameter of a graph $G$ is the length of the longest shortest path between any two vertices in $G$.

The distance to block of a graph $G$ is the size of a smallest vertex subset whose deletion makes $G$ a block graph.

Let $G$ be a graph. Its distance to clique is the minimum number of vertices that have to be deleted from $G$ in order to obtain a clique.

A cluster is a disjoint union of cliques. The distance to cluster of a graph $G$ is the size of a smallest vertex subset whose deletion makes $G$ a cluster graph.

The distance to co-cluster of a graph is the minimum number of vertices that have to be deleted to obtain a co-cluster graph.

The distance to cograph of a graph $G$ is the minimum number of vertices that have to be deleted from $G$ in order to obtain a cograph .

The distance to linear forest of a graph $G = (V, E)$ is the size of a smallest subset $S$ of vertices, such that $G[V \backslash S]$ is a disjoint union of paths and singleton vertices.

The distance to outerplanar of a graph $G = (V,E)$ is the minumum size of a vertex subset $X \subseteq V,ドル such that $G[V \backslash X]$ is a outerplanar graph.

The genus $g$ of a graph $G$ is the minimum number of handles over all surfaces on which $G$ can be embedded without edge crossings.

The max-leaf number of a graph $G$ is the maximum number of leaves in a spanning tree of $G$.

The parameter maximum clique of a graph $G$ is the largest number of vertices in a complete subgraph of $G$.

The maximum degree of a graph $G$ is the largest number of neighbors of a vertex in $G$.

An independent set of a graph $G$ is a subset of pairwise non-adjacent vertices. The parameter maximum independent set of graph $G$ is the size of a largest independent set in $G$.

For a graph $G = (V,E)$ an induced matching is an edge subset $M \subseteq E$ that satisfies the following two conditions: $M$ is a matching of the graph $G$ and there is no edge in $E \backslash M$ connecting any two vertices belonging to edges of the matching $M$. The parameter maximum induced matching of a graph $G$ is the largest size of an induced matching in $G$.

A matching in a graph is a subset of pairwise disjoint edges (any two edges that do not share an endpoint). The parameter maximum matching of a graph $G$ is the largest size of a matching in $G$.

A clique cover of a graph $G = (V, E)$ is a partition $P$ of $V$ such that each part in $P$ induces a clique in $G$. The minimum clique cover of $G$ is the minimum number of parts in a clique cover of $G$. Note that the clique cover number of a graph is exactly the chromatic number of its complement.

A dominating set of a graph $G$ is a subset $D$ of its vertices, such that every vertex not in $D$ is adjacent to at least one member of $D$. The parameter minimum dominating set for graph $G$ is the minimum number of vertices in a dominating set for $G$.

A path decomposition of a graph $G$ is a pair $(P,X)$ where $P$ is a path with vertex set $\{1, \ldots, q\},ドル and $X = \{X_1,X_2, \ldots ,X_q\}$ is a family of vertex subsets of $V(G)$ such that:

• $\bigcup_{p \in \{1,\ldots ,q\}} X_p = V(G)$
• $\forall\{u,v\} \in E(G) \exists p \colon u, v \in X_p$
• $\forall v \in V(G)$ the set of vertices $\{p \mid v \in X_p\}$ is a connected subpath of $P$.

The width of a path decomposition $(P,X)$ is max$\{|X_p| - 1 \mid p \in \{1,\ldots ,q\}\}$. The pathwidth of a graph $G$ is the minimum width among all possible path decompositions of $G$.

Let $M$ be the $|V| \times |V|$ adjacency matrix of a graph $G$. The cut rank of a set $A \subseteq V(G)$ is the rank of the submatrix of $M$ induced by the rows of $A$ and the columns of $V(G) \backslash A$. A rank decomposition of a graph $G$ is a pair $(T,L)$ where $T$ is a binary tree and $L$ is a bijection from $V(G)$ to the leaves of the tree $T$. Any edge $e$ in the tree $T$ splits $V(G)$ into two parts $A_e, B_e$ corresponding to the leaves of the two connected components of $T - e$. The width of an edge $e \in E(T)$ is the cutrank of $A_e$. The width of the rank-decomposition $(T,L)$ is the maximum width of an edge in $T$. The rankwidth of the graph $G$ is the minimum width of a rank-decomposition of $G$.

A tree depth decomposition of a graph $G = (V,E)$ is a rooted tree $T$ with the same vertices $V,ドル such that, for every edge $\{u,v\} \in E,ドル either $u$ is an ancestor of $v$ or $v$ is an ancestor of $u$ in the tree $T$. The depth of $T$ is the maximum number of vertices on a path from the root to any leaf. The tree depth of a graph $G$ is the minimum depth among all tree depth decompositions.

A tree decomposition of a graph $G$ is a pair $(T, X),ドル where $T = (I, F)$ is a tree, and $X = \{X_i \mid i \in I\}$ is a family of subsets of $V(G)$ such that

• the union of all $X_i,ドル $i \in I$ equals $V,ドル
• for all edges $\{v,w\} \in E,ドル there exists $i \in I,ドル such that $v, w \in X_i,ドル and
• for all $v \in V$ the set of nodes $\{i \in I \mid v \in X_i\}$ forms a subtree of $T$.

The width of the tree decomposition is $\max |X_i| - 1$.
The treewidth of a graph is the minimum width over all possible tree decompositions of the graph.

Let $G$ be a graph. Its vertex cover number is the minimum number of vertices that have to be deleted in order to obtain an independent set.

H. Broersma, P.A. Golovach, D. Paulusma, J. Song
Determining the chromatic number of triangle-free 2P3-free graphs in polynomial time
Manuscript (2010)

H. Broersma, P.A. Golovach, D. Paulusma, J. Song
Determining the chromatic number of triangle-free 2P3-free graphs in polynomial time
Manuscript (2010)

J. Stacho
3-colouring of AT-free graphs in polynomial time
21st International Symposium on Algorithms and Computation ISAAC, Lecture Notes in Comp. Sci. LNCS 6507 144-155 (2010)

H. Broersma, P.A. Golovach, D. Paulusma, J. Song
Determining the chromatic number of triangle-free 2P3-free graphs in polynomial time
Manuscript (2010)

B. Randerath, I. Schiermeyer, M. Tewes
Three-colourability and forbidden subgraphs II: polynomial algorithms
Discrete Math. 251 137-212 (2002)

C.T. Hoang, M. Kaminski, V. Lozin, J. Sawada, X. Shu
Deciding k-colorability of P_5-free graphs in polynomial time
Algorithmica Vol.57 No.1 74-81 (2010)

B. Randerath, I. Schiermeyer
3-colorability in P for P_6-free graphs
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M. Kaminski, V. Lozin
Vertex 3-colorability of claw-free graphs
Algorithmic Operations Research Vol.2 15-21 (2007)

H. Broersma, P.A. Golovach, D. Paulusma, J. Song
Determining the chromatic number of triangle-free 2P3-free graphs in polynomial time
Manuscript (2010)

H. Broersma, P.A. Golovach, D. Paulusma, J. Song
Determining the chromatic number of triangle-free 2P3-free graphs in polynomial time
Manuscript (2010)

D. Kral, J. Kratochvil, Z. Tuza, G.J. Woeginger
Complexity of coloring graphs without forbidden induced subgraphs
Proceedings of the 27th International Workshop on Graph-Theoretic Concepts in Computer Science WG'01, LNCS 2204, 254-262 (2001)

D. Kratsch
Domination and total domination in asteroidal triple-free graphs
Discrete Appl. Math. 99 No.1-3, 111-123 (2000)

A. Brandstaedt, F. Dragan
On linear and circular structure of (claw, net)-free graph
To appear in Discrete Appl. Math.

Feedback vertex set on AT-free graphs
D. Kratsch, H. Mueller, I. Todinca
Discrete Appl. Math. 156 No. 10 1936-1947 (2008)

(no preview available)

A. Brandstaedt, F.F. Dragan, E. Koehler
Linear time algorithms for the Hamiltonian problems on (claw,net)-free graphs
SIAM J. Computing 30 1662-1677 (2000)

N. Jedličková, J. Kratochvil
On the structure of Hamiltonian graphs with small independence number
Combinatorial Algorithms. IWOCA 2024. LNCS 14764 180-192 (2024)

N. Jedličková, J. Kratochvil
On the structure of Hamiltonian graphs with small independence number
Combinatorial Algorithms. IWOCA 2024. LNCS 14764 180-192 (2024)

N. Jedličková, J. Kratochvil
On the structure of Hamiltonian graphs with small independence number
Combinatorial Algorithms. IWOCA 2024. LNCS 14764 180-192 (2024)

D. Duffus, R.J. Gould, M.S. Jacobson
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A. Brandstaedt, F.F. Dragan, E. Koehler
Linear time algorithms for the Hamiltonian problems on (claw,net)-free graphs
SIAM J. Computing 30 1662-1677 (2000)

N. Jedličková, J. Kratochvil
On the structure of Hamiltonian graphs with small independence number
Combinatorial Algorithms. IWOCA 2024. LNCS 14764 180-192 (2024)

N. Jedličková, J. Kratochvil
On the structure of Hamiltonian graphs with small independence number
Combinatorial Algorithms. IWOCA 2024. LNCS 14764 180-192 (2024)

N. Jedličková, J. Kratochvil
On the structure of Hamiltonian graphs with small independence number
Combinatorial Algorithms. IWOCA 2024. LNCS 14764 180-192 (2024)

D. Duffus, R.J. Gould, M.S. Jacobson
Forbidden subgraphs and the Hamiltonian theme.
The theory and applications of graphs, 4th int. Conf., Kalamazoo/Mich. 1980, 297-316 (1981).

N.V.R. Mahadev, B.A. Reed
A note on vertex orders for stability number
J. Graph Theory 30 113-120 (1999)

M.U. Gerber, V.V. Lozin
Robust algorithms for the stable set problem
Graphs and Combin., to appear

N.V.R. Mahadev
Vertex deletion and stability number
Research report ORWP 90/2 Dept. of Mathematics, Swiss Fed.Inst. of Technology 1990

N.V.R. Mahadev, B.A. Reed
A note on vertex orders for stability number
J. Graph Theory 30 113-120 (1999)

A. Brandstaedt, V.V. Lozin
A note on \alpha-redundant vertices in graphs.
Discrete Appl. Math. 108 (2001) 301-308

M.U. Gerber, A. Hertz, D. Schindl
P_5-free graphs and the maximum stable set problem
Discrete Appl. Math. 132 109-119 (2004)

V. Alekseev, P. Lozin, R. Mosca
Maximum independent set problem and P_5-free graphs
Manuscript 2004

A. Brandstaedt, V.V. Lozin
A note on \alpha-redundant vertices in graphs.
Discrete Appl. Math. 108 (2001) 301-308

V.E. Alekseev, V.V. Lozin
Augmenting graphs for independent sets
Discrete Appl. Math. 145, No.1 3-10 (2004)

M.U. Gerber, A. Hertz, V.V. Lozin
Stable sets in two subclasses of banner-free graphs
Discrete Appl. Math. 132 121-136 (2004)

M.U. Gerber, V.V. Lozin
Robust algorithms for the stable set problem
Graphs and Combin., to appear

V.L. Lozin, M. Milanic
On finding augmenting graphs
Rutcor Research Report 28-2005

A. Brandstaedt, F. Dragan
On linear and circular structure of (claw, net)-free graph
To appear in Discrete Appl. Math.

P.L. Hammer, N.V.R. Mahadev, D. de Werra
The struction of a graph: application to CN--free graphs
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(decision variant)

V.B. Le, R. Nevries
Complexity and algorithms for recognizing polar and monopolar graphs
Theoretical Computer Science 528 1-11 (2014)

R. Churchley, J. Huang
On the polarity and monopolarity of graphs
J. Graph Theory 76 No. 2 1138-148 (2014)

H. Hempel, D. Kratsch
On claw-free asteroidal triple-free graphs
Discrete Appl. Math. 121, No.1-3, 155-180 (2002)

H. Broersma, T. Kloks, D. Kratsch, H. M\"uller
Independent sets in asteroidal triple-free graphs
SIAM J. Discrete Math. 12, No.2, 276-287 (1999)

A. Brandstaedt, P.L Hammer
On the stability number of claw-free, P_5-free and more general graphs.
Rutcor Research Report 27-97 1997

R. Mosca
Polynomial algorithms for the maximum stable set problem on particular classes of P_5-free graphs
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V. Lozin, R. Mosca
Independent sets and extensions of 2K_2-free graphs
Discrete Appl. Math. 146 74-80 (2005)

N.V.R. Mahadev
Vertex deletion and stability number
Research report ORWP 90/2 Dept. of Mathematics, Swiss Fed.Inst. of Technology 1990

R. Mosca
Polynomial algorithms for the maximum stable set problem on particular classes of P_5-free graphs
Information Processing Letters 61 (1997) 137-144

A. Brandstaedt, C.T. Hoang
On clique separators, nearly chordal graphs, and the Maximum Weight Stable Set Problem
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R. Mosca
Stable sets of maximum weight in (P_7, banner)-free graphs
Discrete Math. 308 Issue 1 20-33 (2008)

V. Alekseev, P. Lozin, R. Mosca
Maximum independent set problem and P_5-free graphs
Manuscript 2004

R. Mosca
Polynomial algorithms for the maximum stable set problem on particular classes of P_5-free graphs
Information Processing Letters 61 (1997) 137-144

R. Mosca
Polynomial algorithms for the maximum stable set problem on particular classes of P_5-free graphs
Information Processing Letters 61 (1997) 137-144

A. Brandstaedt, V.B. Le, H.N. de Ridder
Efficient robust algorithms for the maximum weight stable set problem in chair-free graph classes.
Universitaet Rostock, Fachbereich Informatik, Preprint CS-14-01

D. Lokshantov, M. Vatshelle, Y. Villanger
Independent set in P5-free graphs in polynomial time
Accepted for publication

A. Grzesik, T. Klimosova, M. Pilipczuk
Polynomial-time Algorithm for Maximum Weight Independent Set on P6-free Graphs
ACM Transactions on Algorithms 18 No.1 1-57 (2022)

A. Brandstaedt, C.T. Hoang, V.B. Le
Stability number of bull- and chair-free graphs revisited
to appear in Discrete Appl. Math.

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A. Brandstaedt, V.V. Lozin, R. Mosca
Independent sets of maximum weight in apple-free graphs
SIAM J. Discrete Math. Vol.24 No.1 239-254 (2010)

(decision variant)

M. Kaminski
Max-cut and containment relations in graphs
Proceedings of WG 2010, Lecture Notes in Computer Science 6410, 15-26 (2010)