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The structure of an evolving network contains information about its past. Extracting this information efficiently, however, is, in general, a difficult challenge. We formulate a fast and efficient method to estimate the most likely history…
The tree-depth problem can be seen as finding an elimination tree of minimum height for a given input graph $G$. We introduce a bicriteria generalization in which additionally the width of the elimination tree needs to be bounded by some…
Counterfactual explanations are usually generated through heuristics that are sensitive to the search's initial conditions. The absence of guarantees of performance and robustness hinders trustworthiness. In this paper, we take a…
We construct a sequence of finite graphs that weakly converge to a Cayley graph, but there is no labelling of the edges that would converge to the corresponding Cayley diagram. A similar construction is used to give graph sequences that…
When gene copies are sampled from various species, the resulting gene tree might disagree with the containing species tree. The primary causes of gene tree and species tree discord include lineage sorting, horizontal gene transfer, and gene…
We give a counterexample to a recently conjectured variant of the Penrose inequality.
Recent developments in string duality suggest that the string scale may not be irrevocably tied to the Planck scale. Two explicit but unrealistic examples are described where the ratio of the string scale to the Planck scale is arbitrarily…
For a simple drawing $D$ of the complete graph $K_n$, two (plane) subdrawings are compatible if their union is plane. Let $\mathcal{T}_D$ be the set of all plane spanning trees on $D$ and $\mathcal{F}(\mathcal{T}_D)$ be the compatibility…
We consider two axioms of second-order arithmetic. These axioms assert, in two different ways, that infinite but narrow binary trees always have infinite paths. We show that both axioms are strictly weaker than Weak K\"onig's Lemma, and…
A spanning tree of a graph is a connected subgraph on all vertices with the minimum number of edges. The number of spanning trees in a graph $G$ is given by Matrix Tree Theorem in terms of principal minors of Laplacian matrix of $G$. We…
We introduce the idea of a weakly entangled linear order, and show that it is consistent for a Suslin line to be weakly entangled. We generalize the notion of entangled linear orders to $\omega_1$-trees, and prove that an $\omega_1$-tree is…
In this manuscript, we show that any neural network with any activation function can be represented as a decision tree. The representation is equivalence and not an approximation, thus keeping the accuracy of the neural network exactly as…
This paper considers the enumeration of trees avoiding a contiguous pattern. We provide an algorithm for computing the generating function that counts n-leaf binary trees avoiding a given binary tree pattern t. Equipped with this counting…
Starting with any nondegenerate triangle we can use a well defined interior point of the triangle to subdivide it into six smaller triangles. We can repeat this process with each new triangle, and continue doing so over and over. We show…
Given a set of cycles C of a graph G, the tree graph of G defined by C is the graph T(G,C) whose vertices are the spanning trees of G and in which two trees R and S are adjacent if the union of R and S contains exactly one cycle and this…
Consider a connected graph $G$ and let $T$ be a spanning tree of $G$. Every edge $e \in G-T$ induces a cycle in $T \cup \{e\}$. The intersection of two distinct such cycles is the set of edges of $T$ that belong to both cycles. We consider…
Rotation distance between rooted binary trees is the minimum number of simple rotations needed to transform one tree into the other. Computing the rotation distance between a pair of rooted trees can be quickly reduced in cases where there…
We consider problems to make a given bidirected graph strongly connected with minimum cardinality of additional signs or additional arcs. For the former problem, we show the minimum number of additional signs and give a linear-time…
This note derives asymptotic upper and lower bounds for the number of planted plane trees on $n$ nodes assigned labels from the set $\{1,2,\ldots, k\}$ with the restriction that on any path from the root to a leaf, the labels must strictly…
The existence of greatest lower bounds in the imbalance order of path-length sequences of binary trees is seen to be a consequence of a joint monotonicity property of the greater and lower expension operations. Path length sequences that…