Related papers: Computing bounded-width tree and branch decomposit…

The notion of treewidth plays an important role in theoretical and practical studies of graph problems. It has been recognized that, especially in practical environments, when computing the treewidth of a graph it is invaluable to first…

A graph G is a k-leaf power, for an integer k >= 2, if there is a tree T with leaf set V(G) such that, for all vertices x, y in V(G), the edge xy exists in G if and only if the distance between x and y in T is at most k. Such a tree T is…

Let $G$ be a graph on $n$ vertices and $1 \le k \le n$ a fixed integer. The \textit{$k$-token graph} of $G$ is the graph $F_k(G)$ whose vertex set consists of all $k$-subsets of the vertex set of $G$, where two vertices $A$ and $B$ are…

We prove two results relating the basis number of a graph $G$ to path decompositions of $G$. Our first result shows that the basis number of a graph is at most four times its pathwidth. Our second result shows that, if a graph $G$ has a…

We study the existence and construction of sparse supports for hypergraphs derived from subgraphs of a graph $G$. For a hypergraph $(X,\mathcal{H})$, a support $Q$ is a graph on $X$ s.t. $Q[H]$, the graph induced on vertices in $H$ is…

We introduce the notion of \emph{bounded diameter arboricity}. Specifically, the \emph{diameter-$d$ arboricity} of a graph is the minimum number $k$ such that the edges of the graph can be partitioned into $k$ forests each of whose…

The Tree Decomposition Conjecture by Bar\'at and Thomassen states that for every tree $T$ there exists a natural number $k(T)$ such that the following holds: If $G$ is a $k(T)$-edge-connected simple graph with size divisible by the size of…

We study the parameterized complexity of computing the tree-partition-width, a graph parameter equivalent to treewidth on graphs of bounded maximum degree. On one hand, we can obtain approximations of the tree-partition-width efficiently:…

A k-queue layout is a special type of a linear layout, in which the linear order avoids (k+1)-rainbows, i.e., k+1 independent edges that pairwise form a nested pair. The optimization goal is to determine the queue number of a graph, i.e.,…

Treewidth is a parameter that emerged from the study of minor closed classes of graphs (i.e. classes closed under vertex and edge deletion, and edge contraction). It in some sense describes the global structure of a graph. Roughly, a graph…

Dumas, Foucaud, Perez, and Todinca [SIAM J. Disc. Math., 2024] proved that if the vertex set of a graph $G$ can be covered by $k$ shortest paths, then the pathwidth of $G$ is bounded by $\mathcal{O}(k \cdot 3^k)$. We prove a coarse variant…

Treewidth is arguably the most important structural graph parameter leading to algorithmically beneficial graph decompositions. Triggered by a strongly growing interest in temporal networks (graphs where edge sets change over time), we…

In graph theory, as well as in 3-manifold topology, there exist several width-type parameters to describe how "simple" or "thin" a given graph or 3-manifold is. These parameters, such as pathwidth or treewidth for graphs, or the concept of…

Let v(G) be the number of vertices and t(G,k) the maximum number of disjoint k-edge trees in G. In this paper we show that (a1) if G is a graph with every vertex of degree at least two and at most s, where s > 3, then t(G,2) is at least…

We prove that the combinatorial Weisfeiler-Leman algorithm of dimension $(3k+4)$ is a complete isomorphism test for the class of all graphs of rank width at most $k$. Rank width is a graph invariant that, similarly to tree width, measures…

Motivated by the algorithmic study of 3-dimensional manifolds, we explore the structural relationship between the JSJ decomposition of a given 3-manifold and its triangulations. Building on work of Bachman, Derby-Talbot and Sedgwick, we…

Twin-width is a newly introduced graph width parameter that aims at generalizing a wide range of "nicely structured" graph classes. In this work, we focus on obtaining good bounds on twin-width $\text{tww}(G)$ for graphs $G$ from a number…

Every $k$-tree has book thickness at most $k+1$, and this bound is best possible for all $k\geq3$. Vandenbussche et al. (2009) proved that every $k$-tree that has a smooth degree-3 tree decomposition with width $k$ has book thickness at…

Let $k\geq2$ be an integer. A tree $T$ is called a $k$-tree if $d_T(v)\leq k$ for each $v\in V(T)$, that is, the maximum degree of a $k$-tree is at most $k$. Let $\lambda_1(D(G))$ denote the distance spectral radius in $G$, where $D(G)$…

A problem of practical and theoretical interest is to determine or estimate the diameter of various families of Cayley networks. The previously known estimate for the diameter of Cayley graphs generated by transposition trees is an upper…