Related papers: An Effective Upperbound on Treewidth Using Partial…
We consider a variant of treewidth that we call clique-partitioned treewidth in which each bag is partitioned into cliques. This is motivated by the recent development of FPT-algorithms based on similar parameters for various problems. With…
We present a method for reducing the treewidth of a graph while preserving all the minimal $s-t$ separators. This technique turns out to be very useful for establishing the fixed-parameter tractability of constrained separation and…
Bodlaender and Koster (Discrete Mathematics 2006) introduced the notion of almost-clique separators in the context of computing the treewidth $\tw(G)$ of a given graph $G$. A separator $S \subseteq V(G)$ of $G$ is an \emph{almost-clique…
We consider a variant of the clustering problem for a complete weighted graph. The aim is to partition the nodes into clusters maximizing the sum of the edge weights within the clusters. This problem is known as the clique partitioning…
For a given graph G and integers b,f >= 0, let S be a subset of vertices of G of size b+1 such that the subgraph of G induced by S is connected and S can be separated from other vertices of G by removing f vertices. We prove that every…
In 2020, we initiated a systematic study of graph classes in which the treewidth can only be large due to the presence of a large clique, which we call $(\mathrm{tw},\omega)$-bounded. While $(\mathrm{tw},\omega)$-bounded graph classes are…
Cliquewidth is a dense analogue of treewidth. It can be deduced from recent results by Hickingbotham [arXiv:2501.10840] and Nguyen, Scott, and Seymour [arXiv:2501.09839] that graphs of bounded cliquewidth are quasi-isometric to graphs of…
Many hard graph problems can be solved efficiently when restricted to graphs of bounded treewidth, and more generally to graphs of bounded clique-width. But there is a price to be paid for this generality, exemplified by the four problems…
The treewidth of a graph is an important invariant in structural and algorithmic graph theory. This paper studies the treewidth of line graphs. We show that determining the treewidth of the line graph of a graph $G$ is equivalent to…
Parameterized complexity seeks to use input structure to obtain faster algorithms for NP-hard problems. This has been most successful for graphs of low treewidth: Many problems admit fast algorithms relative to treewidth and many of them…
Graphs with bounded treewidth and bounded maximum degree are known to have tree-partitions of bounded width. What can be said if the bounded treewidth assumption is strengthened to bounded pathwidth? We prove that every graph with bounded…
We are interested in computing the treewidth $\tw(G)$ of a given graph $G$. Our approach is to design heuristic algorithms for computing a sequence of improving upper bounds and a sequence of improving lower bounds, which would hopefully…
We define a special case of tree decompositions for planar graphs that respect a given embedding of the graph. We study the analogous width of the resulting decomposition we call the embedded-width of a plane graph. We show both upper…
A graph has tree-width at most $k$ if it can be obtained from a set of graphs each with at most $k+1$ vertices by a sequence of clique sums. We refine this definition by, for each non-negative integer $\theta$, defining the…
For $n$-vertex graphs with treewidth $k = O(n^{1/2-\epsilon})$ and an arbitrary $\epsilon>0$, we present a word-RAM algorithm to compute vertex separators using only $O(n)$ bits of working memory. As an application of our algorithm, we give…
We prove that the (divisorial) gonality of a finite connected graph is lower bounded by its treewidth. We show that equality holds for grid graphs and complete multipartite graphs. We prove that the treewidth lower bound also holds for…
We continue the study of $(\mathrm{tw},\omega)$-bounded graph classes, that is, hereditary graph classes in which the treewidth can only be large due to the presence of a large clique, with the goal of understanding the extent to which this…
For a tree decomposition $\mathcal{T}$ of a graph $G$, let $\mu(\mathcal{T})$ denote the maximum size of an induced matching in $G$ with the property that some bag of $\mathcal{T}$ contains at least one endpoint of every edge of the…
We present a method for reducing the treewidth of a graph while preserving all of its minimal $s-t$ separators up to a certain fixed size $k$. This technique allows us to solve $s-t$ Cut and Multicut problems with various additional…
For a graph $G$, let $\Pi(G)$ denote the set of all potential maximal cliques of $G$. For each subset $\Pi$ of $\Pi(G)$, let $\tw(G, \Pi)$ denote the smallest $k$ such that there is a tree-decomposition of $G$ of width $k$ whose bags all…