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It is known that any planar graph with diameter D has treewidth O(D), and this fact has been used as the basis for several planar graph algorithms. We investigate the extent to which similar relations hold in other graph families. We show…
As a major step in their proof of Wagner's conjecture, Robertson and Seymour showed that every graph not containing a fixed graph $H$ as a minor has a tree-decomposition in which each torso is almost embeddable in a surface of bounded…
A decomposition of a graph is a set of subgraphs whose edges partition those of $G$. The 3-decomposition conjecture posed by Hoffmann-Ostenhof in 2011 states that every connected cubic graph can be decomposed into a spanning tree, a…
Nash-Williams proved that for an undirected graph $ G $ the set $ E(G) $ can be partitioned into cycles if and only if every cut has either even or infinite number of edges. Later C. Thomassen gave a simpler proof for this and conjectured…
We establish relations between the bandwidth and the treewidth of bounded degree graphs G, and relate these parameters to the size of a separator of G as well as the size of an expanding subgraph of G. Our results imply that if one of these…
A path decomposition of a graph $G$ is a collection of edge-disjoint paths of $G$ that covers the edge set of $G$. Gallai (1968) conjectured that every connected graph on $n$ vertices admits a path decomposition of cardinality at most…
It is known that there is a linear dependence between the treewidth of a graph and its balanced separator number: the smallest integer $k$ such that for every weighing of the vertices, the graph admits a balanced separator of size at most…
A $\overrightarrow{P_{3}}$-decomposition of a directed graph $D$ is a partition of the arcs of $D$ into directed paths of length $2$. In this paper, we give a characterization for a tournament and a bipartite digraph admitting a…
The Koml\'os-S\'ark\"ozy-Szemer\'edi (KSS) theorem establishes that a certain bound on the minimum degree of a graph guarantees it contains all bounded degree trees of the same order. Recently several authors put forward variants of this…
A graph $H$ is an induced subgraph of a graph $G$ if a graph isomorphic to $H$ can be obtained from $G$ by deleting vertices. Recently, there has been significant interest in understanding the unavoidable induced subgraphs for graphs of…
We consider the class of graphs for which the edge connectivity is equal to the maximum number of edge-disjoint spanning trees, and the natural generalization to matroids, where the cogirth is equal to the number of disjoint bases. We…
We present time-efficient distributed algorithms for decomposing graphs with large edge or vertex connectivity into multiple spanning or dominating trees, respectively. As their primary applications, these decompositions allow us to achieve…
We settle the Path Decomposition Conjecture (P.D.C.) due to Tibor Gallai for minimally connected graphs, i.e. trees. We use this validity for trees and settle the P. D. C. using induction on the number of edges for all connected graphs. We…
We prove an ear-decomposition theorem for $4$-edge-connected graphs and use it to prove that for every $4$-edge-connected graph $G$ and every $r\in V(G)$, there is a set of four spanning trees of $G$ with the following property. For every…
Directed graphs can be studied by their associated directed flag complex. The homology of this complex has been successful in applications as a topological invariant for digraphs. Through comparison with path homology theory, we derive a…
We prove that every digraph of circumference $l$ has DAG-width at most $l$ and this is best possible. As a consequence of our result we deduce that the $k$-linkage problem is polynomially solvable for every fixed $k$ in the class of…
We solve the following problem: Can an undirected weighted graph G be parti- tioned into two non-empty induced subgraphs satisfying minimum constraints for the sum of edge weights at vertices of each subgraph? We show that this is possible…
Let $G$ be a group. The directed endomorphism graph, $\dend(G)$ of $G$ is a directed graph with vertex set $G$ and there is a directed edge from the vertex $a$ to the vertex $b$ if $a \neq b$ and there exists an endomorphism on $G$ mapping…
Butterfly minors are a generalisation of the minor containment relation for undirected graphs to directed graphs. Many results in directed structural graph theory use this notion as a central tool next to directed treewidth, a…
We affirm several special cases of a conjecture that first appears in Alspach et al.~(1987) which stipulates that the wreath (lexicographic) product of two hamiltonian decomposable directed graphs is also hamiltonian decomposable.…