Parameter: branchwidth

Definition:

A branch decomposition of a graph $G$ is a pair $(T,\chi)$, 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 $X, E \backslash X$, 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 $G$.

References

[922]
N. Robertson, P.D. Seymour
Graph minors. X. Obstructions to tree--decomposition
J. Comb. Theory (B) 52 1991 153--190
[1699]
R. Sasák
Comparing 17 graph parameters
Master's thesis, University of Bergen 2010

Equivalent parameters

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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

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This parameter is a minimal upper bound for

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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
Cochromatic colouring
[?]
(decision variant)
Input: A graph G in this class and an integer k.
Output: True iff the vertices of G can be partitioned into at most k sets, such that each partite set induces either an independent set or a clique in G.
Unknown to ISGCI [+]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 canonisation
[?]
Input: A graph G in this class.
Output: A graph C(G) such that C(G) = C(H) iff G is isomorphic to H.
XP [+]Details
Graph isomorphism
[?]
Input: Graphs G and H in this class
Output: True iff G and H are isomorphic.
FPT [+]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.
FPT [+]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.
FPT [+]Details
Independent dominating 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, with |S| <= k, such that every vertex in G is either in S or adjacent to a vertex in S.
Unknown to ISGCI [+]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 bisection
[?]
(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 of equal size (or |A|=|B|-1 if G has an odd number of vertices) such that there are at least k edges in G with one endpoint in A and the other endpoint in B.
Unknown to ISGCI [+]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.
FPT [+]Details
Minimum bisection
[?]
(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 of equal size (or |A|=|B|-1 if G has an odd number of vertices) such that there are at most k edges in G with one endpoint in A and the other endpoint in B.
Unknown to ISGCI [+]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
Weighted maximum cut
[?]
(decision variant)
Input: A graph G in this class with weight function on the edges and a real k.
Output: True iff the vertices of G can be partitioned into two sets A,B such that the sum of weights of the edges in G with one endpoint in A and the other endpoint in B is at least k.
Unknown to ISGCI [+]Details

Graph classes

Bounded

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Unbounded

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Open

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Unknown to ISGCI

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