A note on forbidding clique immersions

Robertson and Seymour proved that the relation of graph immersion is well-quasi-ordered for finite graphs. Their proof uses the results of graph minors theory. Surprisingly, there is a very short proof of the corresponding rough structure theorem for graphs without $K_t$-immersions; it is based on the Gomory-Hu theorem. The same proof also works to establish a rough structure theorem for Eulerian digraphs without $\vec{K}_t$-immersions, where $\vec{K}_t$ denotes the bidirected complete digraph of order $t$

Cutwidth: obstructions and algorithmic aspects

Cutwidth is one of the classic layout parameters for graphs. It measures how well one can order the vertices of a graph in a linear manner, so that the maximum number of edges between any prefix and its complement suffix is minimized. As graphs of cutwidth at most $k$ are closed under taking immersions, the results of Robertson and Seymour imply that there is a finite list of minimal immersion obstructions for admitting a cut layout of width at most $k$. We prove that every minimal immersion obstruction for cutwidth at most $k$ has size at most $2^{O(k^3\log k)}$. As an interesting algorithmic byproduct, we design a new fixed-parameter algorithm for computing the cutwidth of a graph that runs in time $2^{O(k^2\log k)}\cdot n$, where $k$ is the optimum width and $n$ is the number of vertices. While being slower by a $\log k$-factor in the exponent than the fastest known algorithm, given by Thilikos, Bodlaender, and Serna in [Cutwidth I: A linear time fixed parameter algorithm, J. Algorithms, 56(1):1--24, 2005] and [Cutwidth II: Algorithms for partial $w$-trees of bounded degree, J. Algorithms, 56(1):25--49, 2005], our algorithm has the advantage of being simpler and self-contained; arguably, it explains better the combinatorics of optimum-width layouts

Multigraphs without large bonds are wqo by contraction

We show that the class of multigraphs with at most $p$ connected components and bonds of size at most $k$ is well-quasi-ordered by edge contraction for all positive integers $p,k$. (A bond is a minimal non-empty edge cut.) We also characterize canonical antichains for this relation and show that they are fundamental

Constructing graphs with no immersion of large complete graphs

In 1989, Lescure and Meyniel proved, for $d=5, 6$, that every $d$-chromatic graph contains an immersion of $K_d$, and in 2003 Abu-Khzam and Langston conjectured that this holds for all $d$. In 2010, DeVos, Kawarabayashi, Mohar, and Okamura proved this conjecture for $d = 7$. In each proof, the $d$-chromatic assumption was not fully utilized, as the proofs only use the fact that a $d$-critical graph has minimum degree at least $d - 1$. DeVos, Dvo\v{r}\'ak, Fox, McDonald, Mohar, and Scheide show the stronger conjecture that a graph with minimum degree $d-1$ has an immersion of $K_d$ fails for $d=10$ and $d\geq 12$ with a finite number of examples for each value of $d$, and small chromatic number relative to $d$, but it is shown that a minimum degree of $200d$ does guarantee an immersion of $K_d$. In this paper we show that the stronger conjecture is false for $d=8,9,11$ and give infinite families of examples with minimum degree $d-1$ and chromatic number $d-3$ or $d-2$ that do not contain an immersion of $K_d$. Our examples can be up to $(d-2)$-edge-connected. We show, using Haj\'os' Construction, that there is an infinite class of non-$(d-1)$-colorable graphs that contain an immersion of $K_d$. We conclude with some open questions, and the conjecture that a graph $G$ with minimum degree $d - 1$ and more than $\frac{|V(G)|}{1+m(d+1)}$ vertices of degree at least $md$ has an immersion of $K_d$

A General Framework for Well-Structured Graph Transformation Systems

Graph transformation systems (GTSs) can be seen as wellstructured transition systems (WSTSs), thus obtaining decidability results for certain classes of GTSs. In earlier work it was shown that wellstructuredness can be obtained using the minor ordering as a well-quasiorder. In this paper we extend this idea to obtain a general framework in which several types of GTSs can be seen as (restricted) WSTSs. We instantiate this framework with the subgraph ordering and the induced subgraph ordering and apply it to analyse a simple access rights management system.Comment: Extended version (including proofs) of a paper accepted at CONCUR 201