Parameter: rankwidth

Definition:

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 $G$.

References

[1723]
Sang-il Oum, P. Seymour
Approximating clique-width and branch-width
Journal of Combinatorial Theory, Series B 96 No.4, 514-528 (2006)

Equivalent parameters

[+]Details

Relations

Minimal/maximal is with respect to the contents of ISGCI. Only references for direct bounds are given. Where no reference is given, check equivalent parameters.

Minimal upper bounds for this parameter

[+]Details

Problems

Problems in italics have no summary page and are only listed when ISGCI contains a result for the current parameter.

3-Colourability
[?]
Input: A graph G in this class.
Output: True iff each vertex of G can be assigned one colour out of 3 such that whenever two vertices are adjacent, they have different colours.
FPT [+]Details
Clique
[?]
Input: A graph G in this class and an integer k.
Output: True iff G contains a set S of pairwise adjacent vertices, with |S| >= k.
FPT [+]Details
Clique cover
[?]
Input: A graph G in this class and an integer k.
Output: True iff the vertices of G can be partitioned into k sets Si, such that whenever two vertices are in the same set Si, they are adjacent.
XP [+]Details
Colourability
[?]
Input: A graph G in this class and an integer k.
Output: True iff each vertex of G can be assigned one colour out of k such that whenever two vertices are adjacent, they have different colours.
FPT [+]Details
Domination
[?]
Input: A graph G in this class and an integer k.
Output: True iff G contains a set S of vertices, with |S| <= k, such that every vertex in G is either in S or adjacent to a vertex in S.
FPT [+]Details
Feedback vertex set
[?]
Input: A graph G in this class and an integer k.
Output: True iff G contains a set S of vertices, with |S| <= k, such that every cycle in G contains a vertex from S.
FPT [+]Details
Graph isomorphism
[?]
Input: Graphs G and H in this class
Output: True iff G and H are isomorphic.
XP [+]Details
Hamiltonian cycle
[?]
Input: A graph G in this class.
Output: True iff G has a simple cycle that goes through every vertex of the graph.
XP [+]Details
Hamiltonian path
[?]
Input: A graph G in this class.
Output: True iff G has a simple path that goes through every vertex of the graph.
XP [+]Details
Independent set
[?]
Input: A graph G in this class and an integer k.
Output: True iff G contains a set S of pairwise non-adjacent vertices, such that |S| >= k.
FPT [+]Details
Maximum cut
[?]
(decision variant)
Input: A graph G in this class and an integer k.
Output: True iff the vertices of G can be partitioned into two sets A,B such that there are at least k edges in G with one endpoint in A and the other endpoint in B.
XP,W-hard [+]Details
Monopolarity
[?]
Input: A graph G in this class.
Output: True iff G is monopolar.
Unknown to ISGCI [+]Details
Polarity
[?]
Input: A graph G in this class.
Output: True iff G is polar.
XP [+]Details
Weighted clique
[?]
Input: A graph G in this class with weight function on the vertices and a real k.
Output: True iff G contains a set S of pairwise adjacent vertices, such that the sum of the weights of the vertices in S is at least k.
FPT [+]Details
Weighted feedback vertex set
[?]
Input: A graph G in this class with weight function on the vertices and a real k.
Output: True iff G contains a set S of vertices, such that the sum of the weights of the vertices in S is at most k and every cycle in G contains a vertex from S.
FPT [+]Details
Weighted independent dominating set
[?]
Input: A graph G in this class with weight function on the vertices and a real k.
Output: True iff G contains a set S of pairwise non-adjacent vertices, with the sum of the weights of the vertices in S at most k, such that every vertex in G is either in S or adjacent to a vertex in S.
FPT [+]Details
Weighted independent set
[?]
Input: A graph G in this class with weight function on the vertices and a real k.
Output: True iff G contains a set S of pairwise non-adjacent vertices, such that the sum of the weights of the vertices in S is at least k.
FPT [+]Details

Graph classes

Bounded

Unbounded

Open

Unknown to ISGCI