Related papers: On spanning tree packings of highly edge connected…
We continue and extend previous work on the parameterized complexity analysis of the NP-hard Stable Roommates with Ties and Incomplete Lists problem, thereby strengthening earlier results both on the side of parameterized hardness as well…
We prove, that every connected graph with $s$ vertices of degree 3 and $t$ vertices of degree at least~4 has a spanning tree with at least ${2\over 5}t +{1\over 5}s+\alpha$ leaves, where $\alpha \ge {8\over 5}$. Moreover, $\alpha \ge 2$ for…
We study structural conditions in dense graphs that guarantee the existence of vertex-spanning substructures such as Hamilton cycles. It is easy to see that every Hamiltonian graph is connected, has a perfect fractional matching and,…
We consider the problem of change-point detection in multivariate time-series. The multivariate distribution of the observations is supposed to follow a graphical model, whose graph and parameters are affected by abrupt changes throughout…
A set $S$ of vertices in a graph is an open packing if (open) neighborhoods of any two distinct vertices in $S$ are disjoint. In this paper, we consider the graphs that have a unique maximum open packing. We characterize the trees with this…
This paper develops a harmonic Galois theory for finite graphs, thereby classifying harmonic branched $G$-covers of a fixed base $X$ in terms of homomorphisms from a suitable fundamental group of $X$ together with $G$-inertia structures on…
Trees fill many extremal roles in graph theory, being minimally connected and serving a critical role in the definition of $n$-good graphs. In this article, we consider the generalization of trees to the setting of $r$-uniform hypergraphs…
We define a range of new coarse geometric invariants based on various graph-theoretic measures of complexity for finite graphs, including: treewidth, pathwidth, cutwidth and bandwidth. We prove that, for bounded degree graphs, these…
The arboricity $\Gamma(G)$ of an undirected graph $G = (V,E)$ is the minimal number such that $E$ can be partitioned into $\Gamma(G)$ forests. Nash-Williams' formula states that $k = \lceil \gamma(G) \rceil$, where $\gamma(G)$ is the…
There are many results asserting the existence of tree-decompositions of minimal width which still represent local connectivity properties of the underlying graph, perhaps the best-known being Thomas' theorem that proves for every graph $G$…
We prove that every oriented tree on $n$ vertices with bounded maximum degree appears as a spanning subdigraph of every directed graph on $n$ vertices with minimum semidegree at least $n/2+o(n)$. This can be seen as a directed graph…
We prove that deciding whether the edge set of a graph can be partitionned into two spanning trees with orientation constraints is NP-complete. If P $\neq$ NP then this disproves a conjecture of Recski.
Working with tree graphs is always easier than with loopy ones and spanning trees are the closest tree-like structures to a given graph. We find a correspondence between the solutions of random K-satisfiability problem and those of spanning…
We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs…
The Matrix-Tree Theorem states that the number of spanning trees of a graph is given by the absolute value of any cofactor of the Laplacian matrix of the graph. We propose a very short proof of this result which amounts to comparing Taylor…
We present a link-by-link rule-based method for constructing all members of the ensemble of spanning trees for any recursively generated, finitely articulated graph, such as the DGM net. The recursions allow for many large-scale properties…
The following theorem is proved: For all $k$-connected graphs $G$ and $H$ each with at least $n$ vertices, the treewidth of the cartesian product of $G$ and $H$ is at least $k(n -2k+2)-1$. For $n\gg k$ this lower bound is asymptotically…
We prove several results showing that every locally finite Borel graph whose large-scale geometry is "tree-like" induces a treeable equivalence relation. In particular, our hypotheses hold if each component of the original graph either has…
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 introduce the problem of finding a spanning tree along with a partition of the tree edges into fewest number of feasible sets, where constraints on the edges define feasibility. The motivation comes from wireless networking, where we…