Related papers: Finding minors in graphs with a given path structu…

In the first paper of the Graph Minors series [JCTB '83], Robertson and Seymour proved the Forest Minor theorem: the $H$-minor-free graphs have bounded pathwidth if and only if $H$ is a forest. In recent years, considerable effort has been…

A graph $G$ contains $H$ as an \emph{immersion} if there is an injective mapping $\phi: V(H)\rightarrow V(G)$ such that for each edge $uv\in E(H)$, there is a path $P_{uv}$ in $G$ joining vertices $\phi(u)$ and $\phi(v)$, and all the paths…

We show that for every fixed undirected graph $H$, there is a $O(|V(G)|^3)$ time algorithm that tests, given a graph $G$, if $G$ contains $H$ as a topological subgraph (that is, a subdivision of $H$ is subgraph of $G$). This shows that…

A graph $H$ is an induced minor of a graph $G$ if it can be obtained from an induced subgraph of $G$ by contracting edges. Otherwise, $G$ is said to be $H$-induced minor-free. Robin Thomas showed that $K_4$-induced minor-free graphs are…

We consider $2$-colourings $f : E(G) \rightarrow \{ -1 ,1 \}$ of the edges of a graph $G$ with colours $-1$ and $1$ in $\mathbb{Z}$. A subgraph $H$ of $G$ is said to be a zero-sum subgraph of $G$ under $f$ if $f(H) := \sum_{e\in E(H)} f(e)…

Given graphs $G$ and $H$, we propose a method to implicitly enumerate topological-minor-embeddings of $H$ in $G$ using decision diagrams. We show a useful application of our method to enumerating subgraphs characterized by forbidden…

The immersion relation is a partial ordering relation on graphs that is weaker than the topological minor relation in the sense that if a graph $G$ contains a graph $H$ as a topological minor, then it also contains it as an immersion but…

We give an algorithm that, given graphs $G$ and $H$, tests whether $H$ is a minor of $G$ in time ${\cal O}_H(n^{1+o(1)})$; here, $n$ is the number of vertices of $G$ and the ${\cal O}_H(\cdot)$-notation hides factors that depend on $H$ and…

Given graphs $G$ and $H$, we consider the problem of decomposing a properly edge-colored graph $G$ into few parts consisting of rainbow copies of $H$ and single edges. We establish a close relation to the previously studied problem of…

The bidimensionality of a set of vertices $X$ in a graph $G$ is the maximum $k$ for which $G$ contains as a $X$-rooted minor the $(k \times k)$-grid. This notion allows for the following version of the Graph Minors Structure Theorem (GMST)…

A graph $G$ is $k$-path-coverable if its vertex set $V(G)$ can be covered by $k$ or fewer vertex disjoint paths. In this paper, using the $Q$-index of a connected graph $G$, we present a tight sufficient condition for $G$ with fixed minimum…

We consider Colouring on graphs that are $H$-subgraph-free for some fixed graph $H$, which are graphs that do not contain $H$ as a subgraph. To classify the complexity of Colouring on $H$-subgraph-free graphs for connected $H$, it remains…

A graph $H$ is an induced minor of a graph $G$ if $H$ can be obtained from $G$ by vertex deletions and edge contractions. We show that there is a function $f(k, d) = O(k^{10} + 2^{d^5})$ so that if a graph has treewidth at least $f(k, d)$…

A graph $G$ contains a graph $H$ as a pivot-minor if $H$ can be obtained from $G$ by applying a sequence of vertex deletions and edge pivots. Pivot-minors play an important role in the study of rank-width. Pivot-minors have mainly been…

Given a finite simple undirected graph $G$, let $T_1(G)$ denote the subset of vertices of $G$ such that every vertex of $T_1(G)$ belongs to at least one subgraph isomorphic to a graph obtained by connecting a single vertex to two vertices…

Let $G = (G_1, G_2, \ldots, G_m)$ be a collection of $m$ graphs on a common vertex set $V$. For a graph $H$ with vertices in $V$, we say that $G$ contains a rainbow $H$ if there is an injection $c: E(H) \to [m]$ such that for every edge $e…

Let $G$ be a graph and $x_1, x_2, \ldots, x_k$ be distinct vertices of $G$. We say $(G,x_1x_2\ldots x_k)$ has a $C_k$-minor or $G$ has a $C_k$-minor rooted at $x_1x_2\ldots x_k$, if there exist pairwise disjoint sets $X_1, X_2, \ldots,…

Hadwiger's Conjecture from 1943 states that every graph with no $K_{t}$ minor is $(t-1)$-colorable; it remains wide open for all $t\ge 7$. For positive integers $t$ and $s$, let $\mathcal{K}_t^{-s}$ denote the family of graphs obtained from…

In a graph $G$, a subset of vertices $S \subseteq V(G)$ is said to be cyclable if there is a cycle containing the vertices in some order. $G$ is said to be $k$-cyclable if any subset of $k \geq 2$ vertices is cyclable. If any $k$…

The Surjective Homomorphism problem is to test whether a given graph G called the guest graph allows a vertex-surjective homomorphism to some other given graph H called the host graph. The bijective and injective homomorphism problems can…