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Designing well-connected graphs is a fundamental problem that frequently arises in various contexts across science and engineering. The weighted number of spanning trees, as a connectivity measure, emerges in numerous problems and plays a…
We introduce the concept of Most, and Least, Compact Spanning Trees - denoted respectively by $T^*(G)$ and $T^\#(G)$ - of a simple, connected, undirected and unweighted graph $G(V, E, W)$. For a spanning tree $T(G) \in \mathcal{T}(G)$ to be…
The problem of spanning trees is closely related to various interesting problems in the area of statistical physics, but determining the number of spanning trees in general networks is computationally intractable. In this paper, we perform…
Tree alignment graphs (TAGs) provide an intuitive data structure for storing phylogenetic trees that exhibits the relationships of the individual input trees and can potentially account for nested taxonomic relationships. This paper…
Graphs provide a universal framework for representing complex relational systems, and inferring their structural properties is a core challenge in graph analysis and reasoning. While large language models have recently demonstrated emerging…
Packing graphs is a combinatorial problem where several given graphs are being mapped into a common host graph such that every edge is used at most once. In the planar tree packing problem we are given two trees T1 and T2 on n vertices and…
A complete description is given of how minimal trees on atoms of the algebra of subsets $\mathfrak{A}_k$ generated by minimal spanning $k$-component forests of a weighted digraph $V$ determine the form of these forests and how forests grow…
Sparse structures are frequently sought when pursuing tractability in optimization problems. They are exploited from both theoretical and computational perspectives to handle complex problems that become manageable when sparsity is present.…
We consider the problem of testing properties of graphs underlying high-dimensional graphical models. We adopt the model of covariance queries introduced by Lugosi, Truszkowski, Velona, and Zwiernik (2021). We study the case when the…
Graphs are a generalized concept that encompasses more complex data structures than trees, such as difference lists, doubly-linked lists, skip lists, and leaf-linked trees. Normally, these structures are handled with destructive assignments…
The Minimum Branch Vertices Spanning Tree problem aims to find a spanning tree $T$ in a given graph $G$ with the fewest branch vertices, defined as vertices with a degree three or more in $T$. This problem, known to be NP-hard, has…
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…
Large graphs abound in machine learning, data mining, and several related areas. A useful step towards analyzing such graphs is that of obtaining certain summary statistics - e.g., or the expected length of a shortest path between two…
Recent advancements in complex network analysis are encouraging and may provide useful insights when applied in software engineering domain, revealing properties and structures that cannot be captured by traditional metrics. In this paper,…
Building on work by Desjarlais, Molina, Faase, and others, a general method is obtained for counting the number of spanning trees of graphs that are a product of an arbitrary graph and either a path or a cycle, of which grid graphs are a…
We study the problem of connecting the parts of a multipartite graph using a minimum number of edges under a matching constraint. We introduce interconnection trees, defined as matchings whose projections onto the quotient graph form a…
Treewidth is a well-known graph invariant with multiple interesting applications in combinatorics. On the practical side, many NP-complete problems are polynomial-time (sometimes even linear-time) solvable on graphs of bounded treewidth. On…
In this work, we introduce TreeWidzard, an engine for developing dynamic programming algorithms that decide graph-theoretic properties parameterized by treewidth and pathwidth. Besides providing a unified framework for algorithms deciding…
In real life, it is always an urge to reach our goal in minimum effort i.e., it should have a minimum constrained path. The path may be shortest route in practical life, either physical or electronic medium. The scenario is to represents…
We continue the study of token sliding reconfiguration graphs of independent sets initiated by the authors in an earlier paper (arXiv:2203.16861). Two of the topics in that paper were to study which graphs $G$ are token sliding graphs and…