Related papers: Well-quasi-ordered classes of bounded clique-width
In the final paper of the Graph Minors series N. Robertson and P. Seymour proved that graphs are well-quasi-ordered under the immersion ordering. A direct implication of this theorem is that each class of graphs that is closed under taking…
Algorithmic decidability is established for two order-theoretic properties of downward closed subsets defined by finitely many obstructions in two infinite posets. The properties under consideration are: (a) being atomic, i.e. not being…
Half graphs and their variants, such as ladders, semi-ladders and co-matchings, are combinatorial objects that encode total orders in graphs. Works by Adler and Adler (Eur. J. Comb.; 2014) and Fabia\'nski et al. (STACS; 2019) prove that in…
The bandwidth of a graph is the labeling of vertices with minimum maximum edge difference. For many graph families this is NP-complete. A classic result computes the bandwidth for the hypercube. We generalize this result to give sharp lower…
We consider the problem of classifying those graphs that arise as an undirected square of an oriented graph by generalising the notion of quasi-transitive directed graphs to mixed graphs. We fully classify those graphs of maximum degree…
Barbieri recently showed that the finite graphs realising any given finite automorphism group have unbounded genus, answering a question of Cornwell et al. In this note we give a short proof of a stronger result: they have unbounded clique…
We resolve the computational complexity of Graph Isomorphism for classes of graphs characterized by two forbidden induced subgraphs $H_1$ and $H_2$ for all but six pairs $(H_1,H_2)$. Schweitzer had previously shown that the number of open…
We prove that the strong immersion order is a well-quasi-ordering on the class of semi-complete digraphs, thereby strengthening a result of Chudnovsky and Seymour that this holds for the class of tournaments.
We consider the connected variant of the classic mixed search game where, in each search step, cleaned edges form a connected subgraph. We consider graph classes with bounded connected (and monotone) mixed search number and we deal with the…
A graph is perfectly divisible if for each of its induced subgraph $H$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $\omega(H[B]) < \omega(H)$, and a graph $G$ is perfectly weight divisible if for every…
An ordered graph is a graph with a total order over its vertices. A linear layout of an ordered graph is a partition of the edges into sets of either non-crossing edges, called stacks, or non-nesting edges, called queues. The stack (queue)…
A (Hasse) diagram of a finite partially ordered set (poset) P will be called quasiplanar if for any two incomparable elements u and v, either v is on the left of all maximal chains containing u, or v is on the right of all these chains.…
The class of bipartite permutation graphs enjoys many nice and important properties. In particular, this class is critically important in the study of clique- and rank-width of graphs, because it is one of the minimal hereditary classes of…
Here we prove that a graph without some three induced subgraphs has chromatic number at the most equal to its maximum clique size plus one. Further we show that the bounds are tight and give examples to show that each of the three forbidden…
Blokhuis showed that all maximum cliques in Paley graphs of square order have a subfield structure. Recently, it has been shown that in Peisert-type graphs, all maximum cliques are affine subspaces, and yet some maximum cliques do not arise…
We characterize clique trees of a chordal graph in their relation to simplicial vertices and perfect sequences of maximal cliques. We investigate boundary cliques defined by Shibata and clarify their relation to endpoints of clique trees.…
A class of graphs closed under taking induced subgraphs is $\chi$-bounded if there exists a function $f$ such that for all graphs $G$ in the class, $\chi(G) \leq f(\omega(G))$. We consider the following question initially studied in [A.…
A contraction sequence of a graph consists of iteratively merging two of its vertices until only one vertex remains. The recently introduced twin-width graph invariant is based on contraction sequences. More precisely, if one puts red edges…
A plane graph $H$ is a {\em plane minor} of a plane graph $G$ if there is a sequence of vertex and edge deletions, and edge contractions performed on the plane, that takes $G$ to $H$. Motivated by knot theory problems, it has been asked if…
A subset of vertices in a graph is called a total dominating set if every vertex of the graph is adjacent to at least one vertex of this set. A total dominating set is called minimal if it does not properly contain another total dominating…