Related papers: Recognizing Union-Find trees is NP-complete
In this note we analyze the performance of a simple root-finding algorithm in uniform attachment trees. The leaf-stripping algorithm recursively removes all leaves of the tree for a carefully chosen number of rounds. We show that, with…
We consider classes of graphs, which we call thick graphs, that have the vertices of a corresponding thin graph replaced by cliques and the edges replaced by cobipartite graphs In particular, we consider the case of thick forests, which we…
Frequent pattern mining is a relevant method to analyse structured data, like sequences, trees or graphs. It consists in identifying characteristic substructures of a dataset. This paper deals with a new type of patterns for tree data:…
In this brief note a straightforward combinatorial proof for an identity directly connecting rooted forests and unordered set partitions is provided. Furthermore, references that put this type of identity in the context of forest volumes…
In this paper, we give a very simple proof that Treewidth is NP-complete; this proof also shows NP-completeness on the class of co-bipartite graphs. We then improve the result by Bodlaender and Thilikos from 1997 that Treewidth is…
A spanning subgraph $F$ of a graph $G$ is called {\em perfect} if $F$ is a forest, the degree $d_F(x)$ of each vertex $x$ in $F$ is odd, and each tree of $F$ is an induced subgraph of $G$. Alex Scott (Graphs \& Combin., 2001) proved that…
The last in-tree recognition problem asks whether a given spanning tree can be derived by connecting each vertex with its rightmost left neighbor of some search ordering. In this study, we demonstrate that the last-in-tree recognition…
Neural networks (NNs) and decision trees (DTs) are both popular models of machine learning, yet coming with mutually exclusive advantages and limitations. To bring the best of the two worlds, a variety of approaches are proposed to…
This work addresses the intrinsic relationship between trees and networks (i.e. graphs). A complete (invertible) mapping is presented which allows trees to be mapped into weighted graphs and then backmapped into the original tree without…
Leaf powers and pairwise compatibility graphs were introduced over twenty years ago as simplified graph models for phylogenetic trees. Despite significant research, several properties of these graph classes remain poorly understood. In this…
It has been known since 1991 that the problem of recognizing grid intersection graphs is NP-complete. Here we use a modified argument of the above result to show that even if we restrict to the class of unit grid intersection graphs…
We study upward planar straight-line embeddings (UPSE) of directed trees on given point sets. The given point set $S$ has size at least the number of vertices in the tree. For the special case where the tree is a path $P$ we show that: (a)…
Decision trees have been a very popular class of predictive models for decades due to their interpretability and good performance on categorical features. However, they are not always robust and tend to overfit the data. Additionally, if…
A uniform attachment tree is a random tree that is generated dynamically. Starting from a fixed "seed" tree, vertices are added sequentially by attaching each vertex to an existing vertex chosen uniformly at random. Upon observing a large…
The status of a vertex $v$ in a connected graph is the sum of the distances from $v$ to all other vertices. The status sequence of a connected graph is the list of the statuses of all the vertices of the graph. In this paper we investigate…
Spanning trees are fundamental for efficient communication in networks. For fault-tolerant communication, it is desirable to have multiple spanning trees to ensure resilience against failures of nodes and edges. To this end, various notions…
We consider the following problem: Given a finite set of straight line segments in the plane, determine the positions of a minimal number of points on the segments, from which guards can see all segments. This problem can be interpreted as…
In the theoretical study of distributed communication networks, "history trees" are a discrete structure that naturally models the concept that anonymous agents become distinguishable upon receiving different sets of messages from…
Combinatorial optimization lies at the core of many real-world problems. Especially since the rise of graph neural networks (GNNs), the deep learning community has been developing solvers that derive solutions to NP-hard problems by…
We characterize the compatibility of a collection of unrooted phylogenetic trees as a question of determining whether a graph derived from these trees --- the display graph --- has a specific kind of triangulation, which we call legal. Our…