Graphclass: (K_2,3,K₄)-minor-free

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.

J. Nesetril, P. Ossona de Mendez
Colorings and homomorphisms of minor closed classes
Discrete and Comput. Geometry 25, The Goodman-Pollack Festschrift 651-664 (2003)

J. Nesetril, P. Ossona de Mendez
Colorings and homomorphisms of minor closed classes
Discrete and Comput. Geometry 25, The Goodman-Pollack Festschrift 651-664 (2003)

J. Nesetril, P. Ossona de Mendez
Colorings and homomorphisms of minor closed classes
Discrete and Comput. Geometry 25, The Goodman-Pollack Festschrift 651-664 (2003)

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 $\min_{i \colon V \rightarrow \mathbb{N}\;}\{\max_{\{u,v\}\in E\;} \{|i(u)-i(v)|\}\mid i\text{ is injective}\}$.

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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 $G$.

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M. Yannakakis
Embedding planar graph in four pages
Journal of Computer and System Sciences 38 36-67 (1989)

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 $\overline{G}$.

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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.

T.V. Wimer
Linear algorithms on $k$--terminal graphs
Ph. D. Thesis, {\sl Dept. of Computer Science, URI-030, Clemson Univ., Clemson, SC} 1987

P. Scheffler
The graphs of tree--width $k$ are exactly the partial $k$--trees
manuscript 1986

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.L. Bodlaender
A tourist guide through treewidth
Acta Cybernetica 11 1993 1--23

H.L. Bodlaender
A linear time algorithm for finding tree-decompositions of small treewidth
SIAM J. Comput. 25 No.6 1305-1317 (1996)

H.L. Bodlaender
A tourist guide through treewidth
Acta Cybernetica 11 1993 1--23

H.L. Bodlaender
A linear time algorithm for finding tree-decompositions of small treewidth
SIAM J. Comput. 25 No.6 1305-1317 (1996)

H.L. Bodlaender
A tourist guide through treewidth
Acta Cybernetica 11 1993 1--23

H.L. Bodlaender
A linear time algorithm for finding tree-decompositions of small treewidth
SIAM J. Comput. 25 No.6 1305-1317 (1996)

H.L. Bodlaender
A tourist guide through treewidth
Acta Cybernetica 11 1993 1--23

H.L. Bodlaender
A linear time algorithm for finding tree-decompositions of small treewidth
SIAM J. Comput. 25 No.6 1305-1317 (1996)

H.L. Bodlaender
A tourist guide through treewidth
Acta Cybernetica 11 1993 1--23

H. Imai, T. Asano
Finding the connected components and a maximum clique of an intersection graph of rectangles in the plane
J. Algorithms 4 1983 310--323

A. Tiskin
Fast distance multiplication of unit-Monge matrices
Proc. of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms SODA 1287-1296 (2010)

A. Pecher, A.K. Wagler
Clique and chromatic number of circular-perfect graphs
Proceedings of ISCO 2010 - International Symposium on Combinatorial Optimization, Elec. Notes in Discrete Math 36 199-206 (2010)

A. Pecher, A.K. Wagler
Clique and chromatic number of circular-perfect graphs
Proceedings of ISCO 2010 - International Symposium on Combinatorial Optimization, Elec. Notes in Discrete Math 36 199-206 (2010)

R.A. Wilson
Graphs, colourings and the four-colour theorem
Oxford University Press, London 2002

I.S. Filotti, J.N. Mayer
A Polynomial-time Algorithm for Determining the Isomorphism of Graphs of Fixed Genus
Proceedings of the Twelfth Annual ACM Symposium on Theory of Computing STOC'80 236-243 (1980)

G. Miller
Isomorphism testing for graphs of bounded genus
Proceedings of the Twelfth Annual ACM Symposium on Theory of Computing STOC'80 225-235 (1980)

I.S. Filotti, J.N. Mayer
A Polynomial-time Algorithm for Determining the Isomorphism of Graphs of Fixed Genus
Proceedings of the Twelfth Annual ACM Symposium on Theory of Computing STOC'80 236-243 (1980)

G. Miller
Isomorphism testing for graphs of bounded genus
Proceedings of the Twelfth Annual ACM Symposium on Theory of Computing STOC'80 225-235 (1980)

M. Vatshelle
New Width Parameters of Graphs
PhD Thesis, Dept. of Informatics, University of Bergen 2012

M. Vatshelle
New Width Parameters of Graphs
PhD Thesis, Dept. of Informatics, University of Bergen 2012

N. Nash, D. Gregg
An output sensitive algorithm for computing a maximum independent set of a circle graph
Inform. Process. Lett. 110 No.16 630-634 (2010)

(decision variant)

J. Diaz, M. Karminski
Max-Cut and Max-Bisection are NP-hard on unit disk graphs
Theo. Comp. Sci 377 271-276 (2007)

S.L. Mitchell
Linear algorithms to recognize outerplanar and maximal outerplanar graphs
Inf. Proc. Letters 9 1979 229--232

C.H. Papadimitriou, M. Yannakakis
The clique problem for planar graphs
Information Proc. Letters 13 131-133 (1981)

M.M. Halldorson, S. Kitaev, A. Pyatkin
Alternation graphs
Proceedings of WG 2011, Lecture Notes in Computer Science 6986, 191-202 (2011)

W.-L. Hsu
Maximum weight clique algorithms for circular-arc graphs and circle graphs
SIAM J. Computing 14 No.1 224-231 (1985)

F. Gavril
Maximum weight independent sets and cliques in intersection graphs of filaments.
Information Processing Letters 73(5-6) 181-188 (2000)

M. Kaufmann, J. Kratochvil, K.A. Lehmann, A.R. Subramanian
Max-tolerance graphs as intersection graphs: cliques, cycles and recognition
Proc. of 17th annual ACM-SIAM symposium on Discrete algorithms SODA'06 832-841 (2006)

F. Maffray, M. Preissmann
Split--neighbourhood graphs and the strong perfect graph conjecture
J. Comb. Theory (B) 63 1995 294--309

F. Gavril
Minimum weight feedback vertex sets in circle graphs
Information Proc. Lett. 107 No.1 1-6 (2008)

A. Apostolico, M.J. Atallah, S.E. Hambrusch
New clique and independent set algorithms for circle graphs.
Discrete Appl. Math 36 (1992) 1-24 Erratum: Discrete Appl. Math 41 (1993) 179-180

F. Gavril
Maximum weight independent sets and cliques in intersection graphs of filaments.
Information Processing Letters 73(5-6) 181-188 (2000)

E. Cenek, L. Stewart
Maximum independent set and maximum clique algorithms for overlap graphs
Discrete Appl. Math. 131, No.1 77-91 (2003)

(decision variant)

W.-k. Shih, S. Wu, Y.S. Kuo
Unifying Maximum Cut and Minimum Cut of a Planar Graph
IEEE Transactions on Computer 39 No.5 694-697 (1990)