Questions tagged [graph-minors]
A graph $H$ is called a minor of a graph $G$ if $H$ can be obtained from $G$ by contracting edges, deleting edges, and deleting isolated vertices.
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Can a problem be simultaneously polynomial time and undecidable?
The Robertson-Seymour theorem on graph minors leads to some interesting conundrums.
The theorem states that any minor-closed class of graphs can be described by a finite number of excluded minors. As ...
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Planar minor graphs
The theorem of Robertson-Seymour about graph minors says that there exists no infinite family of graphs such that none of them is a minor of another one.
Apparently, it came as a generalization of ...
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Obstructions for embedding a graph on a surface of genus g
Kuratowski's theorem tells us the complete graph $K_5$ and the bipartite graph $K_{3,3}$ are the only obstructions to a graph being planar, ie embeddable in the plane with no edge-crossings.
Is the ...
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Two disjoint trees
Let $G$ be a graph and let $A_1, A_2 \subseteq V(G)$ be disjoint sets of vertices. Let us call $(A_1, A_2)$ independent if there exist vertex-disjoint trees $T_1, T_2 \subseteq G$ within $G$ which ...
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Does minimal degree $n$ imply a $K_n$ minor
Is it true that any finite graph has a $K_n$ minor, where $n$ is a minimal vertex degree?
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A separation property of graphs of bounded tree-width
The following separation property of trees is well-known and in fact easy to prove (see e.g. the paper "Covering a hypergraph of subgraphs" by Noga Alon, Lemma 2.2)
Let $T$ be a tree and $r, m$ non-...
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Is the "surface-minor" ordering of plane graphs a well-quasi-ordering?
A plane graph is a finite simple graph with a fixed embedding into the two-sphere. The embedding induces an embedding on a minors of a plane graph (i.e. a graph obtained by successive removal of ...
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Number of edges in linklessly embeddable graphs
What is the maximum number of edges of an $n$-vertex linklessly embeddable graph?
A more general question is the following. What is the maximum number of edges of an $n$-vertex graph with Colin de ...
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Graph minor check
Are there any good algebraic/algorithmic tools available to check if a given graph $H$ is a minor of $G$ from the adjacency matrix of $G$?
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$|G|/\alpha(G) \leq \eta(G)$ where $\eta(G)$ is the Hadwiger number
Let $G=(V,E)$ be a finite, simple, undirected graph. The Hadwiger number $\eta(G)$ is the maximum $n\in\mathbb{N}$ such that $K_n$ is a minor of $G$.
Hadwiger's celebrated conjecture states that $\chi(...
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What is a hypergraph minor?
Is there a theory of hypergraph minors? I could only find some attempts to define them at papers/theses, whose main topic was something else. What would be a useful definition? Does the hypergraph ...
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Minor-closed classes of graphs with large numbers of excluded minors
Robertson and Seymour tell us that any minor-closed family of graphs has a finite collection of excluded minors.
Standard examples include planar graphs with two excluded minors ($K_5$ and $K_{3,3}$) ...
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Does every $4$-connected nonplanar graph contain a $K_5$-minor?
By Kuratowski's theorem, every nonplanar graph contains a (topological) minor of $K_5$ or $K_{3,3}$.
But I observed that every time I construct a $4$-connected nonplanar graph, it always contains not ...
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Disjoint paths between four vertices
Consider the following property of an undirected graph: For any four distinct vertices $a,b,c,d$, there is a path from $a$ to $b$ and a path from $c$ to $d$ such that the two paths do not share any ...
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Does the purported proof of Rota's conjecture provide an algorithm for calculating the forbidden minors of matroids over arbitrary finite fields?
About six years ago there was a proof announced and later outlined in a notice from AMS. However right now I can only seem to find forbidden minor characterizations for matroids linearly ...
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Directed graph minor theorems
In proving the graph minor theorem, Robertson and Seymour proved a stronger statement, namely that the directed graph minor theorem is true, using the definition
A directed graph is a minor of ...
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Characterizing SP-DAGs by Forbidden Minors?
So it's well-known that an alternative way to define a series-parallel (undirected graph) is by the forbidden minor $K_4$. Is there a known analog of this definition for directed graphs — ...
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Graph minors, and Kronecker product
Let $X$ and $Y$ be graphs and consider the Kronecker product: $Z = X \otimes Y$. Is it true that if $X$ excludes an $M$-minor, $Z$ excludes an $M \otimes Y$ minor?
I am particularly interested in the ...
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Bounds on degrees of minors obtained by edge contractions of regular graphs
Given a connected $d$-regular graph $G=(V,E)$, generate a sequence of minors by performing only edge contractions and loop deletions (as, e.g., in Karger's algorithm) until the graph collapses to a ...
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Reference to a definition of a graph homology
Let $G$ be a graph, and define $C_k$ to be the free abelian group on the homomorphisms from graphs $H$ such that $K_k$ is a minor of $H$ without needing to do any vertex deletions, only edge ...
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Generalized graph-minor theorem?
Consider the following generalized graph-minor theorem:
GM($κ,λ$): Given any collection $S$ of $κ$ simple undirected graphs each with less than $λ$ vertices, there are distinct graphs $G,H$ in $S$ ...
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Upper bound on size of obstruction set for wye-delta-wye reducible graphs
A graph is $Y \Delta Y$-reducible if it can be reduced to an empty graph by the following operations:
$Y \leftrightarrow\Delta$ transforms;
Replacing multiple edges with single edges (parallel ...
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Is there a polynomial-time algorithm to check if a signed graph contains an odd-K5 minor?
I suspect this exists, if anyone has a reference please that would be very helpful.
By signed graph, I mean each edge is designated either odd or even (e.g. as in Guenin's result for weakly bipartite ...
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Menger's theorem with restrictions on where the paths can begin and end
Let $k\in\mathbb N$. Given a finite graph with two subsets of vertices $X$ and $Y$, Menger's Theorem gives a criterion for when there are $k$ pairwise disjoint paths starting in $X$ and ending in $Y$.
...
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A claim from "Graph minors - a survey" by Robertson and Seymour
Can someone give me a proof sketch for this:
Let $\mathscr{P}_n$ be the set of all graphs which do not contain a path on $n$ vertices as a subgraph. Define the type of a graph inductively as: the type ...
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Forbidden minors of a graph with treewidth at most 4
I am interested in the graphs with treewidth 5 because of their relationship with the realization dimension of a graph (see here).
In this PhD thesis, 75 minimal forbidden minors of graphs with ...
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4-color theorem for hypergraphs
Question. Does every hypergraph that does not admit a complete minor with $5$ elements have a coloring with $4$ colors?
Below are the definitions to make this precise.
If $H = (V, E)$ is a hypergraph ...
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Ref request: A graph G contains H as a minor iff it contains one of finitely many graphs as a topological minor
For definitions of graph minors and topological minors, see wikipedia's article on graph minors.
Theorem: For every graph H, there is a finite set of graphs, say S(H), such that G contains H as a ...
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Increasing the Hadwiger number by identifying non-adjacent points
This is a specialization of a more general, still unanswered question.
Suppose $G$ is a finite, simple graph. Let $h(G)$ denote the Hadwiger number, that is, the maximum $n\in\mathbb{N}$ such that $...
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Asymptotics of list size in Robertson-Seymour theorem
A planar graph cannot have $K_5$ and $K_{3,3}$ as minors. Robertson-Seymour theorem generalizes this by stating for every genus $g$ there is a finite list of forbidden minor graphs that are ...
user76479
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Do graphs with large number of paths contain large chain minor?
Definition: A "$k$-chain" is a multi-graph obtained from a path of length $k$ by duplicating every edge.
Note that the number of paths between two endpoints of a $k$-chain is $2^k.$
Question: Let $G$...
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Are K_t-minor free graphs on small vertex sets understood?
In a paper on Hadwiger's conjecture, https://web.math.princeton.edu/~pds/papers/hadwiger/paper.pdf, Seymour explains various results on excluding the complete graph as a minor.
In particular, there is ...
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Forbidden minor characterization of polytope skeletons
Say that a graph is "$d$-dimensional" if it is the node-disjoint union of $1$-skeletons of closed convex polytopes in $d$ dimensions, or a subgraph thereof. So the $2$-dimensional graphs are exactly ...
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Contracting a planar graph to a (multiply-edged)-tree
Given a planar graph (no loops, no multiple edge), is it always possible to perform edge contractions* in order to obtain a graph $T$ which has no loops, and if one ignores parallel edges, $T$ is a ...
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Do graphs with large number of cycles always contain large necklace minor?
Let "$k$-necklace" denote the (multi)graph obtained from a cycle of length $k$ by duplicating every edge.
Note that the number of cycles in $k$-necklace is at least $2^k.$
Question : Suppose a ...
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Classes of graphs that are minors of bounded degree graphs in the same class
Notice that every planar graph $G$ is a minor of a planar graph $H$ with maximum degree $\Delta(H)\leq 3$ (replace each vertex of $G$ by a sub-cubic tree to obtain $H$). The same idea can be applied ...
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Increasing the Hadwiger number by making any pair of non-adjacent points adjacent
Let $G=(V,E)$ be a finite, simple, undirected graph. The Hadwiger number $\eta(G)$ of $G$ is defined to be the largest positive integer $n\in\mathbb{N}$ such that the complete graph $K_n$ is a minor ...
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Induced minors and induced topological minors
Question: For which graphs $H$ is the following true?
Every graph that contains $H$ as an induced minor also contains $H$ as an induced topological minor.
Definitions:
Let $G$ and $H$ be graphs.
$H$ ...
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Two types of criticality
Suppose $G$ is a finite simple graph. Let $h(G)$ denote the Hadwiger number (synonym: connected pseudoachromatic number, cf. [Abrams-Berman 2014, p. 315]) of $G$; that is, the maximum $n\in\mathbb{N}$ ...
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Hadwiger's conjecture in the language of graph homomorphisms
Consider the following statement:
(S): If $G$ is not a complete graph, then there is a minor $M$ of $G$ such that
$M \not \cong G$, and
there is a graph homomorphism $f:G\to M$.
Hadwiger's ...
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Is there a version of Robertson-Seymour's graph minor theorem known to apply to signed graphs?
Here a signed graph is one where each edge is signed either odd or even. A cycle is odd or even according to the sum of the signs of its edges. For a given signed graph, a resigning may be performed ...
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Bounding the size of clique minor of the union of two graphs
Suppose that graphs $A$ and $B$ with $V(A)=V(B)$ have Hadwiger numbers $a$ and $b$. That is, $K_a$ and $K_b$ are the largest clique minors of $A$ and $B$, respectively.
Are there upper bounds on the ...
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Let $G$ be a graph of genus $g$. Is the number of (non necessarily disjoint) 5-clique subgraphs at most $f(g)$ for some function $f$?
For a graph of genus $g$, it holds that it cannot have too many disjoint 5-cliques, as each clique requires a new handle. It feels that given a graph of genus $g$, it cannot have an unbounded number ...
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What is the relation between Hadwiger number and Treewidth?
Is there any general relation between Hadwiger number and Treewidth of a graph? Intuitively I think Hadwiger number is greater than or equal to Treewidth, but I couldn't prove it.
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Big binary tree as an induced subgraph
I believe this is true:
Suppose $G$ is a graph. If $G$ has a subdivision of a large binary tree, prove that $G$ has an
induced subgraph which is a subdivision of a large binary tree or the line ...
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Forbidden structures for generalized hypertree width
Generalized hypertree width is a tree-width-like parameter for hypergraphs, which plays an important role in the study of constraint satisfaction problems and related areas. For its more well-known ...
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Hadwiger number of Erdös-Faber-Lovasz graphs
For any set $X$, let $[X]^2 = \big\{\{a,b\}:a,b \in X, a\neq b\big\}$.
We call a finite, simple, undirected graph $G=(V,E)$ an $n$-Erdös-Faber-Lovasz (EFL-) graph if there are $n$ subsets $S_1,\...
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Hadwiger number and minimal degree
Suppose $G$ is a finite simple graph and $\eta(G)$ is the maximum $n\in\mathbb{N}$ such that $K_n$ is a minor of $G$. If $\delta(G)$ is the minimal degree of $G$, do we have $\delta(G)\leq\eta(G)$?
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Are there good ways of relating a minor to the full determinant?
Say $A$ is a $(n-1)\times (n-1)$ matrix and we augment it by a $n^{th}$ row and a column and get a $n \times n$ matrix $B$. Is there a nice way to relate $det(B)$ and $det(A)$ and the added row and ...
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What is the relation between size of maximum clique and branchwidth?
Let $bw(G)$ be the branchwidth of graph $G$ and $\omega(G)$ be the size of maximum clique in $G$. I think the following inequality holds:
$$
\omega(G)\leq bw(G)
$$
Intuition: Assume (in reverse of ...
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