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Functional graphs (FGs) model the graph structures used to analyse the behaviour of functions from a discrete set to itself. In turn, such functions are used to study real complex phenomena evolving in time. As the systems involved can be…
Decomposing a graph into a hierarchical structure via $k$-core analysis is a standard operation in any modern graph-mining toolkit. $k$-core decomposition is a simple and efficient method that allows to analyze a graph beyond its mere…
Many problems in computational geometry are not stated in graph-theoretic terms, but can be solved efficiently by constructing an auxiliary graph and performing a graph-theoretic algorithm on it. Often, the efficiency of the algorithm…
We propose a Reinforcement Learning based approach to approximately solve the Tree Decomposition (TD) problem. TD is a combinatorial problem, which is central to the analysis of graph minor structure and computational complexity, as well as…
Determining whether two graphs are structurally identical is a fundamental problem with applications spanning mathematics, computer science, chemistry, and network science. Despite decades of study, graph isomorphism remains a challenging…
Finding dense components in graphs is of great importance in analyzing the structure of networks. Popular and computationally feasible frameworks for discovering dense subgraphs are core and truss decompositions. Recently, Sariyuce et al.…
A treedepth decomposition of an undirected graph $G$ is a rooted forest $F$ on the vertex set of $G$ such that every edge $uv\in E(G)$ is in ancestor-descendant relationship in $F$. Given a weight function $w\colon V(G)\rightarrow…
Counting non-isomorphic tree-like multigraphs that include self-loops and multiple edges is an important problem in combinatorial enumeration, with applications in chemical graph theory, polymer science, and network modeling. Traditional…
Many multivariate data such as social and biological data exhibit complex dependencies that are best characterized by graphs. Unlike sequential data, graphs are, in general, unordered structures. This means we can no longer use classic,…
Considering the worst-case scenario, junction tree algorithm remains the most general solution for exact MAP inference with polynomial run-time guarantees. Unfortunately, its main tractability assumption requires the treewidth of a…
Deciding whether a graph can be embedded in a grid using only unit-length edges is NP-complete, even when restricted to binary trees. However, it is not difficult to devise a number of graph classes for which the problem is polynomial, even…
The Pathwidth Theorem states that if a class of graphs has unbounded pathwidth, then it contains all trees as graph minors. We prove a similar result for dense graphs. More precisely, we give a finite family of tree-like patterns and prove…
New methods for $D$-decomposition analysis are presented. They are based on topology of real algebraic varieties and computational real algebraic geometry. The estimate of number of root invariant regions for polynomial parametric families…
In the field of parameterized complexity theory, the study of graph width measures has been intimately connected with the development of width-based model checking algorithms for combinatorial properties on graphs. In this work, we…
We generalize some homotopy calculation techniques such as splittings and matching trees that are introduced for the computations in the case of the independence complexes of graphs to arbitrary simplicial complexes, and exemplify their…
The independence number of a tree decomposition is the size of a largest independent set contained in a single bag. The tree-independence number of a graph $G$ is the minimum independence number of a tree decomposition of $G$. As shown…
Fixed parameter tractable algorithms for bounded treewidth are known to exist for a wide class of graph optimization problems. While most research in this area has been focused on exact algorithms, it is hard to find decompositions of…
Connectivity problems like k-Path and k-Disjoint Paths relate to many important milestones in parameterized complexity, namely the Graph Minors Project, color coding, and the recent development of techniques for obtaining kernelization…
In this paper, we investigate the polynomial integrand of an integral formula that yields the expected length of the minimal spanning tree of a graph whose edges are uniformly distributed over the interval [0, 1]. In particular, we derive a…
We introduce a new subclass of chordal graphs that generalizes split graphs, which we call well-partitioned chordal graphs. Split graphs are graphs that admit a partition of the vertex set into cliques that can be arranged in a star…